Tree and graph traversals(5 Questions)Connected components(5 Questions)Spanning trees(5 Questions)Shortest paths(5 Questions)

http://geeksquiz.com/algorithms/graph-traversals/

Question 1

Which of the following algorithms can be used to most efficiently determine the presence of a cycle in

a given graph ?

A Depth First Search

B Breadth First Search

C Prim's Minimum Spanning Tree Algorithm

D Kruskal' Minimum Spanning Tree Algorithm

Discuss it

Question 2

Traversal of a graph is different from tree because

AThere can be a loop in graph so we must maintain a visited flag for every vertex

BDFS of a graph uses stack, but inorrder traversal of a tree is recursive

CBFS of a graph uses queue, but a time efficient BFS of a tree is recursive.

D All of the above

Discuss it

Question 3

What are the appropriate data structures for following algorithms?

1) Breadth First Search                          

2) Depth First Search                           

3) Prim's Minimum Spanning Tree

4) Kruskal' Minimum Spanning Tree

A

1) Stack2) Queue3) Priority Queue4) Union Find

B1) Queue2) Stack3) Priority Queue4) Union Find

C1) Stack2) Queue3) Union Find4) Priority Queue

D1) Priority Queue2) Queue3) Stack4) Union Find

Discuss it

Question 4

The Breadth First Search algorithm has been implemented using the queue data structure. One

possible order of visiting the nodes of the following graph is

A MNOPQR

B NQMPOR

C QMNPRO

D QMNPOR

Discuss it

Question 5

Let G be an undirected graph. Consider a depth-first traversal of G, and let T be the resulting depth-

first search tree. Let u be a vertex in G and let v be the first new (unvisited) vertex visited after visiting

u in the traversal. Which of the following statements is always true? (GATE CS 2000)

A {u,v} must be an edge in G, and u is a descendant of v in T

B {u,v} must be an edge in G, and v is a descendant of u in T

C If {u,v} is not an edge in G then u is a leaf in T

DIf {u,v} is not an edge in G then u and v must have the same parent in T

Discuss it

Question 6

Consider the following graph  

Among the following sequences 

I) a b e g h f 

II) a b f e h g 

III) a b f h g e 

IV) a f g h b e Which are depth first traversals of the above graph? (GATE CS 2003)

A I, II and IV only

B I and IV only

C II, III and IV only

D I, III and IV only

Discuss it

Question 7

Make is a utility that automatically builds executable programs and libraries from source code by

reading files called makefiles which specify how to derive the target program. Which of the following

standard graph algorithms is used by Make.

A Strongly Connected Components

B Topological Sorting

C Breadth First Search

D Dijkstra's Shortest Path

Discuss it

Question 8

Given two vertices in a graph s and t, which of the two traversals (BFS and DFS) can be used to find if

there is path from s to t?

A Only BFS

B Only DFS

C Both BFS and DFS

D Neither BFS nor DFS

Discuss it

Question 9

Which of the following condition is sufficient to detect cycle in a directed graph?

AThere is an edge from currently being visited node to an already visited node.

BThere is an edge from currently being visited node to an ancestor of currently visited node in DFS forest.

C Every node is seen twice in DFS.

D None of the bove

Discuss it

Question 10

Is following statement true/false If a DFS of a directed graph contains a back edge, any other DFS of

the same graph will also contain at least one back edge.

Source:http://courses.csail.mit.edu/6.006/oldquizzes/solutions/q2-s2009-sol.pdf

A True

B False

Discuss it

Question 11

Is following statement true/false? A DFS of a directed graph always produces the same number of

tree edges, i.e., independent of the order in which vertices are considered for DFS.

(Sourcehttp://courses.csail.mit.edu/6.006/oldquizzes/solutions/q2-f2008-sol.pdf)

A True

B False

Discuss it

Question 12

If the DFS finishing time f[u] > f[v] for two vertices u and v in a directed graph G, and u and v are in

the same DFS tree in the DFS forest, then u is an ancestor of v in the depth first tree.

(Source http://courses.csail.mit.edu/6.006/oldquizzes/solutions/q2-f2007-sol.pdf)

A True

B False

Discuss it

Question 13

Consider the DAG with Consider V = {1, 2, 3, 4, 5, 6}, shown below. Which of the following is NOT a

topological ordering? 

A 1 2 3 4 5 6

B 1 3 2 4 5 6

C 1 3 2 4 6 5

D 3 2 4 1 6 5

Discuss it

Question 14

Let G be a graph with n vertices and m edges. What is the tightest upper bound on the running time

on Depth First Search of G? Assume that the graph is represented using adjacency matrix.

A O(n)

B O(m+n)

C O(n2)

D O(mn)

Discuss it

Question 15

Consider the tree arcs of a BFS traversal from a source node W in an unweighted, connected,

undirected graph. The tree T formed by the tree arcs is a data structure for computing.

A the shortest path between every pair of vertices.

B the shortest path from W to every vertex in the graph.

Cthe shortest paths from W to only those nodes that are leaves of T.

D the longest path in the graph

Discuss it

Question 16

Suppose depth first search is executed on the graph below starting at some unknown vertex. Assume

that a recursive call to visit a vertex is made only after first checking that the vertex has not been

visited earlier. Then the maximum possible recursion depth (including the initial call) is _________.

A 17

B 18

C 19

D 20

Discuss it

Question 17

Let T be a depth first search tree in an undirected graph G. Vertices u and n are leaves of this tree T.

The degrees of both u and n in G are at least 2. which one of the following statements is true?

A There must exist a vertex w adjacent to both u and n in G

BThere must exist a vertex w whose removal disconnects u and n in G

C There must exist a cycle in G containing u and n

DThere must exist a cycle in G containing u and all its neighbours in G.

     

http://geeksquiz.com/algorithms/graph-shortest-paths/

 

Graph Shortest Paths

Question 1

Consider the directed graph shown in the figure below. There are multiple shortest paths between

vertices S and T. Which one will be reported by Dijstra?s shortest path algorithm? Assume that, in any

iteration, the shortest path to a vertex v is updated only when a strictly shorter path to v is discovered.

Graph

A SDT

B SBDT

C SACDT

D SACET

Discuss it

Question 2

To implement Dijkstra’s shortest path algorithm on unweighted graphs so that it runs in linear time, the

data structure to be used is:

A Queue

B Stack

C Heap

D B-Tree

Discuss it

Question 3

Dijkstra’s single source shortest path algorithm when run from vertex a in the below graph, computes

the correct shortest path distance to

A only vertex a

B only vertices a, e, f, g, h

C only vertices a, b, c, d

D all the vertices

Discuss it

Question 4

In an unweighted, undirected connected graph, the shortest path from a node S to every other node is

computed most efficiently, in terms of time complexity by

A Dijkstra’s algorithm starting from S.

B Warshall’s algorithm

C Performing a DFS starting from S.

D Performing a BFS starting from S.

Discuss it

Question 5

Suppose we run Dijkstra’s single source shortest-path algorithm on the following edge weighted

directed graph with vertex P as the source. In what order do the nodes get included into the set of

vertices for which the shortest path distances are finalized? (GATE CS 2004)

A P, Q, R, S, T, U

B P, Q, R, U, S, T

C P, Q, R, U, T, S

D P, Q, T, R, U, S

Discuss it

Question 6

Suppose we run Dijkstra’s single source shortest-path algorithm on the following edge weighted

directed graph with vertex P as the source. In what order do the nodes get included into the set of

vertices for which the shortest path distances are finalized? (GATE CS 2004)

A P, Q, R, S, T, U

B P, Q, R, U, S, T

C P, Q, R, U, T, S

D P, Q, T, R, U, S

Discuss it

Question 7

What is the time complexity of Bellman-Ford single-source shortest path algorithm on a complete

graph of n vertices? (A)   (B)   (C)   (D) 

A A

B B

C C

D D

Discuss it

Question 8

In a weighted graph, assume that the shortest path from a source 's' to a destination 't' is correctly

calculated using a shortest path algorithm. Is the following statement true? If we increase weight of

every edge by 1, the shortest path always remains same.

A Yes

B No

Discuss it

Question 9

Following statement is true or false?

If we make following changes to Dijkstra, then it can be used to

find

the longest simple path, assume that the graph is acyclic.

1) Initialize all distances as minus infinite instead of plus

infinite.

2) Modify the relax condition in Dijkstra's algorithm to update

distance

of an adjacent v of the currently considered vertex u only

if "dist[u]+graph[u][v] > dist[v]". In shortest path algo,

the sign is opposite.

A True

B False

Discuss it

Question 10

Which of the following algorithm can be used to efficiently calculate single source shortest paths in a

Directed Acyclic Graph?

A Dijkstra

B Bellman-Ford

C Topological Sort

D Strongly Connected Component

Discuss it

Question 11

Given a directed graph where weight of every edge is same, we can efficiently find shortest path from

a given source to destination using?

A Breadth First Traversal

B Dijkstra's Shortest Path Algorithm

CNeither Breadth First Traversal nor Dijkstra's algorithm can be used

D Depth First Search

Discuss it

Question 12

Is the following statement valid about shortest paths? Given a graph, suppose we have calculated

shortest path from a source to all other vertices. If we modify the graph such that weights of all edges

is becomes double of the original weight, then the shortest path remains same only the total weight of

path changes.

A True

B False

Discuss it

Question 13

Match the following

Group A Group B

a) Dijkstra's single shortest path algo p) Dynamic

Programming

b) Bellmen Ford's single shortest path algo q)

Backtracking

c) Floyd Warshell's all pair shortest path algo. r) Greedy

Algorithm

A a-r, b-q, c-p

B a-p, b-p, c-p

C a-r, b-p, c-p

D a-p, b-r, c-q

Discuss it

Question 14

Is the following statement valid?. 

Given a weighted graph where weights of all edges are unique (no two edge have same weights),

there is always a unique shortest path from a source to destination in such a graph.

A True

B False

Discuss it

Question 15

Is the following statement valid?. 

Given a graph where all edges have positive weights, the shortest paths produced by Dijsktra and

Bellman Ford algorithm may be different but path weight would always be same.

A True

B False

Discuss it

Question 16

Which of the following statement(s)is / are correct regarding Bellman-Ford shortest path algorithm?

P: Always finds a negative weighted cycle, if one exist s.

Q: Finds whether any negative weighted cycle is reachable

from the source.

A P Only

B Q Only

C Both P and Q

D Neither P nor Q

Discuss it

http://geeksquiz.com/algorithms/graph-minimum-spanning-tree/

Graph Minimum Spanning Tree

Question 1

An undirected graph G(V, E) contains n ( n > 2 ) nodes named v1 , v2 ,….vn. Two nodes vi , vj are

connected if and only if 0 < |i – j| <= 2. Each edge (vi, vj ) is assigned a weight i + j. A sample graph

with n = 4 is shown below. What will be the cost of the minimum spanning tree (MST) of such a graph

with n nodes? (GATE CS 2011) 

A 1/12(11n^2 – 5n)

B n^2 – n + 1

C 6n – 11

D 2n + 1

Discuss it

Question 2

The length of the path from v5 to v6 in the MST of previous question with n = 10 is

A 11

B 25

C 31

D 41

Discuss it

Question 3

Consider a complete undirected graph with vertex set {0, 1, 2, 3, 4}. Entry Wij in the matrix W below is

the weight of the edge {i, j}. What is the minimum possible weight of a spanning tree T in this graph

such that vertex 0 is a leaf node in the tree T? (GATE CS 2010) 

A 7

B 8

C 9

D 10

Discuss it

Question 4

In the graph given in above question question, what is the minimum possible weight of a path P from

vertex 1 to vertex 2 in this graph such that P contains at most 3 edges?

A 7

B 8

C 9

D 10

Discuss it

Question 5

An undirected graph G has n nodes. Its adjacency matrix is given by an n × n square matrix whose (i)

diagonal elements are 0‘s and (ii) non-diagonal elements are 1‘s. which one of the following is TRUE?

A Graph G has no minimum spanning tree (MST)

B Graph G has a unique MST of cost n-1

C Graph G has multiple distinct MSTs, each of cost n-1

D Graph G has multiple spanning trees of different costs

Discuss it

Question 6

Consider the following

graph: 

Which one of the following cannot be the sequence of edges added, in that order, to a minimum

spanning tree using Kruskal’s algorithm?

A (a—b),(d—f),(b—f),(d—c),(d—e)

B (a—b),(d—f),(d—c),(b—f),(d—e)

C (d—f),(a—b),(d—c),(b—f),(d—e)

D (d—f),(a—b),(b—f),(d—e),(d—c)

Discuss it

Question 7

Let G be an undirected connected graph with distinct edge weight. Let emax be the edge with

maximum weight and emin the edge with minimum weight. Which of the following statements is false?

(GATE CS 2000)

A Every minimum spanning tree of G must contain emin

BIf emax is in a minimum spanning tree, then its removal must disconnect G

C No minimum spanning tree contains emax

D G has a unique minimum spanning tree

Discuss it

Question 8

Consider a weighted complete graph G on the vertex set {v1,v2 ,v} such that the weight of the edge

(v,,v) is 2|i-j|. The weight of a minimum spanning tree of G is: (GATE CS 2006)

A n — 1

B 2n — 2

C nC2

D 2

Discuss it

Question 9

Let G be a weighted graph with edge weights greater than one and G'be the graph constructed by

squaring the weights of edges in G. Let T and T' be the minimum spanning trees of G and G',

respectively, with total weights t and t'. Which of the following statements is TRUE?

A T' = T with total weight t' = t2

B T' = T with total weight t' < t2

C T' != T but total weight t' = t2

D None of the above

Discuss it

Question 10

Consider the following graph:  Which one of the

following is NOT the sequence of edges added to the minimum spanning tree using Kruskal's

algorithm?

A (b,e)(e,f)(a,c)(b,c)(f,g)(c,d)

B (b,e)(e,f)(a,c)(f,g)(b,c)(c,d)

C (b,e)(a,c)(e,f)(b,c)(f,g)(c,d)

D (b,e)(e,f)(b,c)(a,c)(f,g)(c,d)

Discuss it

Question 11

The number of distinct minimum spanning trees for the weighted graph below is

____ 

A 4

B 5

C 6

D 7

Discuss it

http://www.geeksforgeeks.org/interesting-shortest-path-questions-set-1/

ome interesting shortest path questions | Set 1

Question 1: Given a directed weighted graph. You are also given the shortest path from a source vertex ‘s’ to a destination vertex ‘t’.  If weight of every edge is increased by 10 units, does the shortest path remain same in the modified graph?The shortest path may change. The reason is, there may be different number of edges in different paths from s to t. For example, let shortest path be of weight 15 and has 5 edges. Let there be another path with 2 edges and total weight 25. The weight of the shortest path is increased by 5*10 and becomes 15 + 50. Weight of the other path is increased by 2*10 and becomes 25 + 20. So the shortest path changes to the other path with weight as 45.

Question 2: This is similar to above question. Does the shortest path change when weights of all edges are multiplied by 10?If we multiply all edge weights by 10, the shortest path doesn’t change. The reason is simple, weights of all paths from s to t get multiplied by same amount. The number of edges on a path doesn’t matter. It is like changing unit of weights.

Question 3: Given a directed graph where every edge has weight as either 1 or 2, find the shortest path from a given source vertex ‘s’ to a given destination vertex ‘t’. Expected time complexity is O(V+E).If we apply Dijkstra’s shortest path algorithm, we can get a shortest path in O(E + VLogV) time. How to do it in O(V+E) time? The idea is to use BFS   . One important observation about BFS   is, the path used in BFS always has least number of edges between any two vertices. So if all edges are of same weight, we can use BFS to find the shortest path. For this problem, we can modify the graph and split all edges of weight 2 into two edges of weight 1 each. In the modified graph, we can use BFS to find the shortest path. How is this approach

O(V+E)? In worst case, all edges are of weight 2 and we need to do O(E) operations to split all edges, so the time complexity becomes O(E) + O(V+E) which is O(V+E).

Question 4: Given a directed acyclic weighted graph, how to find the shortest path from a source s to a destination t in O(V+E) time?See: Shortest Path in Directed Acyclic Graph

More Questions See following links for more questions.http://algs4.cs.princeton.edu/44sp/http://geeksquiz.com/graph-shortest-paths/

     

1 A subnet has been assigned a subnet mask of 255.255.255.192.

What is the maximum number of hosts that can belong to this

subnet? 

A) 14

B) 30

C) 62 

D) 126

Answer : (C)

2 Which one of the following is NOT shared by the threads of the

same process ? 

A) Stack 

B) Address Space 

C) File Descriptor Table 

D) Message Queue

Answer : (A)

3 Let G be a weighted undirected graph and e be an edge with

maximum weight in G. Suppose there is a minimum weight spanning

tree in G containing the edge e. Which of the following statements is

always TRUE? 

A) There exists a cutset in G having all edges of maximum weight.

B) There exists a cycle in G having all edges of maximum weight.

C) Edge e cannot be contained in a cycle. 

D) All edges in G have the same weight.

Answer : (C)

4 A software organization has been assessed at SEI CMM Level 4.

Which of the following does the organization need to practice beside

Process Change Management and Technology Change Management

in order to achieve Level 5 ? 

A) Defect Detection

B) Defect Prevention

C) Defect Isolation 

D) Defect Propagation

Answer : (B)

5 Suppose that two parties A and B wish to setup a common secret

key (D-H key) between themselves using the Diffle-Hellman key

exchange technique. They agree on 7 as the modulus and 3 as the

primitive root. Party A chooses 2 and party B chooses 5 as their

respective secrets. Their D-H key is 

A) 3 

B) 4

C) 5

D) 6

Answer : (C)

6 Consider the following message M = 1010001101. The cyclic

redundancy check (CRC) for this message using the divisor

polynomial x5 + x4 + x2 + 1 is : 

A) 01110

B) 01011

C) 10101 

D) 10110

Answer : (C)

7 Which of the following statements is FALSE regarding a bridge

A) Bridge is a layer 2 device 

B) Bridge reduces collision domain

C) Bridge is used to connect two or more LAN segments 

D) Bridge reduces broadcast domain

Answer : (D)

8 What is the availability of a software with the following reliability

figures?

Mean Time Between Failure (MTBF) = 25 days

Mean Time To Repair (MTTR) = 6 hours

A) 1%

B) 24%

C) 99% 

D) 99.009% 

Answer : (B)

9 A channel has a bit rate of 4 kbps and one-way propagation delay

of 20 ms. The channel uses stop and wait protocol. The transmission

time of the acknowledgement frame is negligible. To get a channel

efficiency of at least 50%, the minimum frame size should be

A) 80 bytes

B) 80 bits

C) 160 bytes 

D) 160 bits

Answer : (D)

10 In a depth-first traversal of a graph G with n vertices, k edges are

marked as tree edges. The number of connected components in G is 

A) k

B) k+1

C) n-k-l

D) n-k

Answer : (C)

11 Which of the following statements is TRUE about CSMA/CD 

A) IEEE 802.11 wireless LAN runs CSMA/CD protocol 

B) Ethernet is not based on CSMA/CD protocol 

C) CSMA/CD is not suitable for a high propagation delay network like satellite

network 

D) There is no contention in a CSMA/CD network 

Answer : (C)

12 Which one of the following regular expressions is NOT equivalent

to the regular expression (a + b + c)* ? 

A) (a* + b* + c*)*

B) (a*b*c*)*

C) ((ab)* + c*)* 

D) (a*b* + c*)*

Answer : (C)

http://www.geeksforgeeks.org/find-number-of-islands/

Given a boolean 2D matrix, find the number of islands.

This is an variation of the standard problem: “Counting number of connected components in a

undirected graph”.

Before we go to the problem, let us understand what is a connected component. A connected

component of an undirected graph is a subgraph in which every two vertices are connected to each

other by a path(s), and which is connected to no other vertices outside the subgraph.

For example, the graph shown below has three connected components.

A graph where all vertices are connected with each other, has exactly one connected component,

consisting of the whole graph. Such graph with only one connected component is called as Strongly

Connected Graph.

The problem can be easily solved by applying DFS() on each component. In each DFS() call, a

component or a sub-graph is visited. We will call DFS on the next un-visited component. The number

of calls to DFS() gives the number of connected components. BFS can also be used.

What is an island?

A group of connected 1s forms an island. For example, the below matrix contains 5 islands

{1, 1, 0, 0, 0}, {0, 1, 0, 0, 1}, {1, 0, 0, 1, 1}, {0, 0, 0, 0, 0}, {1, 0, 1, 0, 1}

A cell in 2D matrix can be connected to 8 neighbors. So, unlike standard DFS(), where we recursively

call for all adjacent vertices, here we can recursive call for 8 neighbors only. We keep track of the

visited 1s so that they are not visited again.

// Program to count islands in boolean 2D matrix

 #include <stdio.h>#include <string.h>#include <stdbool.h>

 #define ROW 5#define COL 5

 // A function to check if a given cell (row, col) can be included in DFSint isSafe(int M[][COL], int row, int col, bool visited[][COL]){    return (row >= 0) && (row < ROW) &&     // row number is in range           (col >= 0) && (col < COL) &&     // column number is in range           (M[row][col] && !visited[row][col]); // value is 1 and not yet visited}

 // A utility function to do DFS for a 2D boolean matrix. It only considers// the 8 neighbors as adjacent verticesvoid DFS(int M[][COL], int row, int col, bool visited[][COL]){    // These arrays are used to get row and column numbers of 8 neighbors     // of a given cell    static int rowNbr[] = {-1, -1, -1,  0, 0,  1, 1, 1};    static int colNbr[] = {-1,  0,  1, -1, 1, -1, 0, 1};     // Mark this cell as visited    visited[row][col] = true;

     // Recur for all connected neighbours    for (int k = 0; k < 8; ++k)        if (isSafe(M, row + rowNbr[k], col + colNbr[k], visited) )            DFS(M, row + rowNbr[k], col + colNbr[k], visited);}

 // The main function that returns count of islands in a given boolean// 2D matrixint countIslands(int M[][COL]){    // Make a bool array to mark visited cells.    // Initially all cells are unvisited    bool visited[ROW][COL];    memset(visited, 0, sizeof(visited));

     // Initialize count as 0 and travese through the all cells of    // given matrix    int count = 0;    for (int i = 0; i < ROW; ++i)        for (int j = 0; j < COL; ++j)            if (M[i][j] && !visited[i][j]) // If a cell with value 1 is not            {                              // visited yet, then new island found                DFS(M, i, j, visited);     // Visit all cells in this island.                ++count;                   // and increment island count            }

     return count;}

 // Driver program to test above functionint main(){    int M[][COL]= {  {1, 1, 0, 0, 0},        {0, 1, 0, 0, 1},        {1, 0, 0, 1, 1},        {0, 0, 0, 0, 0},        {1, 0, 1, 0, 1}    };

     printf("Number of islands is: %d\n", countIslands(M));

     return 0;}

Output:

Number of islands is: 5

Time complexity: O(ROW x COL)

Reference:

http://en.wikipedia.org/wiki/Connected_component_%28graph_theory%29

Please write comments if you find anything incorrect, or you want to share more information about the

topic discussed above.

   http://girdhargopalbansal.blogspot.in/2013/05/gate-questions-for-data-structures-and.html

GATE Questions for Data Structures and AlgorithmsFollowing questions have been asked in GATE CS exam

1. Let LASTPOST, LASTIN and LASTPRE denote the last vertex visited in a postorder, inorder and preorder traversal. Respectively, of a complete binary tree. Which of the following is always true? (GATE CS 2000)(a) LASTIN = LASTPOST(b) LASTIN = LASTPRE(c) LASTPRE = LASTPOST(d) None of the aboveAnswer (d)It is given that the given tree is complete binary tree. For a complete binary tree, the last visited node will always be same for inorder and preorder traversal. None of the above is true even for a complete binary tree. The option (a) is incorrect because the last node visited in Inorder traversal is right child and last node visited in Postorder traversal is root.The option (c) is incorrect because the last node visited in Preorder traversal is right child and last node visited in Postorder traversal is root.For option (b), see the following counter example.

1 / \ 2 3 / \ /4 5 6

Inorder traversal is 4 2 5 1 6 3Preorder traversal is 1 2 4 5 3 6

2. The most appropriate matching for the following pairs

X: depth first search 1: heapY: breadth-first search 2: queueZ: sorting 3: stack

is (GATE CS 2000):(a) X—1 Y—2 Z-3(b) X—3 Y—1 Z-2(c) X—3 Y—2 Z-1(d) X—2 Y—3 Z-1Answer: (c)Stack is used for Depth first SearchQueue is used for Breadth First SearchHeap is used for sorting

3. Consider the following nested representation of binary trees: (X Y Z) indicates Y and Z are the left and right sub stress, respectively, of node X. Note that Y and Z may be NULL, or further nested. Which of the following represents a valid binary tree?(a) (1 2 (4 5 6 7))(b) (1 (2 3 4) 5 6) 7)(c) (1 (2 3 4)(5 6 7))(d) (1 (2 3 NULL) (4 5))Answer (c)

4. Let s be a sorted array of n integers. Let t(n) denote the time taken for the most efficient algorithm to determined if there are two elements with sum less than 1000 in s. which of the following statements is true? (GATE CS 2000)a) t (n) is 0 (1)

b) n < t (n) < n

c) n log 2 n < t (n) < 

d) t (n) = Answer (a)Let array be sorted in ascending order, if sum of first two elements is less than 1000 then there are  two elements with sum less than 1000 otherwise not. For array sorted in descending order we need to check last two elements. For an array data structure, number of operations are fixed in both the cases and not dependent on n, complexity is O(1)

5. B+ trees are preferred to binary trees in databases because (GATE CS 2000)(a) Disk capacities are greater than memory capacities(b) Disk access is much slower than memory access(c) Disk data transfer rates are much less than memory data transfer rates(d) Disks are more reliable than memoryAnswer (b)Disk access is slow and  B+ Tree   provide search in less number of disk hits. This is primarily because unlike binary seach trees, B+ trees have very high fanout (typically on the order of 100 or more), which reduces the number of I/O operations required to find an element in the tree.

1. Consider the function f defined below. 

struct item{

  int data;

  struct item * next;};

int f(struct item *p){

  return (          (p == NULL) ||

          (p->next == NULL) ||

          (( P->data &lt;= p->next->data) &amp;&amp; f(p->next))

         );

}

For a given linked list p, the function f returns 1 if and only if (GATE CS 2003)a) the list is empty or has exactly one elementb) the elements in the list are sorted in non-decreasing order of data valuec) the elements in the list are sorted in non-increasing order of data valued) not all elements in the list have the same data value.Answer (b)The function f() works as follows1) If linked list is empty return 12) Else If linked list has only one element return 13) Else if node->data is smaller than equal to node->next->data and same thing holds for rest of the list then return 14) Else return 0

2. Consider the label sequences obtained by the following pairs of traversals on a labeled binary tree. Which of these pairs identify a tree uniquely (GATE CS 2004)?i) preorder and postorder

ii) inorder and postorderiii) preorder and inorderiv) level order and postordera) (i) onlyb) (ii), (iii)c) (iii) onlyd) (iv) onlyAnswer (b)

3. The following numbers are inserted into an empty binary search tree in the given order: 10, 1, 3, 5, 15, 12, 16. What is the height of the binary search tree (the height is the maximum distance of a leaf node from the root)? (GATE CS 2004)a) 2b) 3c) 4d) 6Answer(b)Constructed binary search tree will be..

10 / \ 1 15 \ / \ 3 12 16 \ 5 

4. A data structure is required for storing a set of integers such that each of the following operations can be done in (log n) time, where n is the number of elements in the set. 

o Delection of the smallest element o Insertion of an element if it is not already present in the set

Which of the following data structures can be used for this purpose? (a) A heap can be used but not a balanced binary search tree(b) A balanced binary search tree can be used but not a heap(c) Both balanced binary search tree and heap can be used(d) Neither balanced binary search tree nor heap can be used Answer(b)A self-balancing balancing binary search tree containing n items allows the lookup, insertion, and removal of an item in O(log n) worst-case time. Since it’s a BST, we can easily find out minimum element in O(nlogn).Since Heap   is a balanced binary tree (or almost complete binary tree), insertion complexity for heap is O(logn). Also complexity to get minimum in a min heap is O(logn) because removal of root node causes a call to heapify (after removing the first element from the array) to maintain the heap tree property. But a heap cannot be used for the above purpose as the question says – insert an element if it is not already present. For a heap, we cannot find out in O(logn) if an element is present or not. Thanks to game for providing the correct solution.

5. A circularly linked list is used to represent a Queue. A single variable p is used to access the Queue. To which node should p point such that both the operations enQueue and deQueue can be performed in constant time? (GATE 2004)

a) rear nodeb) front nodec) not possible with a single pointerd) node next to frontAnswer(a)Answer is not “(b) front node”, as we can not get rear from front in O(1), but if p is rear we can implement both enQueue and deQueue in O(1) because from rear we can get front in O(1). Below are sample functions. Note that these functions are just sample are not working. Code to handle base cases is missing.

/* p is pointer to address of rear (double pointer).  This function adds new

   node after rear and updates rear which is *p to point to new node  */

void  enQueue(struct node **p, struct node *new_node)

{

    /* Missing code to handle base cases like *p is NULL */

      

     new_node->next = (*p)->next;

     (*p)->next = new_node;

     (*p) = new_node /* new is now rear */

     /* Note that p is again front and  p->next is rear */

 }

/* p is pointer to rear. This function removes the front element and

    returns the new front */

struct node *deQueue(struct node *p)

{

    /* Missing code to handle base cases like p is NULL,

        p->next is NULL,...  etc */

    struct node *temp = p->next->next;

    p->next = p->next->next;

    return temp;

    /* Note that p is again front and  p->next is rear */

}

 

1. Consider the following C program segment

struct CellNode{

  struct CelINode *leftchild;

  int element;

  struct CelINode *rightChild;}

int Dosomething(struct CelINode *ptr){

    int value = 0;

    if (ptr != NULL)    {

      if (ptr->leftChild != NULL)        value = 1 + DoSomething(ptr->leftChild);

      if (ptr->rightChild != NULL)        value = max(value, 1 + DoSomething(ptr->rightChild));

    }

    return (value);}

The value returned by the function DoSomething when a pointer to the root of a non-empty tree is passed as argument is (GATE CS 2004)a) The number of leaf nodes in the treeb) The number of nodes in the treec) The number of internal nodes in the treed) The height of the treeAnswer: (d)Explanation: DoSomething() returns max(height of left child + 1, height of left child + 1). So given that pointer to root of tree is passed to DoSomething(), it will return height of the tree. Note that this implementation follows the convention where height of a single node is 0. 2. Suppose we run Dijkstra’s single source shortest-path algorithm on the following edge weighted directed graph with vertex P as the source. In what order do the nodes get included into the set of vertices for which the shortest path distances are finalized? (GATE CS 2004)a) P, Q, R, S, T, Ub) P, Q, R, U, S, Tc) P, Q, R, U, T, Sd) P, Q, T, R, U, S

Answer (b)

3. Suppose each set is represented as a linked list with elements in arbitrary order. Which of the operations among union, intersection, membership, cardinality will be the slowest? (GATE CS 2004)a) union only

b) intersection, membershipc) membership, cardinalityd) union, intersectionAnswer (a)Cardinality and membership are definably not the slowest one. For cardinality, just count the number of nodes in a list. For membership, just traverse the list and look for a matchFor getting intersection of L1 and L2, search for each element of L1 in L2 and print the elements we find in L2. There can be many ways for getting union of L1 and L2. One of them is as followsa) Print all the nodes of L1 and print only those which are not present in L2.b) Print nodes of L2.All of these methods will require more operations than intersection as we have to process intersection node plus other nodes.

4. The time complexity of the following C function is (assume n > 0 (GATE CS 2004) 

int recursive (mt n){

   if (n == 1)

     return (1);   else

     return (recursive (n-1) + recursive (n-1));}

a) 0(n)b) 0(nlogn)c) 0(n^2)d) 0(2^n)Answer(d)Explanation:Recursive expression for the above program will be.

T(n) = 2T(n-1) + c T(1) = c1.

Let us solve it.

T(n) = 2(2T(n-2) + c) + c = 4T(n-2) + 3c T(n) = 8T(n-3) + 6c + c = 8T(n-3) + 7c T(n) = 16T(n-4) + 14c + c = 16T(n-4) + 15c ............................................................ ............................................................. T(n) = (2^(n-1))T(1) + (2^(n-1) - 1)c

T(n) = O(2^n)  

1. Suppose you are given an array s[1...n] and a procedure reverse (s,i,j) which reverses the order of elements in a between positions i and j (both inclusive). What does the following sequence do, where 1 < k <= n: reverse (s, 1, k); reverse (s, k + 1, n); reverse (s, 1, n); (GATE CS 2000)(a) Rotates s left by k positions(b) Leaves s unchanged(c) Reverses all elements of s(d) None of the above

Answer: (a)Effect of the above 3 reversals for any k is equivalent to left rotation of the array of size n by k. If we rotate an array n times for k = 1 to n, we get the same array back.

2. The best data structure to check whether an arithmetic expression has balanced parentheses is a (GATE CS 2004)a) queueb) stackc) treed) listAnswer(b)There are three types of parentheses [ ] { } (). Below is an arbit c code segment which has parentheses of all three types.void func(int c, int a[]) { return ((c +2) + arr[(c-2)]) ; } Stack is a straightforward choice for checking if left and right parentheses are balanced. Here is a algorithm to do the same. /*Return 1 if expression has balanced parentheses */ bool areParenthesesBalanced(expression ) { for each character in expression { if(character == ’(’ || character == ’{’ || character == ’[’) push(stack, character); if(character == ’)’ || character == ’}’ || character == ’]’) { if(isEmpty(stack)) return 0; /*We are seeing a right parenthesis without a left pair*/ /* Pop the top element from stack, if it is not a pair bracket of character then there is a mismatch. This will happen for expressions like {(}) */ else if (! isMatchingPair(pop(stack), character) ) return 0; } } if(isEmpty(stack)) return 1; /*balanced*/ else return 0; /*not balanced*/ } /* End of function to check parentheses */ /* Returns 1 if character1 and character2 are matching left and right parentheses */ bool isMatchingPair(character1, character2) { if(character1 == ‘(‘ && character2 == ‘)’) return 1; else If(character1 == ‘{‘ && character2 == ‘}’) return 1; else If(character1 == ‘[‘ && character2 == ‘]’) return 1; else return 0; } 3. Level order traversal of a rooted tree can be done by starting from the root and performing (GATE CS 2004)a) preorder traversalb) in-order traversalc) depth first searchd) breadth first searchAnswer(d)

4. Given the following input (4322, 1334, 1471, 9679, 1989, 6171, 6173, 4199) and the hash function x mod 10, which of the following statements are true?i. 9679, 1989, 4199 hash to the same valueii. 1471, 6171 has to the same valueiii. All elements hash to the same valueiv. Each element hashes to a different value(GATE CS 2004)a) i onlyb) ii onlyc) i and ii onlyd) iii or ivAnswer (c)

5. Postorder traversal of a given binary search tree, T produces the following sequence of keys10, 9, 23, 22, 27, 25, 15, 50, 95, 60, 40, 29Which one of the following sequences of keys can be the result of an in-order traversal of the tree T? (GATE CS 2005)a) 9, 10, 15, 22, 23, 25, 27, 29, 40, 50, 60, 95b) 9, 10, 15, 22, 40, 50, 60, 95, 23, 25, 27, 29c) 29, 15, 9, 10, 25, 22, 23, 27, 40, 60, 50, 95d) 95, 50, 60, 40, 27, 23, 22, 25, 10, 9, 15, 29

Answer (a)Inorder traversal of a BST   always gives elements in increasing order. Among all four options, a) is the only increasing order sequence.

1. Consider the following C function. 

float f(float x, int y) { float p, s; int i; for (s=1, p=1, i=1; i < y; i ++) { p*= x/i; s+=p; } return s; }

For large values of y, the return value of the function f best approximates (GATE CS 2003)a) x^yb) e^xc) ln(1 + x)d) x^x Answer (b)The function f() is implementation of Taylor’s Series to calculates e^x 

e^x = 1 + x + x^2/2! + x^3/3! + ---

More is the value of y more precise value of e^x will be returned by f() References:http://en.wikipedia.org/wiki/E_%28mathematical_constant%29

2. In the worst case, the number of comparisons needed to search a singly linked list of length n for a given element is (GATE CS 2002)a) log 2 nb) n/2c) log 2 n – 1d) n Answer(d)In the worst case, the element to be searched has to be compared with all elements of linked list.3. The elements 32, 15, 20, 30, 12, 25, 16 are inserted one by one in the given order into a Max Heap. The resultant Max Heap is.

Answer (a)4. Consider the following three claimsI (n + k)^m =  (n^m), where k and m are constantsII 2^(n + 1) = 0(2^n)III 2^(2n + 1) = 0(2^n)Which of these claims are correct? (GATE CS 2003)(a) I and II(b) I and III(c) II and III(d) I, II and III Answer(a)

(I) (n+m)^k = n^k + c1*n^(k-1) + ... k^m = (n^k)(II) 2^(n+1) = 2*2^n = O(2^n) 

5. A single array A[1..MAXSIZE] is used to implement two stacks. The two stacks grow from opposite ends of the array. Variables top1 and top2 (topl< top 2) point to the location of the topmost element in each of the stacks. If the space is to be used efficiently, the condition for “stack full” is (GATE CS 2004)a) (top1 = MAXSIZE/2) and (top2 = MAXSIZE/2+1)b) top1 + top2 = MAXSIZEc) (top1= MAXSIZE/2) or (top2 = MAXSIZE)d) top1= top2 -1Answer(d)If we are to use space efficiently then size of the any stack can be more than MAXSIZE/2.Both stacks will grow from both ends and if any of the stack top reaches near to the other top then stacks are full. So the condition will be top1 = top2 -1 (given that top1 < top2)

1. The usual  (n^2) implementation of Insertion Sort to sort an array uses linear search to identify the position where an element is to be inserted into the

already sorted part of the array. If, instead, we use binary search to identify the position, the worst case running time will (GATE CS 2003)(a) remain  (n^2)(b) become  (n(log n)^2)(c) become  (n log n)(d) become  (n) Answer (a)

If we use binary search then there will be   comparisons in the worst case,

which is  (n log n) ( If you want to know how   can be equal to  (n log n), then see this for proof). But the algorithm as a whole will still have a running time of  (n^2) on average because of the series of swaps required for each insertion.Reference:http://en.wikipedia.org/wiki/Insertion_sort

2. The tightest lower bound on the number of comparisons, in the worst case, for comparison-based sorting is of the order ofa) nb) n^2c) nlognd) n(log^2)nAnswer (c)The number of comparisons that a comparison sort algorithm requires increases in proportion to nlog(n), where n is the number of elements to sort. This bound is asymptotically tight:Given a list of distinct numbers (we can assume this because this is a worst-case analysis), there are n factorial permutations exactly one of which is the list in sorted order. The sort algorithm must gain enough information from the comparisons to identify the correct permutations. If the algorithm always completes after at most f(n) steps, it cannot distinguish more than 2^f(n) cases because the keys are distinct and each comparison has only two possible outcomes. Therefore,2^f(n)   n!, or equivalently f(n) [Tex]\log_2[/Tex](n!). References:http://en.wikipedia.org/wiki/Comparison_sorthttp://www.cs.cmu.edu/afs/cs.cmu.edu/academic/class/15451-s07/www/lecture_notes/lect0130.pdf

3. The problem 3-SAT and 2-SAT area) both in Pb) both NP completec) NP-complete and in P respectivelyd) undecidable and NP-complete respectivelyAnswer (c)The Boolean satisfiability problem (SAT) is a decision problem, whose instance is a Boolean expression written using only AND, OR, NOT, variables, and parentheses. The problem is: given the expression, is there some assignment of TRUE and FALSE values to the variables that will make the entire expression true? A formula of propositional logic is said to be satisfiable if logical values can be assigned to its variables in a way that makes the formula true.3-SAT and 2-SAT are special cases of k-satisfiability (k-SAT) or simply satisfiability (SAT), when each clause contains exactly k = 3 and k = 2 literals respectively. 2-SAT is P while 3-SAT is NP Complete. (See this   for explanation)References:http://en.wikipedia.org/wiki/Boolean_satisfiability_problem

4. Consider the following graph 

Among the following sequencesI) a b e g h fII) a b f e h gIII) a b f h g eIV) a f g h b eWhich are depth first traversals of the above graph? (GATE CS 2003)a) I, II and IV onlyb) I and IV onlyc) II, III and IV onlyd) I, III and IV only Answer (d)

1. In a binary max heap containing n numbers, the smallest element can be found in time (GATE CS 2006)(A) 0(n)(B) O(logn)(C) 0(loglogn)(D) 0(1)Answer (A)In a max heap, the smallest element is always present at a leaf node. So we need to check for all leaf nodes for the minimum value. Worst case complexity will be O(n)

12 / \ / \ 8 7 / \ / \ / \ / \2 3 4 5

2. A scheme for storing binary trees in an array X is as follows. Indexing of X starts at 1 instead of 0. the root is stored at X[1]. For a node stored at X[i], the left child, if any, is stored in X[2i] and the right child, if any, in X[2i+1]. To be able to store any binary tree on n vertices the minimum size of X should be. (GATE CS 2006)(A) log2n(B) n(C) 2n + 1(D) 2^n — 1Answer (D)For a right skewed binary tree, number of nodes will be 2^n – 1. For example, in below binary tree, node ‘A’ will be stored at index 1, ‘B’ at index 3, ‘C’ at index 7 and ‘D’ at index 15.

A \ \ B

\ \ C \ \ D

3. Which one of the following in place sorting algorithms needs the minimum number of swaps? (GATE CS 2006)(A) Quick sort(B) Insertion sort(C) Selection sort(D) Heap sortAnswer (C)For selection sort, number of swaps required is minimum (  (n) ).4. An element in an array X is called a leader if it is greater than all elements to the right of it in X. The best algorithm to find all leaders in an array (GATE CS 2006)(A) Solves it in linear time using a left to right pass of the array(B) Solves it in linear time using a right to left pass of the array(C) Solves it using divide and conquer in time 8(nlogn)(D) Solves it in time 8(n2)Answer (B)Please see this   post for explanation.5. Consider a weighted complete graph G on the vertex set {v1,v2 ,v} such that the weight of the edge (v,,v) is 2|i-j|. The weight of a minimum spanning tree of G is: (GATE CS 2006)(A) n — 1(B) 2n — 2(C) nC2(D) 2Answer (B)Minimum spanning tree of such a graph is

v1 \ v2 \ v3 \ v4 . . . vn

Weight of the minimum spanning tree= 2|2 – 1| + 2|3 – 2| + 2|4 – 3| + 2|5 – 4| …. + 2| n – (n-1) |= 2n – 2

1. Consider the following functions

Which of the following is true? (GATE CS 2000)

(a) h(n) is 0(f(n))(b) h(n) is 0(g(n))(c) g(n) is not 0(f(n))(d) f(n) is 0(g(n))Answer (d)

g(n) = 2^  = n^  f(n) and g(n) are of same asymptotic order and following statements are true.f(n) = O(g(n))g(n) = O(f(n)).

(a) and (b) are false because n! is of asymptotically higher order than n^ .

2. Let G be an undirected connected graph with distinct edge weight. Let emax be the edge with maximum weight and emin the edge with minimum weight. Which of the following statements is false? (GATE CS 2000)(a) Every minimum spanning tree of G must contain emin(b) If emax is in a minimum spanning tree, then its removal must disconnect G(c) No minimum spanning tree contains emax(d) G has a unique minimum spanning treeAnswer (c)(a) and (b) are always true.(c) is false because (b) is true.(d) is true because all edge weights are distinct for G.

3. Let G be an undirected graph. Consider a depth-first traversal of G, and let T be the resulting depth-first search tree. Let u be a vertex in G and let v be the first new (unvisited) vertex visited after visiting u in the traversal. Which of the following statements is always true? (GATE CS 2000)(a) {u,v} must be an edge in G, and u is a descendant of v in T(b) {u,v} must be an edge in G, and v is a descendant of u in T(c) If {u,v} is not an edge in G then u is a leaf in T(d) If {u,v} is not an edge in G then u and v must have the same parent in TAnswer (c)

4. Consider an undirected unweighted graph G. Let a breadth-first traversal of G be done starting from a node r. Let d(r, u) and d(r, v) be the lengths of the shortest paths from r to u and v respectively, in G. lf u is visited before v during the breadth-first traversal, which of the following statements is correct? (GATE CS 2001) a) d(r, u) < d (r, v)b) d(r, u) > d(r, v)c) d(r, u) <= d (r, v)d) None of the above Answer (c)d(r, u) and d(r, v) will be equal when u and v are at same level, otherwise d(r, u) will be less than d(r, v)

5. How many undirected graphs (not necessarily connected) can be constructed out of a given set V= {V 1, V 2,…V n} of n vertices ? (GATE CS 2001)a) n(n-l)/2b) 2^nc) n!d) 2^(n(n-1)/2) Answer (d)In an undirected graph, there can be maximum n(n-1)/2 edges. We can choose to have (or not have) any of the n(n-1)/2 edges. So, total number of undirected graphs with n vertices is 2^(n(n-1)/2).

1 In a heap with n elements with the smallest element at the root, the 7th smallest element can be found in time (GATE CS 2003)

a)  (n log n)b)  (n)c)  (log n)d)  (1)Answer(c)Given a Min-heap, to get the 7th smallest element, we need to call Extract-Min   7 times which means  (7logn)( =  (logn)) operations 

2. Suppose the numbers 7, 5, 1, 8, 3, 6, 0, 9, 4, 2 are inserted in that order into an initially empty binary search tree. The binary search tree uses the usual ordering on natural numbers. What is the in-order traversal sequence of the resultant tree? (GATE CS 2003)a) 7 5 1 0 3 2 4 6 8 9b) 0 2 4 3 1 6 5 9 8 7c) 0 1 2 3 4 5 6 7 8 9d) 9 8 6 4 2 3 0 1 5 7 Answer (c)In-order traversal of a BST gives elements in increasing order. So answer c is correct without any doubt.

3. Let S be a stack of size n >= 1. Starting with the empty stack, suppose we push the first n natural numbers in sequence, and then perform n pop operations. Assume that Push and Pop operation take X seconds each, and Y seconds elapse between the end of one such stack operation and the start of the next operation. For m >= 1, define the stack-life of m as the time elapsed from the end of Push(m) to the start of the pop operation that removes m from S. The average stack-life of an element of this stack is (GATE CS 2003)a) n(X+ Y)b) 3Y + 2Xc) n(X + Y)-Xd) Y + 2X Answer(c)We can easily arrive at the result by taking few examples.

1. The height of a binary tree is the maximum number of edges in any root to leaf path. The maximum number of nodes in a binary tree of height h is:(A) 2^h -1(B) 2^(h-1) – 1(C) 2^(h+1) -1(D) 2*(h+1)Answer (C)Maximum number of nodes will be there for a complete tree.Number of nodes in a complete tree of height h = 1 + 2 + 2^2 + 2*3 + …. 2^h = 2^(h+1) – 12: The maximum number of binary trees that can be formed with three unlabeled nodes is:(A) 1(B) 5(C) 4(D) 3Answer (B)

O / \ O O (i)

O / O

/ O (ii)

O / O \ O (iii)

O \ O \ O (iv)

O \ O / O (v)

Note that nodes are unlabeled. If the nodes are labeled, we get more number of trees.3. Which of the following sorting algorithms has the lowest worst-case complexity?(A) Merge sort(B) Bubble sort(C) Quick sort(D) Selection sortAnswer (A)Worst case complexities for the above sorting algorithms are as follows:Merge Sort — nLognBubble Sort — n^2Quick Sort — n^2Selection Sort — n^24. The following postfix expression with single digit operands is evaluated using a stack: 

8 2 3 ^ / 2 3 * + 5 1 * -

Note that ^ is the exponentiation operator. The top two elements of the stack after the first * is evaluated are:(A) 6, 1(B) 5, 7(C) 3, 2(D) 1, 5Answer (A)The algorithm for evaluating any postfix expression is fairly straightforward:

1. While there are input tokens left o Read the next token from input. o If the token is a value + Push it onto the stack. o Otherwise, the token is an operator (operator here includes both operators, and functions). * It is known a priori that the operator takes n arguments. * If there are fewer than n values on the stack (Error) The user has not input sufficient values in the expression. * Else, Pop the top n values from the stack. * Evaluate the operator, with the values as arguments.

* Push the returned results, if any, back onto the stack.2. If there is only one value in the stack o That value is the result of the calculation.3. If there are more values in the stack o (Error) The user input has too many values.

Source for algorithm: http://en.wikipedia.org/wiki/Reverse_Polish_notation#The_postfix_algorithmLet us run the above algorithm for the given expression.First three tokens are values, so they are simply pushed. After pushing 8, 2 and 3, the stack is as follows 

8, 2, 3

When ^ is read, top two are popped and power(2^3) is calculated 

8, 8

When / is read, top two are popped and division(8/8) is performed 

1

Next two tokens are values, so they are simply pushed. After pushing 2 and 3, the stack is as follows 

1, 2, 3

When * comes, top two are popped and multiplication is performed. 

1, 6

5. The inorder and preorder traversal of a binary tree are d b e a f c g and a b d e c f g, respectively. The postorder traversal of the binary tree is:(A) d e b f g c a(B) e d b g f c a(C) e d b f g c a(D) d e f g b c aAnswer (A)

Below is the given tree. a / \ / \ b c / \ / \ / \ / \ d e f g  1. Consider a hash table of size seven, with starting index zero, and a hash function (3x + 4)mod7. Assuming the hash table is initially empty, which of the following is the contents of the table when the sequence 1, 3, 8, 10 is inserted into the table using closed hashing? Note that ‘_’ denotes an empty location in the table.

(A) 8, _, _, _, _, _, 10

(B) 1, 8, 10, _, _, _, 3

(C) 1, _, _, _, _, _,3

(D) 1, 10, 8, _, _, _, 3

Answer (B)Please see http://lcm.csa.iisc.ernet.in/dsa/node38.html for closed hashing and probing.Let us put values 1, 3, 8, 10 in the hash of size 7.Initially, hash table is empty - - - - - - - 0 1 2 3 4 5 6 The value of function (3x + 4)mod 7 for 1 is 0, so let us put the

value at 0 1 - - - - - - 0 1 2 3 4 5 6 The value of function (3x + 4)mod 7 for 3 is 6, so let us put the value at 6 1 - - - - - 3 0 1 2 3 4 5 6 The value of function (3x + 4)mod 7 for 8 is 0, but 0 is already occupied, let us put the value(8) at next available space(1) 1 8 - - - - 3 0 1 2 3 4 5 6 The value of function (3x + 4)mod 7 for 10 is 6, but 6 is already occupied, let us put the value(10) at next available space(2) 1 8 10 - - - 3 0 1 2 3 4 5 6 

2. In an unweighted, undirected connected graph, the shortest path from a node S to every other node is computed most efficiently, in terms of time complexity by(A) Dijkstra’s algorithm starting from S.(B) Warshall’s algorithm(C) Performing a DFS starting from S.(D) Performing a BFS starting from S.Answer(D)* Time Comlexity of the Dijkstra’s algorithm   is O(|V|^2 + E) * Time Comlexity of the Warshall’s algorithm is O(|V|^3) * DFS   cannot be used for finding shortest paths * BFS can be used for unweighted graphs. Time Complexity for BFS is O(|E| + |V|) 

3. A complete n-ary tree is a tree in which each node has n children or no children. Let I be the number of internal nodes and L be the number of leaves in a complete n-ary tree. If L = 41, and I = 10, what is the value of n?(A) 3(B) 4(C) 5(D) 6Answer (C)For an n-ary tree where each node has n children or no children, following relation holdsL = (n-1)*I + 1 Where L is the number of leaf nodes and I is the number of internal nodes.Let us find out the value of n for the given data.L = 41 , I = 10 41 = 10*(n-1) + 1 (n-1) = 4 n = 5 

4. In the following C function, let n >= m.int gcd(n,m) { if (n%m ==0) return m; n = n%m; return gcd(m,n); } How many recursive calls are made by this function?(A) (logn)?(B) (n)(C) (loglogn)(D) (sqrt(n))Answer (A)Above code is implementation of the Euclidean algorithm for finding Greatest Common Divisor (GCD).Please see http://mathworld.wolfram.com/EuclideanAlgorithm.html for time complexity.

5. What is the time complexity of the following recursive function:int DoSomething (int n) { if (n <= 2) return 1; else return (DoSomething (floor(sqrt(n))) + n); } (A)  (n)(B)  (nlogn)(C)  (logn)(D)  (loglogn)Answer (D)Recursive relation for the DoSomething() is

T(n) = T( ) + C1 if n > 2 We have ignored the floor() part as it doesn’t matter

here if it’s a floor or ceiling.Let n = 2^m, T(n) = T(2^m) Let T(2^m) = S(m) From the above two, T(n) = S(m) S(m) = S(m/2) + C1 /* This is simply binary search recursion*/ S(m) = O(logm) = O(loglogn) /* Since n = 2^m */ Now, let us go back to the original recursive function T(n) T(n) = S(m) = O(LogLogn)

1. Consider the following C program segment where CellNode represents a node in a binary tree:

struct CellNode{

  struct CellNOde *leftChild;

  int element;

  struct CellNode *rightChild;};

int GetValue(struct CellNode *ptr){

  int value = 0;

  if (ptr != NULL)  {

   if ((ptr->leftChild == NULL) &&        (ptr->rightChild == NULL))

      value = 1;

   else

      value = value + GetValue(ptr->leftChild)

                   + GetValue(ptr->rightChild);

  }

  return(value);

}

The value returned by GetValue() when a pointer to the root of a binary tree is passed as its argument is:(A) the number of nodes in the tree(B) the number of internal nodes in the tree(C) the number of leaf nodes in the tree(D) the height of the treeAnswer (C)

2. Consider the process of inserting an element into a Max Heap, where the Max Heap is represented by an array. Suppose we perform a binary search on the path from the new leaf to the root to find the position for the newly inserted element, the number of comparisons performed is:(A)  (logn)(B)  (LogLogn )(C)  (n)(D)  (nLogn)Answer (B)The height of a Max Heap is  (logn). If we perform binary search for finding the correct position then we need to do  (LogLogn) comparisons.

3. Let w be the minimum weight among all edge weights in an undirected connected graph. Let e be a specific edge of weight w . Which of the following is FALSE?(A) There is a minimum spanning tree containing e.(B) If e is not in a minimum spanning tree T, then in the cycle formed by adding e to T, all edges have the same weight.(C) Every minimum spanning tree has an edge of weight w .(D) e is present in every minimum spanning tree.Answer (D)(A), (B) and (C) are correct.(D) is incorrect as there may be many edges of wight w in the graph and e may not be picked up in some of the minimum spanning trees.

4. An array of n numbers is given, where n is an even number. The maximum as well as the minimum of these n numbers needs to be determined. Which of the following is TRUE about the number of comparisons needed?(A) At least 2n – c comparisons, for some constant c, are needed.(B) At most 1.5n – 2 comparisons are needed.(C) At least nLog2n comparisons are needed.(D) None of the above.Answer (B)

5. Consider the following C code segment:

int IsPrime(n){

  int i,n;  for(i=2;i<=sqrt(n);i++)

   if(n%i == 0)

     {printf(“Not Prime\n”); return 0;}

  return 1;}

Let T(n) denotes the number of times the for loop is executed by the program on input n. Which of the following is TRUE?(A) T(n) = O(sqrt(n)) and T(n) =  (sqrt(n))(B) T(n) = O(sqrt(n)) and T(n) =  (1)(C) T(n) = O(n) and T(n) =  (sqrt(n))(D) None of the aboveAnswer (B)Big O notation describes the upper bound and Big Omega notation describes the lower bound for an algorithm.The for loop in the question is run maximum sqrt(n) times and minimum 1 time. Therefore, T(n) = O(sqrt(n)) and T(n) =  (1)

1. The number of leaf nodes in a rooted tree of n nodes, with each node having 0 or 3 children is: a) n/2b) (n-1)/3c) (n-1)/2d) (2n+1)/3 Answer(d)

Let L be the number of leaf nodes and I be the number of internal nodes, then following relation holds for above given tree

L = (3-1)I + 1 = 2I + 1

Total number of nodes(n) is sum of leaf nodes and internal nodes

n = L + I

After solving above two, we get L = (2n+1)/3

2. The running time of the following algorithm 

Procedure A(n)

If n <= 2 return(1) else return A( );

is best described by a) O(n)b) O(log n)c) O(1og log n)d) O(1) Answer(c)

3. A weight-balanced tree is a binary tree in which for each node. The number of nodes in the left sub tree is at least half and at most twice the number of nodes in the right sub tree. The maximum possible height (number of nodes on the path from the root to the farthest leaf) of such a tree on n nodes is best described by which of the following? 

a) 

b) 

c) 

d) Answer(d)Let the maximum possible height of a tree with n nodes is represented by H(n).The maximum possible value of H(n) can be approximately written using following recursion

H(n) = H(2n/3) + 1

The solution of above recurrence is  . We can simply get it by drawing a recursion tree. 

4. Consider the following algorithm for searching for a given number x in an unsorted – array A[1..n] having n distinct values: 

1) Choose an i uniformly at random from 1..n; 2) If A[i] = x then Stop else Goto 1;

Assuming that x is present in A, what is the expected number of comparisons made by the algorithm before it terminates? a) nb) n-lc) 2nd) n/2 Answer(a)If you remember the coin and dice questions, you can just guess the answer for the above.Below is proof for the answer.Let expected number of comparisons be E. Value of E is sum of following expression for all the possible cases.

number_of_comparisons_for_a_case * probability_for_the_case

Case 1

If A[i] is found in the first attempt number of comparisons = 1 probability of the case = 1/n

Case 2

If A[i] is found in the second attempt number of comparisons = 2 probability of the case = (n-1)/n*1/n

Case 3

If A[i] is found in the third attempt number of comparisons = 2 probability of the case = (n-1)/n*(n-1)/n*1/n

There are actually infinite such cases. So, we have following infinite series for E. 

E = 1/n + [(n-1)/n]*[1/n]*2 + [(n-1)/n]*[(n-1)/n]*[1/n]*3 + …. (1)

After multiplying equation (1) with (n-1)/n, we get

E (n-1)/n = [(n-1)/n]*[1/n] + [(n-1)/n]*[(n-1)/n]*[1/n]*2 + [(n-1)/n]*[(n-1)/n]*[(n-1)/n]*[1/n]*3 ……….(2)

Subtracting (2) from (1), we get

E/n = 1/n + (n-1)/n*1/n + (n-1)/n*(n-1)/n*1/n + …………

The expression on right side is a GP with infinite elements. Let us apply the sum formula (a/(1-r)) 

E/n = [1/n]/[1-(n-1)/n] = 1 E = n   

1. We have a binary heap on n elements and wish to insert n more elements (not necessarily one after another) into this heap. The total time required for this is

(A)  (logn)

(B)  (n)

(C)  (nlogn)

(D)  (n^2)Answer(C)The worst case time complexity for insertion in a binary heap is O(Logn) (Refer Wiki).

So inserting n elements in a heap of size n should take  (nlogn) time. 

2. The Breadth First Search algorithm has been implemented using the queue data structure. One possible order of visiting the nodes of the following graph is

 (A) MNOPQR(B) NQMPOR(C) QMNPRO(D) QMNPORAnswer (C)

3. Consider the following functions: f(n) = 2^n g(n) = n! h(n) = n^logn Which of the following statements about the asymptotic behaviour of f(n), g(n), and h(n) is true?(A) f(n) = O(g(n)); g(n) = O(h(n))

(B) f(n) =  (g(n)); g(n) = O(h(n))(C) g(n) = O(f(n)); h(n) = O(f(n))

(D) h(n) = O(f(n)); g(n) =  (f(n))Answer (D)According to order of growth: h(n) < f(n) < g(n) (g(n) is asymptotically greater than f(n) and f(n) is asymptotically greater than h(n) )We can easily see above order by taking logs of the given 3 functions 

lognlogn < n < log(n!) (logs of the given f(n), g(n) and h(n)). Note that log(n!) = (nlogn)

4. The minimum number of comparisons required to determine if an integer appears more than n/2 times in a sorted array of n integers is

(A)  (n)

(B)  (logn)

(C)  (log*n)

(D)  (n)Answer (B)

1. The most efficient algorithm for finding the number of connected components in an undirected graph on n vertices and m edges has time complexity.

(A)  (n)

(B)  (m)

(C)  (m + n)

(D)  (mn) Answer (C)Connected components can be found in O(m + n) using Tarjan’s algorithm. Once we have connected components, we can count them. 

2. Consider the Quicksort algorithm. Suppose there is a procedure for finding a

pivot element which splits the list into two sub-lists each of which contains at least one-fifth of the elements. Let T(n) be the number of comparisons required to sort n elements. Then (A) T(n) <= 2T(n/5) + n(B) T(n) <= T(n/5) + T(4n/5) + n(C) T(n) <= 2T(4n/5) + n(D) T(n) <= 2T(n/2) + nAnswer (B)For the case where n/5 elements are in one subset, T(n/5) comparisons are needed for the first subset with n/5 elements, T(4n/5) is for the rest 4n/5 elements, and n is for finding the pivot. If there are more than n/5 elements in one set then other set will have less than 4n/5 elements and time complexity will be less than T(n/5) + T(4n/5) + n because recursion tree will be more balanced.

3 Dijkstra’s single source shortest path algorithm when run from vertex a in the below graph, computes the correct shortest path distance to

(A) only vertex a(B) only vertices a, e, f, g, h(C) only vertices a, b, c, d(D) all the verticesAnswer (D)Dijkstra’s single source shortest path is not guaranteed to work for graphs with negative weight edges, but it works for the given graph.Let us see…Let us run the 1st passb 1b is minimum, so shortest distance to b is 1.After 1st pass, distances arec 3, e -2.e is minimum, so shortest distance to e is -2After 2nd pass, distances arec 3, f 0.f is minimum, so shortest distance to f is 0After 3rd pass, distances arec 3, g 3.Both are same, let us take g. so shortest distance to g is 3.After 4th pass, distances arec 3, h 5c is minimum, so shortest distance to c is 3After 5th pass, distances areh -2h is minimum, so shortest distance to h is -24. The following C function takes a single-linked list of integers as a parameter and rearranges the elements of the list. The function is called with the list containing the integers 1, 2, 3, 4, 5, 6, 7 in the given order. What will be the contents of the list after the function completes execution?

struct node{

  int value;

  struct node *next;};

void rearrange(struct node *list){

  struct node *p, * q;

  int temp;

  if ((!list) || !list->next)      return;

  p = list;

  q = list->next;

  while(q)

  {

     temp = p->value;

     p->value = q->value;

     q->value = temp;

     p = q->next;

     q = p?p->next:0;

  }

}

(A) 1,2,3,4,5,6,7(B) 2,1,4,3,6,5,7(C) 1,3,2,5,4,7,6(D) 2,3,4,5,6,7,1Answer (B)The function rearrange() exchanges data of every node with its next node. It starts exchanging data from the first node itself.

1. Consider a binary max-heap implemented using an array. Which one of the following array represents a binary max-heap?

(A) 25,12,16,13,10,8,14(B) 25,14,13,16,10,8,12(C) 25,14,16,13,10,8,12(D) 25,14,12,13,10,8,16Answer (C)A tree is max-heap if data at every node in the tree is greater than or equal to it’s children’ s data.In array representation of heap tree,  a node at index i has its left child at index 2i + 1 and right child at index 2i + 2. 25 / \ / \ 14 16 / \ / \ / \ / \

13 10 8 12

2. What is the content of the array after two delete operations on the correct answer to the previous question?

(A) 14,13,12,10,8(B) 14,12,13,8,10(C) 14,13,8,12,10(D) 14,13,12,8,10Answer(D)For Heap trees, deletion of a node includes following two operations.1) Replace the root with last element on the last level.2) Starting from root, heapify the complete tree from top to bottom..Let us delete the two nodes one by one:1) Deletion of 25:Replace 25 with 12

12 / \ / \ 14 16 / \ / / \ /13 10 8

Since heap property is violated for root (16 is greater than 12), make 16 as root of the tree.

16 / \ / \ 14 12 / \ / / \ /13 10 8

2) Deletion of 16:Replace 16 with 8

8 / \ / \ 14 12 / \ / \ 13 10

Heapify from root to bottom.

14 / \ / \ 8 12 / \ / \ 13 10 14 / \ / \ 13 12 / \ / \ 8 10

3. In quick sort, for sorting n elements, the (n/4)th smallest element is selected as pivot using an O(n) time algorithm. What is the worst case time complexity of the quick sort?

(A)  (n)(B)  (nLogn)(C)  (n^2)(D)  (n^2 log n)Answer(B)The recursion expression becomes:T(n) = T(n/4) + T(3n/4) + cnAfter solving the above recursion, we get  (nLogn).

4. What is the maximum height of any AVL-tree with 7 nodes? Assume that the height of a tree with a single node is 0.(A) 2(B) 3(C) 4(D) 5Answer(B)AVL trees are binary trees with the following restrictions.1) the height difference of the children is at most 1.2) both children are AVL trees

a / \ / \ b c / \ / / \ / d e g / / h

References:http://en.wikipedia.org/wiki/AVL_tree

1. An implementation of a queue Q, using two stacks S1 and S2, is given below:

void insert(Q, x) { push (S1, x);}

void delete(Q){ if(stack-empty(S2)) then if(stack-empty(S1)) then { print(“Q is empty”); return; } else while (!(stack-empty(S1))){ x=pop(S1); push(S2,x); } x=pop(S2);}

Let n insert and m (<=n) delete operations be performed in an arbitrary order on an empty queue Q. Let x and y be the number of push and pop operations performed respectively in the process. Which one of the following is true for all m and n?

(A) n+m <= x < 2n and 2m <= y <= n+m(B) n+m <= x < 2n and 2m<= y <= 2n(C) 2m <= x < 2n and 2m <= y <= n+m(D) 2m <= x <2n and 2m <= y <= 2nAnswer(A)The order in which insert and delete operations are performed matters here.The best case: Insert and delete operations are performed alternatively. In every delete operation, 2 pop and 1 push operations are performed. So, total m+ n push (n push for insert() and m push for delete()) operations and 2m pop operations are performed.

The worst case: First n elements are inserted and then m elements are deleted. In first delete operation, n + 1 pop operations and n push operation are performed. Other than first, in all delete operations, 1 pop operation is performed. So, total m + n pop operations and 2n push operations are performed (n push for insert() and m push for delete())2. Consider the following graph:

Which one of the following cannot be the sequence of edges added, in that order, to a minimum spanning tree using Kruskal’s algorithm?(A) (a—b),(d—f),(b—f),(d—c),(d—e)(B) (a—b),(d—f),(d—c),(b—f),(d—e)(C) (d—f),(a—b),(d—c),(b—f),(d—e)(D) (d—f),(a—b),(b—f),(d—e),(d—c)Answer (D)The edge (d-e) cannot be considered before (d-c) in Kruskal’s minimum spanning tree algorithmbecause Kruskal’s algorithm picks the edge with minimum weight from the current set of edges at each step.3. The median of n elements can be found in O(n)time. Which one of the following is correct about the complexity of quick sort, in which median is selected as pivot?(A)  (n)(B)  (nlogn)(C)  (n^2)(D)  (n^3)Answer (B)If median is always used as pivot, then recursion remains T(n) = 2T(n/2) + cn for all the cases where cn is combined time for median finding and partition. So, worst case time complexity of this quick sort becomes  (nlogn). In practical implementations, however, this variant is considerably slower on average (see http://en.wikipedia.org/wiki/Quicksort#Selection-based_pivoting)

1. Consider the polynomial p(x) = a0 + a1x + a2x^2 +a3x^3, where ai != 0, for all i. The minimum number of multiplications needed to evaluate p on an input

x is:(A) 3(B) 4(C) 6(D) 9Answer (A)Multiplications can be minimized using following order for evaluation of the given expression.p(x) = a0 + x(a1 + x(a2 + a3x))2. To implement Dijkstra’s shortest path algorithm on unweighted graphs so that it runs in linear time, the data structure to be used is:(A) Queue(B) Stack(C) Heap(D) B-TreeAnswer(A)The shortest path in an un-weighted graph means the smallest number of edges that must be traversed in order to reach the destination in the graph. This is the same problem as solving the weighted version where all the weights happen to be 1. If we use Queue (FIFO) instead of Priority Queue (Min Heap), we get the shortest path in linear time O(|V| + |E|). Basically we do BFS   traversal of the graph to get the shortest paths.3.  A 3-ary max heap is like a binary max heap, but instead of 2 children, nodes have 3 children. A 3-ary heap can be represented by an array as follows: The root is stored in the first location, a[0], nodes in the next level, from left to right, is stored from a[1] to a[3]. The nodes from the second level of the tree from left to right are stored from a[4] location onward. An item x can be inserted into a 3-ary heap containing n items by placing x in the location a[n] and pushing it up the tree to satisfy the heap property.Which one of the following is a valid sequence of elements in an array representing 3-ary max heap?(A) 1, 3, 5, 6, 8, 9(B) 9, 6, 3, 1, 8, 5(C) 9, 3, 6, 8, 5, 1(D) 9, 5, 6, 8, 3, 1Answer (D)

9 / | \ / | \ 5 6 8 / | / | 3 1

4.  Suppose the elements 7, 2, 10 and 4 are inserted, in that order, into the valid 3- ary max heap found in the above question, Which one of the following is the sequence of items in the array representing the resultant heap?(A) 10, 7, 9, 8, 3, 1, 5, 2, 6, 4(B) 10, 9, 8, 7, 6, 5, 4, 3, 2, 1(C) 10, 9, 4, 5, 7, 6, 8, 2, 1, 3(D) 10, 8, 6, 9, 7, 2, 3, 4, 1, 5Answer(A)

After insertion of 7 9 / | \ / | \ 7 6 8 / | \ / | \ 3 1 5

After insertion of 2

9 / | \ / | \ 7 6 8 / | \ / / | \ / 3 1 5 2

After insertion of 10

10 / | \ / | \ 7 9 8 / | \ / | / | \ / | 3 1 5 2 6

After insertion of 4

10 / | \ / | \ 7 9 8 / | \ / | \ / | \ / | \ 3 1 5 2 6 4  

1. Let X be a problem that belongs to the class NP. Then which one of the following is TRUE?(A) There is no polynomial time algorithm for X.(B) If X can be solved deterministically in polynomial time, then P = NP.(C) If X is NP-hard, then it is NP-complete.(D) X may be undecidable.Answer (C)(A) is incorrect because set NP includes both P(Polynomial time solvable) and NP-Complete .(B) is incorrect because X may belong to P (same reason as (A))(C) is correct because NP-Complete set is intersection of NP and NP-Hard sets.(D) is incorrect because all NP problems are decidable in finite set of operations.2. What is the number of swaps required to sort n elements using selection sort, in the worst case?(A)  (n)(B)  (n log n)(C)  (n^2 )(D)  (n^2 log n)Answer (A)Here is Selection Sort algorithm for sorting in ascending order.1. Find the minimum value in the list 2. Swap it with the value in the first position 3. Repeat the steps above for the remainder of the list (starting at the second position and advancing each time) As we can see from the algorithm, selection sort performs swap only after finding the appropriate position of the current picked element. So there are O(n) swaps performed in selection sort.Because swaps require writing to the array, selection sort is preferable if writing to memory is significantly more expensive than reading. This is generally the case if the items are huge but the keys are small. Another example where writing times are crucial is an array stored in EEPROM or Flash. There is no other algorithm with less data movement.References:http://en.wikipedia.org/wiki/Selection_sort

3. The running time of an algorithm is represented by the following recurrence relation:if n <= 3 then T(n) = n else T(n) = T(n/3) + cn Which one of the following represents the time complexity of the algorithm?(A)  (n)(B)  (n log n)(C)  (n^2)(D)  (n^2log n)Answer(A)T(n) = cn + T(n/3) = cn + cn/3 + T(n/9) = cn + cn/3 + cn/9 + T(n/27) Taking the sum of infinite GP series. The value of T(n) will be less than this sum. T(n) <= cn(1/(1-1/3)) <= 3cn/2 or we can say cn <= T(n) <= 3cn/2 Therefore T(n) =  (n) This can also be solved using Master Theorem for solving recurrences. The given expression lies in Case 3   of the theorem. 4. The keys 12, 18, 13, 2, 3, 23, 5 and 15 are inserted into an initially empty hash table of length 10 using open addressing with hash function h(k) = k mod 10 and linear probing. What is the resultant hash table?

Answer (C)To get the idea of open addressing concept, you can go through below lines from Wikipedia.Open addressing, or closed hashing, is a method of collision resolution in hash tables. With this method a hash collision is resolved by probing, or searching through alternate locations in the array (the probe sequence) until either the target record is found, or an unused array slot is found, which indicates that there is no such key in the table. Well known probe sequences include:linear probing in which the interval between probes is fixed–often at 1.quadratic probing in which the interval between probes increases linearly (hence, the indices are described by a quadratic function).double hashing in which the interval between probes is fixed for each record but is computed by another hash function.

1. Let S be an NP-complete problem and Q and R be two other problems not known to be in NP. Q is polynomial time reducible to S and S is polynomial-time reducible to R. Which one of the following statements is true?(A) R is NP-complete(B) R is NP-hard(C) Q is NP-complete(D) Q is NP-hardAnswer (B)(A) Incorrect because R is not in NP. A NP Complete problem has to be in both NP and NP-hard.(B) Correct because a NP Complete problem S is polynomial time educable to R.(C) Incorrect because Q is not in NP.

(D) Incorrect because there is no NP-complete problem that is polynomial time Turing-reducible to Q. 

2) A set X can be represented by an array x[n] as follows:

Consider the following algorithm in which x,y and z are Boolean arrays of size n:

algorithm zzz(x[] , y[], z []){ int i; for (i=O; i<n; ++i) z[i] = (x[i] ^ ~y[i]) V (~x[i] ^ y[i])}

The set Z computed by the algorithm is:(A) (X Intersection Y)(B) (X Union Y)(C) (X-Y) Intersection (Y-X)(D) (X-Y) Union (Y-X)Answer (D)The expression x[i] ^ ~y[i]) results the only 1s in x where corresponding entry in y is 0. An array with these set bits represents set X – YThe expression ~x[i] ^ y[i]) results the only 1s in y where corresponding entry in x is 0. An array with these set bits represents set Y – X.The operator “V” results in Union of the above two sets.

3. Consider the following recurrence:

Which one of the following is true?(A) T(n) =  (loglogn)(B) T(n) =  (logn)(C) T(n) =  (sqrt(n))(D) T(n) =  (n)Answer (B)

Let n = 2^m T(2^m) = T(2^(m/2)) + 1 Let T(2^m) = S(m) S(m) = 2S(m/2) + 1

Above expression is a binary tree traversal recursion whose time complexity is  (m). You can also prove using Master theorem.

S(m) = (m) = (logn) /* Since n = 2^m */

Now, let us go back to the original recursive function T(n)

T(n) = T(2^m) = S(m) = (Logn)  

1. The subset-sum problem is defined as follows. Given a set of n positive integers, S = {a1 ,a2 ,a3 ,…,an} and positive integer W, is there a subset of S whose elements sum to W? A dynamic program for solving this problem uses a

2-dimensional Boolean array X, with n rows and W+1 columns. X[i, j],1 <= i <= n, 0 <= j <= W, is TRUE if and only if there is a subset of {a1 ,a2 ,...,ai} whose elements sum to j. Which of the following is valid for 2 <= i <= n and ai <= j <= W?(A) X[i, j] = X[i - 1, j] V X[i, j -ai](B) X[i, j] = X[i - 1, j] V X[i - 1, j - ai](C) X[i, j] = X[i - 1, j] V X[i, j - ai](D) X[i, j] = X[i - 1, j] V X[i -1, j - ai]Answer (B)X[I, j] (2 <= i <= n and ai <= j <= W), is true if any of the following is true1) Sum of weights excluding ai is equal to j, i.e., if X[i-1, j] is true.2) Sum of weights including ai is equal to j, i.e., if X[i-1, j-ai] is true so that we get (j – ai) + ai as j.2. In question 1, which entry of the array X, if TRUE, implies that there is a subset whose elements sum to W?(A) X[1, W](B) X[n ,0](C) X[n, W](D) X[n -1, n]Answer (C)If we get the entry X[n, W] as true then there is a subset of {a1, a2, .. an} that has sum as W.Reference: http://en.wikipedia.org/wiki/Subset_sum_problem3. Consider the following C program that attempts to locate an element x in an array Y[] using binary search. The program is erroneous.

1.   f(int Y[10], int x) {

2.     int i, j, k;3.     i = 0; j = 9;

4.     do {5.             k =  (i + j) /2;

6.             if( Y[k] < x)  i = k; else j = k;7.         } while(Y[k] != x && i < j);

8.     if(Y[k] == x) printf ("x is in the array ") ;

9.     else printf (" x is not in the array ") ;10. }

On which of the following contents of Y and x does the program fail?(A) Y is [1 2 3 4 5 6 7 8 9 10] and x < 10(B) Y is [1 3 5 7 9 11 13 15 17 19] and x < 1(C) Y is [2 2 2 2 2 2 2 2 2 2] and x > 2(D) Y is [2 4 6 8 10 12 14 16 18 20] and 2 < x < 20 and x is evenAnswer (C)The above program doesn’t work for the cases where element to be searched is the last element of Y[] or greater than the last element (or maximum element) in Y[]. For such cases, program goes in an infinite loop because i is assigned value as k in all iterations, and i never becomes equal to or greater than j. So while condition never becomes false.

4. In question 3, the correction needed in the program to make it work properly is(A) Change line 6 to: if (Y[k] < x) i = k + 1; else j = k-1;(B) Change line 6 to: if (Y[k] < x) i = k - 1; else j = k+1;(C) Change line 6 to: if (Y[k] <= x) i = k; else j = k;(D) Change line 7 to: } while ((Y[k] == x) && (i < j));

Answer (A)Below is the corrected function

f(int Y[10], int x) {

   int i, j, k;   i = 0; j = 9;

   do {           k =  (i + j) /2;

           if( Y[k] < x)  i = k + 1; else j = k – 1;       } while(Y[k] != x && i < j);

   if(Y[k] == x) printf ("x is in the array ") ;

   else printf (" x is not in the array ") ;}

Reference: http://en.wikipedia.org/wiki/Binary_search_algorithm#Implementations

1) A program P reads in 500 integers in the range [0..100] exepresenting the scores of 500 students. It then prints the frequency of each score above 50. What would be the best way for P to store the frequencies?(a) An array of 50 numbers(b) An array of 100 numbers(c) An array of 500 numbers(d) A dynamically allocated array of 550 numbersAnswer (a)An array of size 50 looks the best option to store number of students for each score. We need to store frequencies of scores above 50. We can ignore scores below 50 and to index the scores above 50, we can subtract 50 from the score value/

2) An undirected graph G has n nodes. Its adjacency matrix is given by an n × n square matrix whose (i) diagonal elements are 0‘s and (ii) non-diagonal elements are 1‘s. which one of the following is TRUE?(a) Graph G has no minimum spanning tree (MST)(b) Graph G has a unique MST of cost n-1(c) Graph G has multiple distinct MSTs, each of cost n-1(d) Graph G has multiple spanning trees of different costsAnswer (c)If all non diagonal elements are 1, then every vertex is connected to every other vertex in the graph with an edge of weight 1. Such a graph has multiple distinct MSTs with cost n-1. See the below example.The connected graph:

Below are three Minimum Spanning trees each of cost 2.0.Minimum Spanning Tree 1

Minimum Spanning Tree 2

Minimum Spanning Tree 3

3) The time complexity of computing the transitive closure of a binary relation on a set of n elements is known to be:a) O(n)b) O(nLogn)c) O(n^(3/2))d) O(n^3)Answer (d)In mathematics, the transitive closure of a binary relation R on a set X is the smallest transitive relation on X that contains R. If the original relation is transitive, the transitive closure will be that same relation; otherwise, the transitive closure will be a different relation. In computer science the concept of transitive closure can be thought of as constructing a data structure that makes it possible to answer reachability questions. That is, can one get from node a to node other node b in one or more hops? A binary relation tells you only that node a is connected to node b, and that node b is connected to node c, etc. After the transitive closure is constructed in an O(1) operation one may determine that node c is reachable from node a.Warshall’s algorithm can be used to construct the Transitive closure of directed graphs (). In Warshall’s original formulation of the algorithm, the graph is unweighted and represented by a Boolean adjacency matrix. Then the addition operation is replaced by logical conjunction (AND) and the minimum operation by logical disjunction (OR).References:http://en.wikipedia.org/wiki/Floyd%E2%80%93Warshall_algorithmhttp://en.wikipedia.org/wiki/Transitive_closure4. A Priority-Queue is implemented as a Max-Heap. Initially, it has 5 elements. The level-order traversal of the heap is given below:10, 8, 5, 3, 2Two new elements ”1‘ and ”7‘ are inserted in the heap in that order. The level-order traversal of the heap after the insertion of the elements is:(a) 10, 8, 7, 5, 3, 2, 1(b) 10, 8, 7, 2, 3, 1, 5(c) 10, 8, 7, 1, 2, 3, 5(d) 10, 8, 7, 3, 2, 1, 5

Answer (D)Original Max-Heap is:

10 / \ 8 5 / \ 3 2

After Insertion of 1.

10 / \ 8 5 / \ / 3 2 1

After Insertion of 7.

10 / \ 8 7 / \ / \ 3 2 1 5  

1. Which one of the following is a key factor for preferring B-trees to binary search trees for indexing database relations?(a) Database relations have a large number of records(b) Database relations are sorted on the primary key(c) B-trees require less memory than binary search trees(d) Data transfer form disks is in blocks.Answer (d)A disk block contains fairly large number of keys. Unlike BST where each node contains only one key, B-Tree is designed to contain large number of keys so that tree height is small. 2. How many distinct binary search trees can be created out of 4 distinct keys?(a) 5(b) 14(c) 24(d) 42Answer (b)Here is a systematic way to enumerate these BSTs. Consider all possible binary search trees with each element at the root. If there are n nodes, then for each choice of root node, there are n – 1 non-root nodes and these non-root nodes must be partitioned into those that are less than a chosen root and those that are greater than the chosen root.Let’s say node i is chosen to be the root. Then there are i – 1 nodes smaller than i and n – i nodes bigger than i. For each of these two sets of nodes, there is a certain number of possible subtrees. Let t(n) be the total number of BSTs with n nodes. The total number of BSTs with i at the root is t(i – 1) t(n – i). The two terms are multiplied together because the arrangements in the left and right subtrees are independent. That is, for each arrangement in the left tree and for each arrangement in the right tree, you get one BST with i at the root.Summing over i gives the total number of binary search trees with n nodes.

The base case is t(0) = 1 and t(1) = 1, i.e. there is one empty BST and there is one BST with one node.

3. In a complete k-ary tree, every internal node has exactly k children. The number of leaves in such a tree with n internal nodes is:(a) nk(b) (n – 1) k+ 1(c) n( k – 1) + 1(d) n(k – 1)Answer (c)4) Suppose T(n) = 2T(n/2) + n, T(0) = T(1) = 1Which one of the following is false.a) T(n) = O(n^2)b) T(n) =  (nLogn)c) T(n) =  (n^2)d) T(n) = O(nLogn)Answer (c)The given recurrence relation can be solved using Master Theorem. It lies in case 2 of Master Theorem. Or, if you remember recurrence relation of Merge Sort or best case Quick Sort, you can guess the value of T(n).T(n) =  (nLogn) By definition of Big O notation, we can say.

(nLogn) = O(nLogn) = O(n^2) (nLogn) ca be equal to  (n) or  (nLogn), but not  (n^2)

1. The following C function takes a simply-linked list as input argument. It modifies the list by moving the last element to the front of the list and returns the modified list. Some part of the code is left blank.

typedef struct node { int value; struct node *next;}Node;

Node *move_to_front(Node *head) { Node *p, *q; if ((head == NULL: || (head->next == NULL)) return head; q = NULL; p = head; while (p-> next !=NULL) { q = p; p = p->next; } _______________________________ return head;}

Choose the correct alternative to replace the blank line.(A) q = NULL; p->next = head; head = p;(B) q->next = NULL; head = p; p->next = head;(C) head = p; p->next = q; q->next = NULL;(D) q->next = NULL; p->next = head; head = p;

Answer(D)When the while loop ends, q contains address of second last node and p contains address of last node. So we need to do following things after while loop.i) Set next of q as NULL (q->next = NULL).ii) Set next of p as head (p->next = head).iii) Make head as p ( head = p)Step (ii) must be performed before step (iii). If we change head first, then we lose track of head node in the original linked list.

2. A hash table of length 10 uses open addressing with hash function h(k)=k mod 10, and linear probing. After inserting 6 values into an empty hash table, the table is as shown below.

Which one of the following choices gives a possible order in which the key values could have been inserted in the table?(A) 46, 42, 34, 52, 23, 33(B) 34, 42, 23, 52, 33, 46(C) 46, 34, 42, 23, 52, 33(D) 42, 46, 33, 23, 34, 52Answer (C)The sequence (A) doesn’t create the hash table as the element 52 appears before 23 in this sequence.The sequence (B) doesn’t create the hash table as the element 33 appears before 46 in this sequence.The sequence (C) creates the hash table as 42, 23 and 34 appear before 52 and 33, and 46 appears before 33.The sequence (D) doesn’t create the hash table as the element 33 appears before 23 in this sequence.3. How many different insertion sequences of the key values using the same hash function and linear probing will result in the hash table shown above?(A) 10(B) 20(C) 30(D) 40Answer (C)In a valid insertion sequence, the elements 42, 23 and 34 must appear before 52 and 33, and 46 must appear before 33.Total number of different sequences = 3! x 5 = 30In the above expression, 3! is for elements 42, 23 and 34 as they can appear in any order, and 5 is for element 46 as it can appear at 5 different places.

1 Consider a complete undirected graph with vertex set {0, 1, 2, 3, 4}. Entry Wij in the matrix W below is the weight of the edge {i, j}.

What is the minimum possible weight of a spanning tree T in this graph such that vertex 0 is a leaf node in the tree T?(A) 7(B) 8(C) 9(D) 10Answer (D)To get the minimum spanning tree with vertex 0 as leaf, first remove 0th row and 0th column and then get the minimum spanning tree (MST) of the remaining graph. Once we have MST of the remaining graph, connect the MST to vertex 0 with the edge with minimum weight (we have two options as there are two 1s in 0th row). 

2. In the graph given in question 1, what is the minimum possible weight of a path P from vertex 1 to vertex 2 in this graph such that P contains at most 3 edges?(A) 7(B) 8(C) 9(D) 10Answer (B)Path: 1 -> 0 -> 4 -> 2Weight: 1 + 4 + 33. The degree sequence of a simple graph is the sequence of the degrees of the nodes in the graph in decreasing order. Which of the following sequences can not be the degree sequence of any graph?I. 7, 6, 5, 4, 4, 3, 2, 1II. 6, 6, 6, 6, 3, 3, 2, 2III. 7, 6, 6, 4, 4, 3, 2, 2IV. 8, 7, 7, 6, 4, 2, 1, 1(A) I and II(B) III and IV(C) IV only(D) II and IVAnswer (D)In sequence IV, we have a vertex with degree 8 which is not possible in a simple graph (no self loops and no multiple edges) with total vertex count as 8. Maximum possible degree in such a graph is 7.In sequence II, four vertices are connected to 6 other vertices, but remaining 4 vertices have degrees as 3, 3, 2 and 2 which are not possible in a simple graph (no self loops and no multiple edges). 4. Consider a B+-tree in which the maximum number of keys in a node is 5. What is the minimum number of keys in any non-root node?(A) 1(B) 2(C) 3(D) 4Answer (B)Since the maximum number of keys is 5, maximum number of children a node can have is 6. Bydefinition of B Tree, minimum children that a node can have would be 6/2 = 3. Therefore, minimum number of keys that a node can have becomes 2 (3-1).

1) A max-heap is a heap where the value of each parent is greater than or equal

to the values of its children. Which of the following is a max-heap?

Answer: (B)A binary tree is max-heap if it is a complete binary tree (A complete binary tree is a binary tree in which every level, except possibly the last, is completely filled, and all nodes are as far left as possible) and it follows the max-heap property (value of each parent is greater than or equal to the values of its children).A) is not a max-heap because it is not a complete binary treeB) is a max-heap because it is complete binary tree and follows max-heap property.C) is not a max-heap because 8 is a chile of 5 in this tree, so violates the max-heap property.D) is not a max-heap because 8 is a chile of 5 in this tree, so violates the max-heap property. There are many other nodes in this tree which violate max-heap property in this tree.2) Four matrices M1, M2, M3 and M4 of dimensions pxq, qxr, rxs and sxt respectively can be multiplied is several ways with different number of total scalar multiplications. For example, when multiplied as ((M1 X M2) X (M3 X M4)), the total number of multiplications is pqr + rst + prt. When multiplied as (((M1 X M2) X M3) X M4), the total number of scalar multiplications is pqr + prs + pst.If p = 10, q = 100, r = 20, s = 5 and t = 80, then the number of scalar multiplications needed isA) 248000B) 44000C) 19000D) 25000Answer (C)We get minimum number of multiplications using ((M1 X (M2 X M3)) X M4).Total number of multiplications = 100x20x5 (for M2 x M3) + 10x100x5 + 10x5x80 = 19000.

3) Which of the given options provides the increasing order of asymptotic complexity of functions f1, f2, f3 and f4?

f1(n) = 2^nf2(n) = n^(3/2)f3(n) = nLognf4(n) = n^(Logn)A) f3, f2, f4, f1B) f3, f2, f1, f4C) f2, f3, f1, f4

D) f2, f3, f4, f1Answer (A)

4) We are given a set of n distinct elements and an unlabeled binary tree with n nodes. In how many ways can we populate the tree with the given set so that it becomes a binary search tree?A) 0B) 1C) n!D) (1/(n+1)).2nCnAnswer (B)See this   explanation from PeddaBoku.5) An algorithm to find the length of the longest monotonically increasing sequence of numbers in an array A[0 :n-1] is given below.Let Li denote the length of the longest monotonically increasing sequence starting at index i in the array

Which of the following statements is TRUE?(A) The algorithm uses dynamic programming paradigm(B) The algorithm has a linear complexity and uses branch and bound paradigm(C) The algorithm has a non-linear polynomial complexity and uses branch and bound paradigm(D) The algorithm uses divide and conquer paradigm.Answer: (A)

1) An undirected graph G(V, E) contains n ( n > 2 ) nodes named v1 , v2 ,….vn. Two nodes vi , vj are connected if and only if 0 < |i – j| <= 2. Each edge (vi, vj ) is assigned a weight i + j. A sample graph with n = 4 is shown below. 

What will be the cost of the minimum spanning tree (MST) of such a graph with n nodes?(A) 1/12(11n^2 – 5n)(B) n^2 – n + 1(C) 6n – 11(D) 2n + 1Answer: (B)Minimum spanning tree for 2 nodes would be

(v1) _ (v2)

Total weight 3Minimum spanning tree for 3 nodes would be

(v1) _ (v2) | (v3)

Total weight= 3 + 4 = 7Minimum spanning tree for 4 nodes would be

(v1) _ (v2) _ (v4) | (v3)

Total weight= 3 + 4 + 6 = 13Minimum spanning tree for 5 nodes would be

(v1) _ (v2) _ (v4) | (v3) | (v5)

Total weight= 3 + 4 + 6 + 8 = 21Minimum spanning tree for 6 nodes would be

(v1) _ (v2) _ (v4) _ (v6) | (v3) | (v5)

Total weight= 3 + 4 + 6 + 8 + 10 = 31We can observe from above examples that when we add kth node, the weight of spanning tree increases by 2k-2. Let T(n) be the weight of minimum spanning tree. T(n) can be written asT(n) = T(n-1) + (2n-2) for n > 2T(1) = 0, T(2) = 0 and T(2) = 3The recurrence can be written as sum of series (2n – 2) + (2n-4) + (2n-6) + (2n-8) + …. 3 and solution of this recurrence is n^2 – n + 1. 

2) The length of the path from v5 to v6 in the MST of previous question with n = 10 is(A) 11(B) 25(C) 31(D) 41Answer: (C)Any MST which has more than 5 nodes will have the same distance between v5 and v6 as the basic structure of all MSTs (with more than 5 nodes) would be following.

(v1) _ (v2) _ (v4) _ (v6) _ . . (more even numbered nodes) | (v3) | (v5) | . .(more odd numbered nodes)

Distance between v5 and v6 = 3 + 4 + 6 + 8 + 10 = 31

3) Consider two binary operators '  ' and ' ' with the precedence of

operator   being lower than that of the   operator. Operator   is right

associative while operator   is left associative. Which one of the following

represents the parse tree for expression (7  3  4  3   2)?

Answer: (B)

Let us consider the given expression ( ).

Since the precedence of   is higher, the sub-expression ( ) will be

evaluated first. In this sub-expression,   would be evaluated first because   is right to left associative. So the expression is evaluated

as  . Also, note that among the two   operators, first one

is evaluated before the second one because the associativity of   is left to right.

1) Let w(n) and A(n) denote respectively, the worst case and average case running time of an algorithm executed on an input of size n. which of the following is ALWAYS TRUE?

(A) 

(B) 

(C) 

(D) Answer (C)The worst case time complexity is always greater than or same as the average case time complexity.

2) The worst case running time to search for an element in a balanced in a binary search tree with n2^n elements is

(A) 

(B) 

(C) 

(D) Answer (C)

Time taken to search an element is   where h is the height of Binary Search Tree (BST). The growth of height of a balanced BST is logerthimic in terms of number of

nodes. So the worst case time to search an element would be   

which is   Which is   which can be written

as  . 

3) Assuming P != NP, which of the following is true ?(A) NP-complete = NP(B) NP-complete   P = (C) NP-hard = NP(D) P = NP-completeAnswer (B)The answer is B (no NP-Complete problem can be solved in polynomial time). Because, if one NP-Complete problem can be solved in polynomial time, then all NP problems can solved in polynomial time. If that is the case, then NP and P set become same which contradicts the given condition.

4) The height of a tree is defined as the number of edges on the longest path in the tree. The function shown in the pseudocode below is invoked as height (root) to compute the height of a binary tree rooted at the tree pointer root.

The appropriate expression for the two boxes B1 and B2 are(A) B1 : (1 + height(n->right)), B2 : (1 + max(h1,h2))(B) B1 : (height(n->right)), B2 : (1 + max(h1,h2))(C) B1 : height(n->right), B2 : max(h1,h2)(D) B1 : (1 + height(n->right)), B2 : max(h1,h2)Answer (A)The box B1 gets exected when left subtree of n is NULL and right sbtree is not NULL. In this case, height of n will be height of right subtree plus one.The box B2 gets executed when both left and right sbtrees of n are not NULL. In this case, height of n will be max of heights of left and right sbtrees of n plus 1.

5) A list of n string, each of length n, is sorted into lexicographic order using the merge-sort algorithm. The worst case running time of this computation is

(A) 

(B) 

(C) 

(D) Answer (B)The recurrence tree for merge sort will have height n. And O(n^2) work will be done at each level of the recurrence tree (Each level involves n comparisons and a comparison takes O(n) time in worst case). So time complexity of this Merge Sort will

be  .

1) The recurrence relation capturing the optimal time of the Tower of Hanoi problem with n discs is(A) T(n) = 2T(n – 2) + 2(B) T(n) = 2T(n – 1) + n(C) T(n) = 2T(n/2) + 1(D) T(n) = 2T(n – 1) + 1Answer (D)Following are the steps to follow to solve Tower of Hanoi problem recursively.

Let the three pegs be A, B and C. The goal is to move n pegs from A to C.To move n discs from peg A to peg C: move n-1 discs from A to B. This leaves disc n alone on peg A move disc n from A to C move n?1 discs from B to C so they sit on disc n

The recurrence function T(n) for time complexity of the above recursive solution can be written as following.T(n) = 2T(n-1) + 12) Consider the directed graph shown in the figure below. There are multiple shortest paths between vertices S and T. Which one will be reported by Dijstra?s shortest path algorithm? Assume that, in any iteration, the shortest path to a vertex v is updated only when a strictly shorter path to v is discovered.

(A) SDT(B) SBDT(C) SACDT(D) SACETAnswer (D)3) Suppose a circular queue of capacity (n – 1) elements is implemented with an array of n elements. Assume that the insertion and deletion operation are carried out using REAR and FRONT as array index variables, respectively. Initially, REAR = FRONT = 0. The conditions to detect queue full and queue empty are(A) Full: (REAR+1) mod n == FRONT, empty: REAR == FRONT(B) Full: (REAR+1) mod n == FRONT, empty: (FRONT+1) mod n == REAR

(C) Full: REAR == FRONT, empty: (REAR+1) mod n == FRONT(D) Full: (FRONT+1) mod n == REAR, empty: REAR == FRONTAnswer (A)

1) Which of the following statements is/are TRUE for an undirected graph?P: Number of odd degree vertices is evenQ: Sum of degrees of all vertices is evenA) P OnlyB) Q OnlyC) Both P and QD) Neither P nor QAnswer (C)Q is true: Since the graph is undirected, every edge increases the sum of degrees by 2.P is true: If we consider sum of degrees and subtract all even degrees, we get an even number (because Q is true). So total number of odd degree vertices must be even.

2) Consider an undirected random graph of eight vertices. The probability that there is an edge between a pair of vertices is 1/2. What is the expected number of unordered cycles of length three? (A) 1/8(B) 1(C) 7(D) 8 Answer (C)A cycle of length 3 can be formed with 3 vertices. There can be total 8C3 ways to pick 3 vertices from 8. The probability that there is an edge between two vertices is 1/2. So expected number of unordered cycles of length 3 = (8C3)*(1/2)^3 = 7

3) What is the time complexity of Bellman-Ford single-source shortest path algorithm on a complete graph of n vertices? 

(A) 

(B) 

(C) 

(D) Answer (C).

Time complexity of Bellman-Ford algorithm is   where V is number of vertices and E is number edges (See this). If the graph is complete, the value of E

becomes  . So overall time complexity becomes 

4) Which of the following statements are TRUE? (1) The problem of determining whether there exists a cycle in an undirected graph is in P.(2) The problem of determining whether there exists a cycle in an undirected graph is in NP.(3) If a problem A is NP-Complete, there exists a non-deterministic polynomial time algorithm to solve A. (A) 1,2 and 3(B) 1 and 2 only(C) 2 and 3 only(D) 1 and 3 onlyAnswer (A)1 is true because cycle detection can be done in polynomial time using DFS (See this).2 is true because P is a subset of NP.3 is true because NP complete is also a subset of NP and NP means Non-

deterministic Polynomial time solution exists. (See this)

5) Which one of the following is the tightest upper bound that represents the time complexity of inserting an object into a binary search tree of n nodes? (A) O(1)(B) O(log n)(C) O(n)(D) O(n log n)Answer (C)The worst case occurs for a skewed tree. In a skewed tree, when a new node is inserted as a child of bottommost node, the time for insertion requires traversal of all node. For example, consider the following tree and the case when something smaller than 70 is inserted.

100 / 90 / 80 / 70

6) Which one of the following is the tightest upper bound that represents the number of swaps required to sort n numbers using selection sort? (A) O(log n)(B) O(n)(C) O(n log n)(D) O(n^2)Answer (B)Selection sort requires only O(n) swaps. See this   for details. 

7) Consider the following operation along with Enqueue and Dequeue operations onqueues, where k is a global parameter 

MultiDequeue(Q){ m = k while (Q is not empty and m > 0) { Dequeue(Q) m = m - 1 }}

What is the worst case time complexity of a sequence of n MultiDequeue() operations on an initially empty queue? 

(A) 

(B) 

(C) 

(D) Answer (A)Since the queue is empty initially, the condition of while loop never becomes true. So the time complexity is 

1) What is the return value of f(p, p) if the value of p is initialized to 5 before the call? Note that the first parameter is passed by reference, whereas the second parameter is passed by value. 

int f(int &x, int c) { c = c - 1; if (c == 0) return 1; x = x + 1; return f(x, c) * x;}

(A) 3024(B) 6561(C) 55440(D) 161051 Answer (B)Since c is passed by value and x is passed by reference, all functions will have same copy of x, but different copies of c.f(5, 5) = f(x, 4)*x = f(x, 3)*x*x = f(x, 2)*x*x*x = f(x, 1)*x*x*x*x = 1*x*x*x*x = x^4Since x is incremented in every function call, it becomes 9 after f(x, 2) call. So the value of expression x^4 becomes 9^4 which is 6561. 

#include <stdio.h>

int f(int &x, int c){ c = c - 1; if (c == 0) return 1; x = x + 1; return f(x, c) * x;}int main(){ int p = 5; printf("%d", f(p, p));}

1) The preorder traversal sequence of a binary search tree is 30, 20, 10, 15, 25, 23, 39, 35, 42. Which one of the following is the postorder traversal sequence of the same tree? (A) 10, 20, 15, 23, 25, 35, 42, 39, 30(B) 15, 10, 25, 23, 20, 42, 35, 39, 30(C) 15, 20, 10, 23, 25, 42, 35, 39, 30(D) 15, 10, 23, 25, 20, 35, 42, 39, 30Ans (D)The following is the constructed tree 

30 / \ 20 39 / \ / \ 10 25 35 42 \ / 15 23

3) Consider the following function 

int unknown(int n) { int i, j, k = 0;

for (i = n/2; i <= n; i++) for (j = 2; j <= n; j = j * 2) k = k + n/2; return k; }

What is the returned value of the above function?

(A) 

(B) 

(C) 

(D) Answer (B)

The outer loop runs n/2 or   times. The inner loop runs   times (Note that j is divide by 2 in every iteration). So the statement "k = k + n/2;"

runs   times. The statement increases value of k by n/2. So the value of k

becomes n/2*  which is 

4) The number of elements that can be sorted in   time using heap sort is 

(A) 

(B) 

(C) 

(d) Answer (C)

Time complexity of Heap Sort is   for m input elements. For m

=  , the value of   will be

 which will

be   which is 

5) The procedure given below is required to find and replace certain characters inside an input character string supplied in array A. The characters to be replaced are supplied in array oldc, while their respective replacement characters are supplied in array newc. Array A has a fixed length of five characters, while arrays oldc and newc contain three characters each. However, the procedure is flawed 

void find_and_replace(char *A, char *oldc, char *newc) { for (int i = 0; i < 5; i++) for (int j = 0; j < 3; j++) if (A[i] == oldc[j]) A[i] = newc[j];}

The procedure is tested with the following four test cases(1) oldc = "abc", newc = "dab"(2) oldc = "cde", newc = "bcd"(3) oldc = "bca", newc = "cda"(4) oldc = "abc", newc = "bac"The tester now tests the program on all input strings of length five consisting of characters ‘a’, ‘b’, ‘c’, ‘d’ and ‘e’ with duplicates allowed. If the tester carries out this testing with the four test cases given above, how many test cases will be able to capture the flaw? (A) Only one(B) Only two

(C) Only three(D) All fourAnswer (B)The test cases 3 and 4 are the only cases that capture the flaw. The code doesn't work properly when an old character is replaced by a new character and the new character is again replaced by another new character. This doesn't happen in test cases (1) and (2), it happens only in cases (3) and (4).6) If array A is made to hold the string “abcde”, which of the above four test cases will be successful in exposing the flaw in this procedure? (A) None(B) 2 only(C) 3 and 4 only(D) 4 onlyAnswer (C)

#include <stdio.h>#include <string.h>

void find_and_replace(char *A, char *oldc, char *newc) { for (int i = 0; i < 5; i++) for (int j = 0; j < 3; j++) if (A[i] == oldc[j]) A[i] = newc[j];}

int main(){ char *oldc1 = "abc", *newc1 = "dab"; char *oldc2 = "cde", *newc2 = "bcd"; char *oldc3 = "bca", *newc3 = "cda"; char *oldc4 = "abc", *newc4 = "bac";

char test[] = "abcde";

printf("Test 2\n"); printf("%s\n", test); find_and_replace(test, oldc2, newc2); printf ("%s\n", test);

printf("\nTest 3\n"); strcpy(test, "abcde"); printf("%s\n", test); find_and_replace(test, oldc3, newc3); printf ("%s\n", test);

printf("\nTest 4\n"); strcpy(test, "abcde"); printf("%s\n", test); find_and_replace(test, oldc4, newc4); printf ("%s\n", test);}

Output: 

Test 2abcdeabbcd

Test 3abcdeaddde

Test 4abcdeaacde  

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Binary Search Trees

Question 1

What is the worst case time complexity for search, insert and delete operations in a general Binary

Search Tree?

A O(n) for all

B O(Logn) for all

C O(Logn) for search and insert, and O(n) for delete

D O(Logn) for search, and O(n) for insert and delete

Discuss it

Question 2

In delete operation of BST, we need inorder successor (or predecessor) of a node when the node to

be deleted has both left and right child as non-empty. Which of the following is true about inorder

successor needed in delete operation?

A Inorder Successor is always a leaf node

BInorder successor is always either a leaf node or a node with empty left child

C Inorder successor may be an ancestor of the node

DInorder successor is always either a leaf node or a node with empty right child

Discuss it

Question 3

We are given a set of n distinct elements and an unlabeled binary tree with n nodes. In how many

ways can we populate the tree with the given set so that it becomes a binary search tree? (GATE CS

2011)

A 0

B 1

C n!

D (1/(n+1)).2nCn

Discuss it

Question 4

How many distinct binary search trees can be created out of 4 distinct keys?

A 4

B 14

C 24

D 42

Discuss it

Question 5

Which of the following traversal outputs the data in sorted order in a BST?

A Preorder

B Inorder

C Postorder

D Level order

Discuss it

Question 6

Suppose the numbers 7, 5, 1, 8, 3, 6, 0, 9, 4, 2 are inserted in that order into an initially empty binary

search tree. The binary search tree uses the usual ordering on natural numbers. What is the in-order

traversal sequence of the resultant tree?

A 7 5 1 0 3 2 4 6 8 9

B 0 2 4 3 1 6 5 9 8 7

C 0 1 2 3 4 5 6 7 8 9

D 9 8 6 4 2 3 0 1 5 7

Discuss it

Question 7

The following numbers are inserted into an empty binary search tree in the given order: 10, 1, 3, 5, 15,

12, 16. What is the height of the binary search tree (the height is the maximum distance of a leaf node

from the root)? (GATE CS 2004)

A 2

B 3

C 4

D 6

Discuss it

Question 8

The preorder traversal sequence of a binary search tree is 30, 20, 10, 15, 25, 23, 39, 35, 42. Which

one of the following is the postorder traversal sequence of the same tree?

A 10, 20, 15, 23, 25, 35, 42, 39, 30

B 15, 10, 25, 23, 20, 42, 35, 39, 30

C 15, 20, 10, 23, 25, 42, 35, 39, 30

D 15, 10, 23, 25, 20, 35, 42, 39, 30

Discuss it

Question 9

Consider the following Binary Search Tree

10 / \ 5 20 / / \ 4 15 30 / 11

If we randomly search one of the keys present in above BST, what would be the expected number of

comparisons?

A 2.75

B 2.25

C 2.57

D 3.25

Discuss it

Question 10

Which of the following traversals is sufficient to construct BST from given traversals 1) Inorder 2)

Preorder 3) Postorder

A Any one of the given three traversals is sufficient

B Either 2 or 3 is sufficient

C 2 and 3

D 1 and 3

Discuss it

Question 11

Consider the following code snippet in C. The function print() receives root of a Binary Search Tree

(BST) and a positive integer k as arguments.// A BST nodestruct node {    int data;    struct node *left, *right;};

 int count = 0; void print(struct node *root, int k){    if (root != NULL && count <= k)    {        print(root->right, k);        count++;        if (count == k)

          printf("%d ", root->data);       print(root->left, k);    }}

What is the output of print(root, 3) where root represent root of the following BST. 15 / \ 10 20 / \ / \ 8 12 16 25

A 10

B 16

C 20

D 20 10

Discuss it

Question 12

Consider the same code as given in above question. What does the function print() do in general?

The function print() receives root of a Binary Search Tree (BST) and a positive integer k as

arguments.// A BST nodestruct node {    int data;    struct node *left, *right;};

 int count = 0; void print(struct node *root, int k){    if (root != NULL && count <= k)    {        print(root->right, k);        count++;        if (count == k)          printf("%d ", root->data);       print(root->left, k);    }}

A Prints the kth smallest element in BST

B Prints the kth largest element in BST

C Prints the leftmost node at level k from root

D Prints the rightmost node at level k from root

Discuss it

Question 13

You are given the postorder traversal, P, of a binary search tree on the n elements 1, 2, ..., n. You

have to determine the unique binary search tree that has P as its postorder traversal. What is the time

complexity of the most efficient algorithm for doing this?

A O(Logn)

B O(n)

C O(nLogn)

D none of the above, as the tree cannot be uniquely determined.

Discuss it

Question 14

Suppose we have a balanced binary search tree T holding n numbers. We are given two numbers L

and H and wish to sum up all the numbers in T that lie between L and H. Suppose there are m such

numbers in T. If the tightest upper bound on the time to compute the sum is O(nalogb n + mc logd n),

the value of a + 10b + 100c + 1000d is ____.

A 60

B 110

C 210

D 50

Discuss it

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Binary Trees

Question 1

Which of the following is a true about Binary Trees

A Every binary tree is either complete or full.

B Every complete binary tree is also a full binary tree.

C Every full binary tree is also a complete binary tree.

D No binary tree is both complete and full.

E None of the above

Discuss it

Question 2

If arity of operands is fixed, then which of the following notations can be used to parse expressions

without parentheses? a) Infix Notation (Inorder traversal of a expression tree) b) Postfix Notation

(Postorder traversal of a expression tree) c) Prefix Notation (Preorder traversal of a expression tree)

A b and c

B Only b

C a, b and c

D None of them

Discuss it

Question 3

What are the main applications of tree data structure? 1) Manipulate hierarchical data 2) Make

information easy to search (see tree traversal). 3) Manipulate sorted lists of data 4) Router algorithms

5) Form of a multi-stage decision-making, like Chess Game. 6) As a workflow for compositing digital

images for visual effects

A 1, 2, 3, 4 and 6

B 1, 2, 3, 4 and 5

C 1, 3, 4, 5 and 6

D 1, 2, 3, 4, 5 and 6

Discuss it

Question 4

Level of a node is distance from root to that node. For example, level of root is 1 and levels of left and

right children of root is 2. The maximum number of nodes on level i of a binary tree is 

In the following answers, the operator '^' indicates power.

A 2^(i-1)

B 2^i

C 2^(i+1)

D 2^[(i+1)/2]

Discuss it

Question 5

In a complete k-ary tree, every internal node has exactly k children or no child. The number of leaves

in such a tree with n internal nodes is:

A nk

B (n – 1) k+ 1

C n( k – 1) + 1

D n(k – 1)

Discuss it

Question 6

The maximum number of binary trees that can be formed with three unlabeled nodes is:

A 1

B 5

C 4

D 3

Discuss it

Question 7

In a complete k-ary tree, every internal node has exactly k children. The number of leaves in such a

tree with n internal nodes is: (GATE CS 2005)

A nk

B (n – 1) k+ 1

C n( k – 1) + 1

D n( k – 1)

Discuss it

Question 8

The number of leaf nodes in a rooted tree of n nodes, with each node having 0 or 3 children is:

A n/2

B (n-1)/3

C (n-1)/2

D (2n+1)/3

Discuss it

Question 9

A weight-balanced tree is a binary tree in which for each node. The number of nodes in the left sub

tree is at least half and at most twice the number of nodes in the right sub tree. The maximum

possible height (number of nodes on the path from the root to the farthest leaf) of such a tree on n

nodes is best described by which of the following? a)   b)   c)   

d) 

A A

B B

C C

D D

Discuss it

Question 10

A complete n-ary tree is a tree in which each node has n children or no children. Let I be the number

of internal nodes and L be the number of leaves in a complete n-ary tree. If L = 41, and I = 10, what is

the value of n?

A 6

B 3

C 4

D 5

Discuss it

Question 11

The height of a binary tree is the maximum number of edges in any root to leaf path. The maximum

number of nodes in a binary tree of height h is:

A 2^h -1

B 2^(h-1) – 1

C 2^(h+1) -1

D 2*(h+1)

Discuss it

Question 12

A scheme for storing binary trees in an array X is as follows. Indexing of X starts at 1 instead of 0. the

root is stored at X[1]. For a node stored at X[i], the left child, if any, is stored in X[2i] and the right

child, if any, in X[2i+1]. To be able to store any binary tree on n vertices the minimum size of X should

be. (GATE CS 2006)

A log2n

B n

C 2n + 1

D 2^n — 1

Discuss it

Question 13

Postorder traversal of a given binary search tree, T produces the following sequence of keys 10, 9,

23, 22, 27, 25, 15, 50, 95, 60, 40, 29 Which one of the following sequences of keys can be the result

of an in-order traversal of the tree T? (GATE CS 2005)

A 9, 10, 15, 22, 23, 25, 27, 29, 40, 50, 60, 95

B 9, 10, 15, 22, 40, 50, 60, 95, 23, 25, 27, 29

C 29, 15, 9, 10, 25, 22, 23, 27, 40, 60, 50, 95

D 95, 50, 60, 40, 27, 23, 22, 25, 10, 9, 15, 29

Discuss it

Question 14

Consider the following nested representation of binary trees: (X Y Z) indicates Y and Z are the left and

right sub stress, respectively, of node X. Note that Y and Z may be NULL, or further nested. Which of

the following represents a valid binary tree?

A (1 2 (4 5 6 7))

B (1 (2 3 4) 5 6) 7)

C (1 (2 3 4)(5 6 7))

D (1 (2 3 NULL) (4 5))

Discuss it

Question 15

Consider a node X in a Binary Tree. Given that X has two children, let Y be Inorder successor of X.

Which of the following is true about Y?

A Y has no right child

B Y has no left child

C Y has both children

D None of the above

Discuss it

Question 16

In a binary tree with n nodes, every node has an odd number of descendants. Every node is

considered to be its own descendant. What is the number of nodes in the tree that have exactly one

child?

A 0

B 1

C (n-1)/2

D n-1

Discuss it

Question 17

The height of a binary tree is the maximum number of edges in any root to leaf path. The maximum

number of nodes in a binary tree of height h is:

A 2h−1

B 2h−1 -1

C 2h+1-1

D 2h+1

Discuss it

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Balanced Binary Search Trees

Question 1

The worst case running time to search for an element in a balanced in a binary search tree with n2^n

elements is 

(A) 

(B) 

(C) 

(D) 

A A

B B

C C

D D

Discuss it

Question 2

What is the maximum height of any AVL-tree with 7 nodes? Assume that the height of a tree with a

single node is 0.

A 2

B 3

C 4

D 5

Discuss it

Question 3

What is the worst case possible height of AVL tree?

A2Logn Assume base of log is 2

B1.44log n Assume base of log is 2

C Depends upon implementation

D Theta(n)

Discuss it

Question 4

Which of the following is AVL Tree?A 100 / \ 50 200 / \ 10 300

B 100 / \ 50 200 / / \ 10 150 300 /5

C 100 / \ 50 200 / \ / \ 10 60 150 300 / \ \5 180 400

A Only A

B A and C

C A, B and C

D Only B

Discuss it

Question 5

Consider the following AVL tree.

60 / \ 20 100 / \ 80 120

Which of the following is updated AVL tree after insertion of 70A 70 / \ 60 100 / / \ 20 80 120

B 100 / \ 60 120 / \ / 20 70 80

C 80 / \ 60 100 / \ \ 20 70 120

D 80 / \ 60 100 / / \ 20 70 120

A A

B B

C C

D D

Discuss it

Question 6

Which of the following is a self-adjusting or self-balancing Binary Search Tree

A Splay Tree

B AVL Tree

C Red Black Tree

D All of the above

Discuss it

Question 7

Consider the following left-rotate and right-rotate functions commonly used in self-adjusting BSTsT1, T2 and T3 are subtrees of the tree rooted with y (on left side) or x (on right side) y x / \ Right Rotation / \ x T3 – - – - – - – > T1 y / \ < - - - - - - - / \ T1 T2 Left Rotation T2 T3

Which of the following is tightest upper bound for left-rotate and right-rotate operations.

A O(1)

B O(Logn)

C O(LogLogn)

D O(n)

Discuss it

Question 8

Which of the following is true

AThe AVL trees are more balanced compared to Red Black Trees, but they may cause more rotations during insertion and deletion.

BHeights of AVL and Red-Black trees are generally same, but AVL Trees may cause more rotations during insertion and deletion.

CRed Black trees are more balanced compared to AVL Trees, but may cause more rotations during insertion and deletion.

DHeights of AVL and Red-Black trees are generally same, but Red Black rees may cause more rotations during insertion and deletion.

Discuss it

Question 9

Which of the following is true about Red Black Trees?

AThe path from the root to the furthest leaf is no more than twice as long as the path from the root to the nearest leaf

B At least one children of every black node is red

C Root may be red

D A leaf node may be red

Discuss it

Question 10

Which of the following is true about AVL and Red Black Trees?

AIn AVL tree insert() operation, we first traverse from root to newly inserted node and then from newly inserted node to root. While in Red Black tree insert(), we only traverse once from root to newly inserted node.

BIn both AVL and Red Black insert operations, we traverse only once from root to newly inserted node,

CIn both AVL and Red Black insert operations, we traverse twiceL first traverse root to newly inserted node and then from newly inserted node to root.

D None of the above

Discuss it

Question 11

What is the worst case possible height of Red-Black tree? Assume base of Log as 2 in all options

A 2Log(n+1)

B 1.44 Logn

C 4Logn

D None of the above

Discuss it

Question 12

Is the following statement valid? A Red-Black Tree which is also a perfect Binary Tree can have all

black nodes

A Yes

B No

Discuss it

Question 13

Which of the following operations are used by Red-Black trees to maintain balance during insertion/deletion?

a) Recoloring of nodesb) Rotation (Left and Right)

A Only a

B Only b

C Both a and b

D Neither a nor b

Discuss it

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Graph

Question 1

Which of the following is an advantage of adjacency list representation over adjacency matrix

representation of a graph?

AIn adjacency list representation, space is saved for sparse graphs.

BDFS and BSF can be done in O(V + E) time for adjacency list representation. These operations take O(V^2) time in adjacency list representation. Here is V and E are number of vertices and edges respectively.

CAdding a vertex in adjacency list representation is easier than adjacency matrix representation.

D All of the above

Discuss it

Question 2

The degree sequence of a simple graph is the sequence of the degrees of the nodes in the graph in

decreasing order. Which of the following sequences can not be the degree sequence of any graph? I.

7, 6, 5, 4, 4, 3, 2, 1 II. 6, 6, 6, 6, 3, 3, 2, 2 III. 7, 6, 6, 4, 4, 3, 2, 2 IV. 8, 7, 7, 6, 4, 2, 1, 1

A I and II

B III and IV

C IV only

D II and IV

Discuss it

Question 3

The time complexity of computing the transitive closure of a binary relation on a set of n elements is

known to be:

A O(n)

B O(nLogn)

C O(n ^ (3/2))

D O(n^3)

Discuss it

Question 4

The most efficient algorithm for finding the number of connected components in an undirected graph

on n vertices and m edges has time complexity. (A)  (n) (B)  (m) (C)  (m + n) (D) 

(mn)

A A

B B

C C

D D

Discuss it

Question 5

Consider an undirected unweighted graph G. Let a breadth-first traversal of G be done starting from a

node r. Let d(r, u) and d(r, v) be the lengths of the shortest paths from r to u and v respectively, in G. lf

u is visited before v during the breadth-first traversal, which of the following statements is correct?

(GATE CS 2001)

A d(r, u) < d (r, v)

B d(r, u) > d(r, v)

C d(r, u) <= d (r, v)

D None of the above

Discuss it

Question 6

How many undirected graphs (not necessarily connected) can be constructed out of a given set V= {V

1, V 2,…V n} of n vertices ?

A n(n-l)/2

B 2^n

C n!

D 2^(n(n-1)/2)

Discuss it

Question 7

Which of the following statements is/are TRUE for an undirected graph? P: Number of odd degree

vertices is even Q: Sum of degrees of all vertices is even

A P Only

B Q Only

C Both P and Q

D Neither P nor Q

Discuss it

Question 8

Consider an undirected random graph of eight vertices. The probability that there is an edge between

a pair of vertices is 1/2. What is the expected number of unordered cycles of length three?

A 1/8

B 1

C 7

D 8

Discuss it

Question 9

Given an undirected graph G with V vertices and E edges, the sum of the degrees of all vertices is

A E

B 2E

C V

D 2V

Discuss it

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Tree Traversals

Question 1

Following function is supposed to calculate the maximum depth or height of a Binary tree -- the

number of nodes along the longest path from the root node down to the farthest leaf node.int maxDepth(struct node* node){   if (node==NULL)       return 0;   else   {       /* compute the depth of each subtree */       int lDepth = maxDepth(node->left);       int rDepth = maxDepth(node->right);         /* use the larger one */       if (lDepth > rDepth)           return X;       else return Y;   }}

What should be the values of X and Y so that the function works correctly?

A X = lDepth, Y = rDepth

B X = lDepth + 1, Y = rDepth + 1

C X = lDepth - 1, Y = rDepth -1

D None of the above

Discuss it

Question 2

What is common in three different types of traversals (Inorder, Preorder and Postorder)?

A Root is visited before right subtree

B Left subtree is always visited before right subtree

C Root is visited after left subtree

D All of the above

E None of the above

Discuss it

Question 3

The inorder and preorder traversal of a binary tree are d b e a f c g and a b d e c f g, respectively. The

postorder traversal of the binary tree is:

A d e b f g c a

B e d b g f c a

C e d b f g c a

D d e f g b c a

Discuss it

Question 4

What does the following function do for a given binary tree?int fun(struct node *root){   if (root == NULL)      return 0;   if (root->left == NULL && root->right == NULL)      return 0;   return 1 + fun(root->left) + fun(root->right);}

A Counts leaf nodes

B Counts internal nodes

CReturns height where height is defined as number of edges on the path from root to deepest node

DReturn diameter where diameter is number of edges on the longest path between any two nodes.

Discuss it

Question 5

Which of the following pairs of traversals is not sufficient to build a binary tree from the given

traversals?

A Preorder and Inorder

B Preorder and Postorder

C Inorder and Postorder

D None of the Above

Discuss it

Question 6

Consider two binary operators '  ' and ' ' with the precedence of operator   being lower

than that of the   operator. Operator   is right associative while operator   is left

associative. Which one of the following represents the parse tree for expression (7   3  4 -

 3   2)? (GATE CS 2011)

A A

B B

C C

D D

Discuss it

Question 7

Which traversal of tree resembles the breadth first search of the graph?

A Preorder

B Inorder

C Postorder

D Level order

Discuss it

Question 8

Which of the following tree traversal uses a queue data structure?

A Preorder

B Inorder

C Postorder

D Level order

Discuss it

Question 9

Which of the following cannot generate the full binary tree?

A Inorder and Preorder

B Inorder and Postorder

C Preorder and Postorder

D None of the above

Discuss it

Question 10

Consider the following C program segmentstruct CellNode{  struct CelINode *leftchild;  int element;  struct CelINode *rightChild;}

 int Dosomething(struct CelINode *ptr){    int value = 0;    if (ptr != NULL)    {      if (ptr->leftChild != NULL)        value = 1 + DoSomething(ptr->leftChild);      if (ptr->rightChild != NULL)        value = max(value, 1 + DoSomething(ptr->rightChild));    }    return (value);}

The value returned by the function DoSomething when a pointer to the root of a non-empty tree is

passed as argument is (GATE CS 2004)

A The number of leaf nodes in the tree

B The number of nodes in the tree

C The number of internal nodes in the tree

D The height of the tree

Discuss it

Question 11

Let LASTPOST, LASTIN and LASTPRE denote the last vertex visited in a postorder, inorder and

preorder traversal. Respectively, of a complete binary tree. Which of the following is always true?

(GATE CS 2000)

A LASTIN = LASTPOST

B LASTIN = LASTPRE

C LASTPRE = LASTPOST

D None of the above

Discuss it

Question 12

The array representation of a complete binary tree contains the data in sorted order. Which traversal

of the tree will produce the data in sorted form?

A Preorder

B Inorder

C Postorder

D Level order

Discuss it

Question 13

Consider the following rooted tree with the vertex P labeled as root

 The order in which the nodes are visited during in-

order traversal is

A SQPTRWUV

B SQPTURWV

C SQPTWUVR

D SQPTRUWV

Discuss it

Question 14

Consider the pseudocode given below. The function DoSomething() takes as argument a pointer to

the root of an arbitrary tree represented by the leftMostChild-rightSibling representation. Each node of

the tree is of type treeNode.typedef struct treeNode* treeptr;struct treeNode{    treeptr leftMostChild, rightSibling;};int DoSomething (treeptr tree){    int value=0;    if (tree != NULL)    {        if (tree->leftMostChild == NULL)            value = 1;        else            value = DoSomething(tree->leftMostChild);        value = value + DoSomething(tree->rightSibling);    }    return(value);

}

When the pointer to the root of a tree is passed as the argument to DoSomething, the value returned

by the function corresponds to the

A number of internal nodes in the tree.

B height of the tree.

C number of nodes without a right sibling in the tree.

D number of leaf nodes in the tree.

Discuss it

http://geeksquiz.com/algorithms/graph-shortest-paths/

Graph Shortest Paths

Question 1

Consider the directed graph shown in the figure below. There are multiple shortest paths between

vertices S and T. Which one will be reported by Dijstra?s shortest path algorithm? Assume that, in any

iteration, the shortest path to a vertex v is updated only when a strictly shorter path to v is discovered.

Graph

A SDT

B SBDT

C SACDT

D SACET

Discuss it

Question 2

To implement Dijkstra’s shortest path algorithm on unweighted graphs so that it runs in linear time, the

data structure to be used is:

A Queue

B Stack

C Heap

D B-Tree

Discuss it

Question 3

Dijkstra’s single source shortest path algorithm when run from vertex a in the below graph, computes

the correct shortest path distance to

A only vertex a

B only vertices a, e, f, g, h

C only vertices a, b, c, d

D all the vertices

Discuss it

Question 4

In an unweighted, undirected connected graph, the shortest path from a node S to every other node is

computed most efficiently, in terms of time complexity by

A Dijkstra’s algorithm starting from S.

B Warshall’s algorithm

C Performing a DFS starting from S.

D Performing a BFS starting from S.

Discuss it

Question 5

Suppose we run Dijkstra’s single source shortest-path algorithm on the following edge weighted

directed graph with vertex P as the source. In what order do the nodes get included into the set of

vertices for which the shortest path distances are finalized? (GATE CS 2004)

A P, Q, R, S, T, U

B P, Q, R, U, S, T

C P, Q, R, U, T, S

D P, Q, T, R, U, S

Discuss it

Question 6

Suppose we run Dijkstra’s single source shortest-path algorithm on the following edge weighted

directed graph with vertex P as the source. In what order do the nodes get included into the set of

vertices for which the shortest path distances are finalized? (GATE CS 2004)

A P, Q, R, S, T, U

B P, Q, R, U, S, T

C P, Q, R, U, T, S

D P, Q, T, R, U, S

Discuss it

Question 7

What is the time complexity of Bellman-Ford single-source shortest path algorithm on a complete

graph of n vertices? (A)   (B)   (C)   (D) 

A A

B B

C C

D D

Discuss it

Question 8

In a weighted graph, assume that the shortest path from a source 's' to a destination 't' is correctly

calculated using a shortest path algorithm. Is the following statement true? If we increase weight of

every edge by 1, the shortest path always remains same.

A Yes

B No

Discuss it

Question 9

Following statement is true or false?

If we make following changes to Dijkstra, then it can be used to find the longest simple path, assume that the graph is acyclic.

1) Initialize all distances as minus infinite instead of plus infinite.

2) Modify the relax condition in Dijkstra's algorithm to update distance of an adjacent v of the currently considered vertex u only if "dist[u]+graph[u][v] > dist[v]". In shortest path algo, the sign is opposite.

A True

B False

Discuss it

Question 10

Which of the following algorithm can be used to efficiently calculate single source shortest paths in a

Directed Acyclic Graph?

A Dijkstra

B Bellman-Ford

C Topological Sort

D Strongly Connected Component

Discuss it

Question 11

Given a directed graph where weight of every edge is same, we can efficiently find shortest path from

a given source to destination using?

A Breadth First Traversal

B Dijkstra's Shortest Path Algorithm

CNeither Breadth First Traversal nor Dijkstra's algorithm can be used

D Depth First Search

Discuss it

Question 12

Is the following statement valid about shortest paths? Given a graph, suppose we have calculated

shortest path from a source to all other vertices. If we modify the graph such that weights of all edges

is becomes double of the original weight, then the shortest path remains same only the total weight of

path changes.

A True

B False

Discuss it

Question 13

Match the following Group A Group Ba) Dijkstra's single shortest path algo p) Dynamic Programmingb) Bellmen Ford's single shortest path algo q) Backtracking c) Floyd Warshell's all pair shortest path algo. r) Greedy Algorithm

A a-r, b-q, c-p

B a-p, b-p, c-p

C a-r, b-p, c-p

D a-p, b-r, c-q

Discuss it

Question 14

Is the following statement valid?. 

Given a weighted graph where weights of all edges are unique (no two edge have same weights),

there is always a unique shortest path from a source to destination in such a graph.

A True

B False

Discuss it

Question 15

Is the following statement valid?. 

Given a graph where all edges have positive weights, the shortest paths produced by Dijsktra and

Bellman Ford algorithm may be different but path weight would always be same.

A True

B False

Discuss it

Question 16

Which of the following statement(s)is / are correct regarding Bellman-Ford shortest path algorithm?P: Always finds a negative weighted cycle, if one exist s.Q: Finds whether any negative weighted cycle is reachable from the source.

A P Only

B Q Only

C Both P and Q

D Neither P nor Q

Discuss it

http://geeksquiz.com/algorithms/graph-minimum-spanning-tree/

Graph Minimum Spanning Tree

Question 1

An undirected graph G(V, E) contains n ( n > 2 ) nodes named v1 , v2 ,….vn. Two nodes vi , vj are

connected if and only if 0 < |i – j| <= 2. Each edge (vi, vj ) is assigned a weight i + j. A sample graph

with n = 4 is shown below. What will be the cost of the minimum spanning tree (MST) of such a graph

with n nodes? (GATE CS 2011) 

A 1/12(11n^2 – 5n)

B n^2 – n + 1

C 6n – 11

D 2n + 1

Discuss it

Question 2

The length of the path from v5 to v6 in the MST of previous question with n = 10 is

A 11

B 25

C 31

D 41

Discuss it

Question 3

Consider a complete undirected graph with vertex set {0, 1, 2, 3, 4}. Entry Wij in the matrix W below is

the weight of the edge {i, j}. What is the minimum possible weight of a spanning tree T in this graph

such that vertex 0 is a leaf node in the tree T? (GATE CS 2010) 

A 7

B 8

C 9

D 10

Discuss it

Question 4

In the graph given in above question question, what is the minimum possible weight of a path P from

vertex 1 to vertex 2 in this graph such that P contains at most 3 edges?

A 7

B 8

C 9

D 10

Discuss it

Question 5

An undirected graph G has n nodes. Its adjacency matrix is given by an n × n square matrix whose (i)

diagonal elements are 0‘s and (ii) non-diagonal elements are 1‘s. which one of the following is TRUE?

A Graph G has no minimum spanning tree (MST)

B Graph G has a unique MST of cost n-1

C Graph G has multiple distinct MSTs, each of cost n-1

D Graph G has multiple spanning trees of different costs

Discuss it

Question 6

Consider the following

graph: 

Which one of the following cannot be the sequence of edges added, in that order, to a minimum

spanning tree using Kruskal’s algorithm?

A (a—b),(d—f),(b—f),(d—c),(d—e)

B (a—b),(d—f),(d—c),(b—f),(d—e)

C (d—f),(a—b),(d—c),(b—f),(d—e)

D (d—f),(a—b),(b—f),(d—e),(d—c)

Discuss it

Question 7

Let G be an undirected connected graph with distinct edge weight. Let emax be the edge with

maximum weight and emin the edge with minimum weight. Which of the following statements is false?

(GATE CS 2000)

A Every minimum spanning tree of G must contain emin

BIf emax is in a minimum spanning tree, then its removal must disconnect G

C No minimum spanning tree contains emax

D G has a unique minimum spanning tree

Discuss it

Question 8

Consider a weighted complete graph G on the vertex set {v1,v2 ,v} such that the weight of the edge

(v,,v) is 2|i-j|. The weight of a minimum spanning tree of G is: (GATE CS 2006)

A n — 1

B 2n — 2

C nC2

D 2

Discuss it

Question 9

Let G be a weighted graph with edge weights greater than one and G'be the graph constructed by

squaring the weights of edges in G. Let T and T' be the minimum spanning trees of G and G',

respectively, with total weights t and t'. Which of the following statements is TRUE?

A T' = T with total weight t' = t2

B T' = T with total weight t' < t2

C T' != T but total weight t' = t2

D None of the above

Discuss it

Question 10

Consider the following graph:  Which one of the

following is NOT the sequence of edges added to the minimum spanning tree using Kruskal's

algorithm?

A (b,e)(e,f)(a,c)(b,c)(f,g)(c,d)

B (b,e)(e,f)(a,c)(f,g)(b,c)(c,d)

C (b,e)(a,c)(e,f)(b,c)(f,g)(c,d)

D (b,e)(e,f)(b,c)(a,c)(f,g)(c,d)

Discuss it

Question 11

The number of distinct minimum spanning trees for the weighted graph below is

____ 

A 4

B 5

C 6

D 7

Discuss it

http://geeksquiz.com/algorithms/graph-traversals/

Graph Traversals

Question 1

Which of the following algorithms can be used to most efficiently determine the presence of a cycle in

a given graph ?

A Depth First Search

B Breadth First Search

C Prim's Minimum Spanning Tree Algorithm

D Kruskal' Minimum Spanning Tree Algorithm

Discuss it

Question 2

Traversal of a graph is different from tree because

AThere can be a loop in graph so we must maintain a visited flag for every vertex

BDFS of a graph uses stack, but inorrder traversal of a tree is recursive

CBFS of a graph uses queue, but a time efficient BFS of a tree is recursive.

D All of the above

Discuss it

Question 3

What are the appropriate data structures for following algorithms?1) Breadth First Search                           2) Depth First Search                            3) Prim's Minimum Spanning Tree 4) Kruskal' Minimum Spanning Tree

A

1) Stack2) Queue3) Priority Queue4) Union Find

B1) Queue2) Stack3) Priority Queue4) Union Find

C1) Stack2) Queue3) Union Find4) Priority Queue

D1) Priority Queue2) Queue3) Stack4) Union Find

Discuss it

Question 4

The Breadth First Search algorithm has been implemented using the queue data structure. One

possible order of visiting the nodes of the following graph is

A MNOPQR

B NQMPOR

C QMNPRO

D QMNPOR

Discuss it

Question 5

Let G be an undirected graph. Consider a depth-first traversal of G, and let T be the resulting depth-

first search tree. Let u be a vertex in G and let v be the first new (unvisited) vertex visited after visiting

u in the traversal. Which of the following statements is always true? (GATE CS 2000)

A {u,v} must be an edge in G, and u is a descendant of v in T

B {u,v} must be an edge in G, and v is a descendant of u in T

C If {u,v} is not an edge in G then u is a leaf in T

DIf {u,v} is not an edge in G then u and v must have the same parent in T

Discuss it

Question 6

Consider the following graph  

Among the following sequences 

I) a b e g h f 

II) a b f e h g 

III) a b f h g e 

IV) a f g h b e Which are depth first traversals of the above graph? (GATE CS 2003)

A I, II and IV only

B I and IV only

C II, III and IV only

D I, III and IV only

Discuss it

Question 7

Make is a utility that automatically builds executable programs and libraries from source code by

reading files called makefiles which specify how to derive the target program. Which of the following

standard graph algorithms is used by Make.

A Strongly Connected Components

B Topological Sorting

C Breadth First Search

D Dijkstra's Shortest Path

Discuss it

Question 8

Given two vertices in a graph s and t, which of the two traversals (BFS and DFS) can be used to find if

there is path from s to t?

A Only BFS

B Only DFS

C Both BFS and DFS

D Neither BFS nor DFS

Discuss it

Question 9

Which of the following condition is sufficient to detect cycle in a directed graph?

AThere is an edge from currently being visited node to an already visited node.

BThere is an edge from currently being visited node to an ancestor of currently visited node in DFS forest.

C Every node is seen twice in DFS.

D None of the bove

Discuss it

Question 10

Is following statement true/false If a DFS of a directed graph contains a back edge, any other DFS of

the same graph will also contain at least one back edge.

Source:http://courses.csail.mit.edu/6.006/oldquizzes/solutions/q2-s2009-sol.pdf

A True

B False

Discuss it

Question 11

Is following statement true/false? A DFS of a directed graph always produces the same number of

tree edges, i.e., independent of the order in which vertices are considered for DFS.

(Sourcehttp://courses.csail.mit.edu/6.006/oldquizzes/solutions/q2-f2008-sol.pdf)

A True

B False

Discuss it

Question 12

If the DFS finishing time f[u] > f[v] for two vertices u and v in a directed graph G, and u and v are in

the same DFS tree in the DFS forest, then u is an ancestor of v in the depth first tree.

(Source http://courses.csail.mit.edu/6.006/oldquizzes/solutions/q2-f2007-sol.pdf)

A True

B False

Discuss it

Question 13

Consider the DAG with Consider V = {1, 2, 3, 4, 5, 6}, shown below. Which of the following is NOT a

topological ordering? 

A 1 2 3 4 5 6

B 1 3 2 4 5 6

C 1 3 2 4 6 5

D 3 2 4 1 6 5

Discuss it

Question 14

Let G be a graph with n vertices and m edges. What is the tightest upper bound on the running time

on Depth First Search of G? Assume that the graph is represented using adjacency matrix.

A O(n)

B O(m+n)

C O(n2)

D O(mn)

Discuss it

Question 15

Consider the tree arcs of a BFS traversal from a source node W in an unweighted, connected,

undirected graph. The tree T formed by the tree arcs is a data structure for computing.

A the shortest path between every pair of vertices.

B the shortest path from W to every vertex in the graph.

Cthe shortest paths from W to only those nodes that are leaves of T.

D the longest path in the graph

Discuss it

Question 16

Suppose depth first search is executed on the graph below starting at some unknown vertex. Assume

that a recursive call to visit a vertex is made only after first checking that the vertex has not been

visited earlier. Then the maximum possible recursion depth (including the initial call) is _________.

A 17

B 18

C 19

D 20

Discuss it

Question 17

Let T be a depth first search tree in an undirected graph G. Vertices u and n are leaves of this tree T.

The degrees of both u and n in G are at least 2. which one of the following statements is true?

A There must exist a vertex w adjacent to both u and n in G

BThere must exist a vertex w whose removal disconnects u and n in G

C There must exist a cycle in G containing u and n

DThere must exist a cycle in G containing u and all its neighbours in G.

Discuss it

http://mcqquestion.blogspot.in/2012/08/data-structures-and-algorithm-analysis.html

1. If all c(i, j )’s and r(i, j)’s are calculated, then OBST algorithm in worst case takes one of the following time.

(a)  O(n log n)          (b)  O(n3)                 (c)  O(n2)                  (d)  O(log n)             (e)  O(n4).

2. The following is a weighted binary tree, then what is the weighted array for the TVS problem?

(a)  [9, 2, 7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 4]  (b)  [9, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7, 4, 6]

(c)  [9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 6, 7, 4]  (d)  [9, 2, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 6, 4]

(e)  [9, 2, 0, 0, 0, 7, 0, 0, 0, 0, 6, 4, 0, 0].

3. What are the entries of the array TREE[ ] for the above weighted binary tree for the TVS problem.

(a)   [1, 2, 3, 4, 5, 6, 0, 0]

(b)   [1, 2, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 4, 5, 6]

(c)   [1, 2, 3, 0, 0, 0, 0, 4, 5, 6]

(d)   [1, 2, 3, 0, 0, 0, 0, 0, 4, 5, 6]

(e)   [1, 2, 3, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 5, 6]. 

4. For a 15-puzzle problem let the initial arrangement be the following one, then answer the questions 4 – 7 with the following arrangement.

10 13 15 7

9 1 4 14

12 8 6

11 2 5 3

What is the value of ‘x’ used to find the reachability of the solution?

(a)  1                       (b)  0                       (c)  8                       (d)  10                     (e)  13.

5. The values for the position(i), position(j) where i = 14 and j =11, are

(a)  2, 8                   (b)  8, 2                   (c)  8, 13                 (d)  2, 13                 (e)  13, 8.

6. The values for less(i), less(j) where i =5, j =7 are

(a)  0, 6                   (b)  6, 0                   (c)  2, 4                   (d)  2, 6                   (e)  1, 6.

7. What is the value to find the reachability of the problem with which you can say the solution is reachable or not?

(a)  71                     (b)  72                    (c)  73                     (d)  69                     (e)  68.

8. The upper bound on the time complexity of the nondeterministic sorting algorithm is

(a)  O(n)                  (b)  O(n log n)          (c)  O(1)                  (d)  O( log n)           (e)  O(n2).

9. The worst case time complexity of the nondeterministic dynamic knapsack algorithm is

(a)  O(n log n)          (b)  O( log n)           (c)  O(n2)                 (d)  O(n)                  (e)  O(1).

10. For the LCBB solution of knapsack problem with the data (p1–p4) = (10,10,12,18) and (w1–w4) = (2, 4, 6, 9) and m = 18, then the values of u(1) and ĉ(1) respectively are

(a)  -38, -44              (b)  -44, -38             (c)  -44, -32              (d)  -32, -44             (e)  -32, -38.

Set 1  Set 2  Set 3  Set 4  Set 5  Set 6  Set 7  Set 8  Set 9  Set 10  Set

11  Set 12  Set 13  Set 14  Set 15  Set 16  Set 17  Set 18  Set 19  Set

20  Set 21  Set 22  Set 23  Set 24  Set 25  Set 26  

Set 27  Set 28  Set 29  Set 30 

Answers

1. Answer : (b)

Reason:  As we have to calculate c(i,j), w(i,j) and r(i,j).

2. Answer : (d)

Reason:  If we create a full binary tree then the non existing nodes will have the corresponding edge values to zero.

3. Answer : (e)

Reason:  If we create a full binary tree then the non existing nodes will have the corresponding node values to zero.

4. Answer : (a)

Reason:  ‘x’ takes the value 0 if it is there in shaded position which starts from second square and goes across every alternative square in the initial arrangement, otherwise it takes one. In this question it is there in shaded square.

5. Answer : (c)

Reason:  position( x ) = the square number with in which value ‘x’ is there.

6. Answer : (e)

Reason:  less( x ) is  the number of all js, such that  j < x and  position( j ) > position(x).

7. Answer : (b)

Reason:  The value to define reachable is  “sum of less(i) + x”, where i takes the values from 1 to 15, and ‘x’ is value defined in question(4).

8. Answer : (a)

Reason:  In the algorithm there is only one for loop which runs from 1 to n.

9. Answer : (d)

Reason:  In the algorithm there is only one for loop which runs from 1 to n.

10. Answer : (d)

Reason:  u(1) is the negation of sum of all profits so long as the weights as whole can be taken [ -(10+10+12)]. ĉ(1) is the value of u(1) and the negation  of profit of  the fraction of the next element taken to fill the knapsack [ u(1) + -( 6/9*18) ].

DATA STRUCTURE AND ALGORITHM ANALYSIS SET 2

Data Structure and Algorithm AnalysisQuestions 11 To 20

11. If for the above problem, the tree is created following the FIFOBB, then the values for ĉ (5) and u(5) respectively are

(a)  -40, -40              (b)  -40, -38             (c)  -40, -32              (d)  -22, -32             (e)  -32, -22.

12. There is a chain of 20 stores; each of the store having 5 departments. Then, which of the following is a correct representation of array size?

(a)  5 * 20                (b)  6 * 21                (c)  20 * 5                (d)  21 * 6                (e)  19 * 4.

13. From the following, select the valid data structure which reflects hierarchical relationship between the elements.

(a)  Graph                (b)  Queue               (c)  Linked list          (d)  Stack                (e)  Tree.

14. Which of the following matrix does have high proportions of zero entries as the elements?

(a)  Inverse Matrix                                  (b)  Sparse Matrix                 (c)  Determinant Matrix

(d)  Square Matrix                                  (e)  Transpose Matrix.

15. Let there is a queue whose initial values for Front and Rear are 0 and 1 respectively. Later queue found to be having the elements 3, 2, 1, 5. If a new element is inserted in the queue, then which of the following statements is correct?

(a)   Front = 0 and Rear = 5

(b)   Front = 0 and Rear = 4

(c)   Front = 1 and Rear = 4

(d)   Front = 1 and Rear = 5

(e)   Insertion is not possible as there is no space in the queue.

16. One from the following is also known as Pre-Order traversal. What that one is?

(a)  LNR                  (b)  LRN                  (c)  RLN                  (d)  RNL                  (e)  NLR.

17. Which of the following data structure is the best example for the multitasking systems?

(a)  Tree                  (b)  Queue               (c)  Stack                (d)  Linked List        (e)  Graph.

18. The initial configuration of the queue is having the elements ‘x, y, z, a’ forming a queue. To get the new configuration ‘a, z, y, x’ one needs a minimum of

(a)  3 additions and 4 deletion                (b)  3 deletions and 4 additions

(c)  3 deletions and 2 additions               (d)  3 deletions and 3 additions

(e)  4 deletions and 4 additions.

19. To form the new configuration in the above problem, what best data structure is seems to be used?

(a)  Tree                  (b)  Queue               (c)  Linked List         (d)  Circular Queue   (e)  Stack.

20. Recursive algorithms are based on

(a)  Divide and conquer approach           (b)  Top-down approach

(c)  Bottom-up approach                        (d)  Hierarchical approach

(e)  Heuristic approach.

Answers

11. Answer : (a)

Reason:  Node 5 is a node generated if first element is taken and the second object is rejected. And just calculate as question(10).

12. Answer : (c)

Reason:  Its  array [20][5] and hence 20 * 5.

13. Answer : (e)

Reason:  Tree, according to the organization of tree a node may have a child and a parent also.

14. Answer : (b)

Reason:  As sparse matrix is a matrix where the number of zeros are much than that of non zero elements.

15. Answer : (a)

Reason:  As a new element is inserted at a position pointed by Rear, And Rear is incremented after insertion.

16. Answer : (e)

Reason:  As in Pre-Order the traverse is as Node( root node), then Left node and Right node.

17. Answer : (b)

Reason:  Because a queue is used to hold the jobs which are waiting for their turn to work with the common processor.

18. Answer : (d)

Reason:  Because, first we have to delete ‘x’, ‘y’, ‘z’ and then insert ‘z’, ‘y’, ‘x’. And hence three deletions first followed by three additions.

19. Answer : (e)

Reason:  As in the above discussion ‘x’ is deleted first but inserted at last. Hence there should be a stack to hold the deleted entries from above queue.

20. Answer : (b)

Reason:  As in case of recursions the ARIs are created, and the recursive  function is called by other function first, then it calls itself util a condition is met.

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DATA STRUCTURE AND ALGORITHM ANALYSIS SET 3

Data Structure and Algorithm AnalysisQuestions 21 To 30

21. What do you call the selected keys in the quick sort method?

(a)  Outer key (b)  Inner Key       (c)  Partition key

(d)  Pivot key (e)  Recombine key.

22. Primary clustering occurs in

(a)  Linear probing         (b)  Quadratic probing

(c)  Mid-square method  (d)  Order probing                                    (e)  Chaining.

23. How many nodes do a full binary tree with ‘N’ leaves contain?

(a)  2N nodes        (b)  N nodes (c)  2N-1 nodes

(d)  N-1 nodes       (e)  2(N-1) nodes.

24. How many nodes does a complete binary tree of level 5 have?

(a)  16     (b)  15     (c)  32           (d)  31   (e)  64.

25. How do you determine the cost of a spanning tree?

(a)   By the sum of the costs of the edges of the tree

(b)   By the sum of the costs of the edges and vertices of the tree

(c)   By the sum of the costs of the vertices of the tree

(d)   By the sum of the costs of the edges of the graph

(e)   By the sum of the costs of the edges and vertices of the graph.

26. What would be the depth of tree whose level is 9?

(a)  10     (b)  8      (c)  9            (d)  11   (e)  7.

27. A node of a directed graph G having no out-degree and a positive in-degree is called

(a)  Source node    (b)  Sink node           (c)  Sibling node

(d)  Null node (e)  In-node.

28. The time complexity of the normal quick sort, randomized quick sort algorithms in the worst case is

(a)  O(n2), O(n log n)                  (b)  O(n2), O(n2)

(c)  O(n log n), O(n2)                  (d)  O(n log n), O(n log n)

(e)  O(n log n), O(n2 log n).

29. Let there be an array of length ‘N’, and the selection sort algorithm is used to sort it, how many times a swap function is called to complete the execution?

(a)  N log N times          (b)  log N times (c)  N2 times

(d)  N-1 times        (e)  N times. 

30. The Sorting method which is used for external sort is

(a)  Bubble sort     (b)  Quick sort                       (c)  Merge sort

(d)  Radix sort       (e)  Selection sort.

Answers

21. Answer : (d)

Reason:  As in quick sort the division is made at an element which is pivotal element.

22. Answer : (a)

Reason:  As there is no chance for the other probing techniques to have primary clustering.

23. Answer : (c)

Reason:  According to full binary tree, if there are 2n-1 nodes, then there would be n leaves exactly.

24. Answer : (d)

Reason:  This is 25 – 1. (2level – 1)

25. Answer : (a)

Reason:  As the other options are wrong, and the cost of a spanning tree is the cost of edges of the tree only.

26. Answer : (c)

Reason:  As both the depth and level are same.

27. Answer : (a)

Reason:  Because the source vertex is a vertex which contains only leaving edges, but no any incoming edge.

28. Answer : (b)

Reason:  Because, there is no difference between normal quick sort and randomized quick sort

algorithms for their efficiency, except the picking of a pivotal element.

29. Answer : (d)

Reason:  Because, every time we swap the ith smallest element with the ith position. And we do this for i=1 to n-1 times.

30. Answer : (c)

Reason:  As Merge sort is the only one from the given sorting techniques which is used for the external sorting.

DATA STRUCTURE AND ALGORITHM ANALYSIS SET 4

Data Structure and Algorithm AnalysisQuestions 31 To 40

31. In analysis of algorithm, approximate relationship between the size of the job and the amount of work required to do is expressed by using _________

(a)   Central tendency                                                            

(b) Differential equation

(c)   Order of execution                          (d) Order of magnitude

(e)   Order of Storage.

32. Pick the correct prefix form to the given infix expression   

               {a*[b/(c-d)*f]/g}/[e+h]

(a)   //*a/b*-cdfg+ch (b)  abcd-f*/g/*eh+/            

(c)   //*a*/b-cdfg+eh (d) //*ab*/-cdfg+eh                                

(e)   -//*a*/bcdfg+eh.

33. For recursive MinMax algorithm, the average, worst cases are __________

(a)                                     (b) 

(c)                                      (d) 

(e)  

34. P, Q and R are pointer variables. The statements below are intended to swap the contents of the nodes pointed to by P and Q. rewrite it so that it will work as intended.P = Q;    R = Q;   Q = R;

(a)   R=Q;  P=R;   Q=R;                          (b)   R=P;   P=P;   Q=Q;

(c)   P=P;   P=Q;   R=Q;                         (d)   R=P;   P=Q;   Q=R;

(e)   P=R;   R=Q;   Q=R;

35. For defining the best time complexity, let f (n) = log n and g (n) = √n, _________

(a)   f (n) Ω(g(n)), but g(n)  Ω (f(n))               (b) f (n) Ω(g(n)), but g(n)  Ω (f(n))

(c)   f (n) Ω(g(n)), and g(n)  Ω (f(n))              (d) f (n) Ω(g(n)), and g(n)  Ω (f(n))

(e)   Can’t be defined for best case.

36. If f, r are the front, rear pointers respectively in a circular queue then the condition    (( f – r + size) % size == 1) && ( (f != 0) || (r != -1) ) represents

(a)  Queue has only one element                                          (b)  Two Elements   

(c)  More than 2 elements but not full                                    (d)  Impossible values

(e)  Queue is full.

37. For the following two statements, choose the correct answer.

I.     n2O(n3).

II.     n = θ ((n+1)!).

(a)   (I) and (II) are TRUE

(b)   (I), (II) are FALSE

(c)   (I) is FALSE, but (II) is TRUE

(d)   (I) is TRUE and (II) is FALSE

(e)   (I) is TRUE but (II) can’t be defined.

38. The statement head->Link->Link->Link = NULL terminates a linked list after its __________ node.

(a)  2nd                     (b)  4th                     (c)  5th                     (d)  3rd                    

(e) first.

39. For analyzing an algorithm, which is better computing time?

(a)   O (100 Log N)                                 (b) O (N)

(c)   O (2N)                                             (d) O (N log N)                                       

(e)  O (N2).

40. To implement the Round Robin algorithm, which of the following data structure is used?

(a)   Stack                                             (b)   Linear Queue

(c)   Circular Queue                                (d)   Priority Queue 

(e)   Double Stack.

Answers

31. Answer : (d)

Reason : The order of the magnitude is the approximate relationship between the size of the job and the amount of work required to do. All the other options are not related to the asked question.

32. Answer : (c)

Reason : According to the priority of the operators ( as first inner round brackets, then the inner square brackets and hence so on.

33. Answer : (a)

Reason : According to the time analysis of the algorithm

34. Answer : (d)

Reason : According to the pointer manipulations of the variables

35. Answer : (b)

Reason : According to the limit rule for the omega notation ( Using the duality rule )

36. Answer : (e)

Reason : This is the one of the conditions to define the circular queue is full. All other are invalid choices.

37. Answer : (d)

Reason : According to the limit rule for the big Oh notation and the theta notation

38. Answer : (d)

Reason : As the are only three nodes, and the list is pointed by head.

39. Answer : (a)

Reason : As this is a very small value when compared with all the other values.

40. Answer : (c)

Reason : According to the concepts of OS at process scheduling, circular queue is the best suitable one when compared with all the remaining one.

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DATA STRUCTURE AND ALGORITHM ANALYSIS SET 5

Data Structure and Algorithm AnalysisQuestions 41 To 50

41. Let f, t: N→R  0, and t (n) O (f (n)) iff  t(n) ≤  c.f (n) where c is positive real constant and n ≥  no, then no is ___________

(a)   Upper bound                                   (b) Lower bound

(c)   Duality value                                   (d) Threshold value                      (e) Maximum value.

42. In a circular queue, we can disambiguate empty from full queue by

(a)   Using a gap in the array

(b)   Incrementing queue positions by 2 instead of 1

(c)   Keeping a count of the number of elements

(d)   (a) and (c) above

(e)   (a), (b), and (c) above.

43. In a graph, a vertex with degree one is known as ___________

(a)   Pendant vertex  (b) Leaf

(c)   Root                                               (d) End vertex                                        (e) Internal.

44. Consider the usual algorithm for determining whether a sequence of parentheses is balanced. What is the maximum number of parentheses that will appear on the stack AT ANY ONE TIME when the algorithm analyzes: (()

(())(()))

(a) 1                        (b) 2                        (c) 3                        (d) 4                  (e) 5 or more.

45. If a graph G = (N, E), where E= {}, then the corresponding adjacency matrix is _____

(a)   Unit Matrix

(b)   Identity Matrix

(c)   Having Diagonal elements Zero

(d)   Matrix with all 1’s

(e)     Zero Matrix.

46. Suppose we have an array implementation of the stack class, with ten items in the stack stored at data[0] through data[9]. The SIZE is 42. Where does the push method place the new entry in the array?

(a) data[0]               (b) data[1]               (c) data[9]               (d) data[10]             (e) data[11].

47. Breadth first search __________

(a)   Scans each incident node along with its children.

(b)   Scans all incident edges before moving to other node.

(c)   Is same as backtracking

(d)   Scans all the nodes in random order.

(e)   Scans all the nodes in pre-order manner.

48. Provided the space is available, then to insert an element in the queue, we can use for the following structure

       struct queue

       {

              int Q[20];

              int f, r;

       }Q;

(a)   ++Q.Q[Q.r] = x;                                                             (b)   Q.Q[++Q.r] = x;             (c)   Q.Q[Q.r]++ = x;

(d)   Syntax error                                    (e)   Q.Q[Q.r++] = x;

49. Which method of traversal does not use stack to hold nodes that are waiting to be processed?

(a)   Dept First                                       (b)  D-search           (c) Breadth first

(d)   Back-tracking                                  (e)  Bounding.

50. If the characters 'D', 'C', 'B', 'A' are placed in a queue (in that order), and then

removed one at a time, in what order will they be removed?

(a)   ABCD               (b) ABDC                (c) DCAB                (d) DCBA (e) ACDB.

Answers

41. Answer : (d)

Reason : According to the mathematical definition of the big Oh notation, this is the Threshold value defines the upper limit on the size of the instance.

42. Answer : (d)

Reason : The option B is not the suitable one and not correct also, while the other two(a, c ) can be used

43. Answer : (a)

Reason : According to the definition of the graph, and the degrees of the vertex.

44. Answer : (c)

Reason : Use stack and increment counter by one if opening bracket is encountered, and decrement it by one if a closing bracket is encountered.

45. Answer : (e)

Reason : As E = empty set, means there is no any edge between any two set of vertices. And hence it is zero matrix.

46. Answer : (d)

Reason : As it is Stack always the items are pushed at the top of the stack if there is any space. Also data[10] is the next space as data[0] to data[9] are already filled.

47. Answer : (b)

Reason : As BFS always make use of the Queue.

48. Answer : (b)

Reason : Because first we have to increment the rear pointer and at that place we can insert an element.

49. Answer : (c)

Reason : As D search, DFS uses stack, so BFS only uses queue and not the stack.

50. Answer : (d)

Reason : As Queue follows FIFO structure.

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DATA STRUCTURE AND ALGORITHM ANALYSIS SET 6

Data Structure and Algorithm AnalysisQuestions 51 To 60

51. The Knapsack problem where the objective function is to minimize the profit is ______

(a)   Greedy           (b)  Dynamic 0 / 1                 (c)  Back tracking

(d)   Branch & Bound 0/1           (e)  NP Knapsack.

52. I have implemented the queue with a linked list, keeping track of a front node and a rear node with two reference variables. Which of these reference variables will change during an insertion into a NONEMPTY queue?

(a) Neither changes        (b) Only front changes            (c) Only rear changes

(d) Both change            (e) (a) or (d) above.

53. For the given 5 jobs with penalties 6,3,4,8,5 respectively, the static state space tree is generated, then if the first job is selected but not the second job for the first two jobs consideration, then the rank of such situational node is ______

(a)   6      (b)  3      (c)  0            (d)  9    (e) 20.

54. Which of the following statements is/are TRUE?

        i)     ADTs define how data is stored and manipulated.  

       ii)     Every linked list has to have at least one external pointer.

       iii)    Recursive solutions may be easier to understand.

(a)   (i),(ii) and (iii) above            (b)  Only (i) above                    

(c)  Only (ii) above         (d)  Only (iii)above        

(e)  Only (ii) and (iii) above.

55. Choose the correct answer for the following statements:

I.     The theory of NP–completeness provides a method of obtaining a polynomial time for NPalgorithms.

II.     All NP-complete problem are NP-Hard.

(a)   I is FALSE and II is TRUE

(b)   I is TRUE and II is FALSE

(c)   Both are TRUE

(d)   Both are FALSE

(e)   II is TRUE, I can’t be determined.

56. Having the address of the node to be deleted in doubly linked list, the node can be deleted

(a)   Without traversing the list   

(b)  Only after traversing the list from the head

(c)   Only after traversing the list from the tail     

(d)   (b) or (c) above     

(e)   Can’t be deleted.

57. The Hamiltonian cycles problem uses the following line of code to generate a next vertex, provided x[ ] is a global array and kth vertex is under consideration:

(a)   x[k]  (x[k] + 1) mod n                   (b) x[k]  (x[k]) mod (n)

(c)   x[k]  (x[k] + 1) mod (n+1)             (d) x[k]  x[k+1] mod n

(e)   x[k]  x[k+1] mod (n + 1)

58. Suppose cursor refers to a node in a linked list, what statement changes cursor so that it refers to the next node?

(a)   cursor++;

(b)   cursor = link;

(c)   cursor += link;

(d)   cursor = cursor->link;

(e)   (a) and (c) above.

(Note:

 struct cursor {

                     int data;

                     struct cursor *link;

                  };

59. The graph coloring algorithm’s time can be bounded by _________

(a)   O(mnm)    (b) O(nm)  (c)  O(nm. 2n)      (d)  O(mn.2n)      (e) O(nmn).

60. For 0/1 KNAPSACK problem, the algorithm takes ________ amount of time for memory table, and ______time to determine the optimal load, for N objects and W as the capacity of KNAPSACK.

(a)   O(N+W), O(NW)      (b)  (NW), O(N+W)

(c)   O(N), O(NW)           (d)  O(NW), O(N)                            (e) O(N+W), O(N+W).

Answers

51. Answer : (d)

Reason : As there for the generation of the state space tree we use minimization Of the profit.

52. Answer : (c)

Reason : As the queue is non empty and there is rear pointer pointing, this only must be incremented.

53. Answer : (b)

Reason : As we have considered first two jobs and have not considered the second job so we have to pay the penalty of second job.

54. Answer : (c)

Reason : As the other two options are not true and correct.

55. Answer : (a)

Reason : Because the theory of NP does not provides a way to obtain a polynomial time for NP algorithms. Hence the I statement is wrong while the II statement is correct.

56. Answer : (a)

Reason : That is the advantage of doubly linked list. No need to traverse if the address of the desired node is provided.

57. Answer : (c)

Reason : As to find the next node to be traversed this is the formula.

58. Answer : (d)

Reason : All the other options are invalid operations.

59. Answer : (e)

Reason : As all the n nodes are required to be colored, and there are mn ways of coloring a specific node.

60. Answer : (b)

Reason : As the memory table ( 2 –D matrix ) contains N rows and W columns, hence it is NW, and for the composition of the optimal load it is up to N+W time it takes.

61. List out the areas in which data structures are applied extensively.

(a) Compiler Design

(b) Operating System

(c) Database Management System

(d) Both (a) and (b) above

(e) (a), (b) and (c) above.

62. What is the major data structure used in the Network data model?

(a) Stack

(b) Array

(c) Tree

(d) Graph

(e) Queue.

63. The data structure used in the Hierarchical data model is

(a) Array

(b) Tree

(c) Graph

(d) Stack

(e) Queue.

64. Minimum number of queue(s) needed to implement the priority queue is(are)

(a) Two

(b) One

(c) Three

(d) Four

(e) Depends upon the application.

65. What data structure is used to perform recursion?

(a) Queue

(b) Linked List

(c) Stack

(d) Double Linked List

(e) Circular Queue.

66. For the expression ((A + B) * C – (D – E) ^ (F + G)), the equivalent Postfix notation is

(a) AB + C * DE - - ^ FG +

(b) AB + CDE - - * FG + ^

(c) AB + C * DE - FG + - ^

(d) AB + C * DE - - FG + ^

(e) AB + C - DE - * FG + ^.

67. The Prefix notation for the above infix expression is

(a) - * +ABC - DE ^ + FG

(b) ^ - * +ABC - DE + FG

(c) ^ - +AB * C - DE + FG

(d) ^ - + * ABC - DE + FG

(e) - ^ * + ABC - DE + FG.

68. Sorting is not possible by using which of the following methods?

(a) Insertion

(b) Selection

(c) Deletion

(d) Exchange

(e) Partitioning.

69. The number of null branches for a binary tree with 20 nodes is

(a) 21

(b) 20

(c) 22

(d) 19

(e) 18.

70. What is the type of the algorithm used in solving the 8 Queens problem?

(a) Greedy

(b) Dynamic

(c) Branch and Bound

(d) Divide and Conquer

(e) Backtracking.

Answers

61. Answer : (e)

Reason : As the data structure can be used in all the application areas right from the compiler design to operating systems and even in database management system design.

62. Answer : (d)

Reason : As the graph is the suitable data structure to represent the connectivity between the nodes as edges.

63. Answer : (b)

Reason : As tree data structure is used to represent the Hierarchical relationship between the nodes as parent and children.

64. Answer : (a)

Reason : Two : One queue is used for actual storing of data and another for storing priorities.

65. Answer : (c)

Reason : Because of its LIFO (Last In First Out) property it remembers its ‘caller’ so knows whom to return when the function has to return. Recursion makes use of system stack for storing the return addresses of the function calls.

66. Answer : (d)

Reason : As per the precedence and associativity of the operators, brackets are evaluated first and then multiplication operators followed by addition, subtraction operators.

67. Answer : (b)

Reason : As per the precedence and associativity of the operators, brackets are evaluated first and then multiplication operators followed by addition, subtraction operators.

68. Answer : (c)

Reason : Using insertion we can perform insertion sort, using selection we can perform selection sort, using exchange we can perform the bubble sort (and other similar sorting methods) and using the portioning we can perform merge sort(and other similar sorting methods) But no sorting method can be done just using deletion.

69. Answer : (a)

Reason : A binary tree with n nodes has exactly n+1 null nodes.

70. Answer : (e)

Reason : As the other types of algorithms are not suitable to find the solution for the

given problem.

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1 comment :

1.

KalyanDecember 14, 2013 at 3:24 AM

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Data Structure and Algorithm AnalysisQuestions 91 To 100

DATA STRUCTURE AND ALGORITHM ANALYSIS SET 8

Data Structure and Algorithm AnalysisQuestions 71 To 80

71. The Following is a binary tree. Answer questions (11 – 13) considering the below given tree.

The In-order traversal of the above tree is

(a) H D E B F I J G C A

(b) D H B E A F C I G J

(c) A B D H E C F G I J

(d) H D B E A F C I G J

(e) D H B E A I G J F C.

72. The Pre-order traversal for the tree in question -11 is

(a) A B D H E C F G J I

(b) A B D E H C F G I J

(c) D H B E A F C I G J

(d) A B D H E C F G I J

(e) H D E B F I J G C A.

73. The Post-order traversal for the tree in question -11 is

(a) H D E B I J G F C A

(b) D H B E A F C I G J

(c) H D E B F I J G C A

(d) H D E B F I J G A C

(e) A B D H E C F G I J.

74. There are four trees named A, B, C and D having 8, 15, 13, 14 nodes in them respectively. Which of them could have formed a full binary tree?

(a) A

(b) B

(c) C

(d) D

(e) No any tree can be a full binary tree.

75. In the given binary tree if the nodes are stored in an array, then where the node 4 can be stored?

(a) At location 6

(b) At location 5

(c) At location 4

(d) At location 3

(e) Can’t be stored at any location.

76. Of the following tree structure, which is efficient, considering space and time complexities?

(a) Incomplete Binary Tree

(b) Complete Binary Tree

(c) Full Binary Tree

(d) Binary Search Tree

(e) AVL Tree.

77. Consider the following two statements and choose the correct option:

I.     According to Access strategies Linked List is a linear one.

II.     According to Storage Linked List is a Non-linear one.

(a) (I) is true but (II) is false

(b) (I) is false but (II) is true

(c) Both (I) and (II) are true

(d) Both (I) and (II) are false

(e) Can’t be determined.

78. Which of the following are differences between structures and arrays?

(a) Structures and arrays use different operators to access their elements

(b) Structures are not limited to containing variables of a single data type

(c) Structures can contain elements that are pointers but arrays not

(d) Structures are passed by value while arrays are passed by reference

(e) Structures and arrays contains the different elements.

79. Which of the following pairs of statements are identical?

(a) (*ptr) element AND ptr.element

(b) *ptr.element AND ptrelement

(c) *ptrelement AND ptr.element

(d) *(ptr.element) AND ptrelement

(e) (*ptr).element AND ptrelement.

80. A "stack" is also known as what?

(a) A FIFO

(b) A Queue

(c) A LIFO

(d) A Linked List

(e) A FOLI.

Answers

71. Answer : (b)

Reason : Because the rules for traversing the binary tree in In-order are:

Traverse the left sub tree in In-order

Traverse the root

     iii.    Traverse the right sub tree in In-order

72. Answer : (d)

Reason : Because the rules for traversing the binary tree in Pre-order are:

Traverse the root

Traverse the left sub tree in Pre-order

     iii.    Traverse the right sub tree in Pre-order

73. Answer : (c)

Reason : Because the rules for traversing the binary tree in Post-order are:

Traverse the left sub tree in Post-order

Traverse the right sub tree in Post-order

     iii.    Traverse the root

74. Answer : (b)

Reason : In general there are 2n-1 nodes in a full binary tree.

     By the method of elimination: Full binary trees contain odd number of nodes. So there cannot be full binary trees with 8 or 14 nodes, so rejected. With 13 nodes you can form a complete binary tree but not a full binary tree. So the correct answer is 15.

75. Answer : (a)

Reason : At location  6

1 2 3 - - 4 - - 5

Root

1

LC1

2

RC1

3

LC2

4

RC2

5

LC3

6

RC3

7

LC4

8

RC4

9

     where LCn means Left Child of node ‘n’ and RCn means Right Child  of node ‘n’. Top row represents the node values and bottom row represents the positions.

76. Answer : (b)

Reason : Complete Binary Tree.

     By the method of elimination: Full binary tree loses its nature when operations of insertions and deletions are done. For incomplete binary trees, extra storage is required and overhead of NULL node checking takes place. So complete binary tree is the better one since the property of complete binary tree is maintained even after operations like additions and deletions are done on it

77. Answer : (c)

Reason : As the linked list nodes do not have any index and hence they must be accessed in linear order always. Also they are stored in discrete locations and not in contiguous memory locations.

78. Answer : (d)

Reason : As this is the most suitable option when compared to the other options. Also option c, e are wrong.

79. Answer : (e)

Reason : As both the operators [ and (*). ] are used to access the members of the structure through pointers. All the other options have the syntax errors.

80. Answer : (c)

Reason : As the Stack behaves in Last In First Out manner.

Answers:

71. Answer : (b)

Reason : Because the rules for traversing the binary tree in In-order are:

Traverse the left sub tree in In-order

Traverse the root

     iii.    Traverse the right sub tree in In-order

72. Answer : (d)

Reason : Because the rules for traversing the binary tree in Pre-order are:

Traverse the root

Traverse the left sub tree in Pre-order

     iii.    Traverse the right sub tree in Pre-order

73. Answer : (c)

Reason : Because the rules for traversing the binary tree in Post-order are:

Traverse the left sub tree in Post-order

Traverse the right sub tree in Post-order

     iii.    Traverse the root

74. Answer : (b)

Reason : In general there are 2n-1 nodes in a full binary tree.

     By the method of elimination: Full binary trees contain odd number of nodes. So there cannot be full binary trees with 8 or 14 nodes, so rejected. With 13 nodes you can form a complete binary tree but not a full binary tree. So the correct answer is 15.

75. Answer : (a)

Reason : At location  6

1 2 3 - - 4 - - 5

Root

1

LC1

2

RC1

3

LC2

4

RC2

5

LC3

6

RC3

7

LC4

8

RC4

9

     where LCn means Left Child of node ‘n’ and RCn means Right Child  of node ‘n’. Top row represents the node values and bottom row represents the positions.

76. Answer : (b)

Reason : Complete Binary Tree.

     By the method of elimination: Full binary tree loses its nature when operations of insertions and deletions are done. For incomplete binary trees, extra storage is required and overhead of NULL node checking takes place. So complete binary tree is the better one since the property of complete binary tree is maintained even after operations like additions and deletions are done on it

77. Answer : (c)

Reason : As the linked list nodes do not have any index and hence they must be accessed in linear order always. Also they are stored in discrete locations and not in contiguous memory locations.

78. Answer : (d)

Reason : As this is the most suitable option when compared to the other options. Also option c, e are wrong.

79. Answer : (e)

Reason : As both the operators [ and (*). ] are used to access the members of the structure through pointers. All the other options have the syntax errors.

80. Answer : (c)

Reason : As the Stack behaves in Last In First Out manner.

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DATA STRUCTURE AND ALGORITHM ANALYSIS SET 9

Data Structure and Algorithm AnalysisQuestions 81 To 90

81. A "queue" is also known as what?

(a) A FIFO

(b) A stack

(c) A LIFO

(d) A Linked List

(e) A FOFI.

82. A "linked list" always contains elements that can be described as

(a) Redundant

(b) Recursive

(c) Sentinel

(d) Bidirectional

(e) Self-referential.

83. In tree construction, which is the suitable and efficient data structure?

(a) Array

(b) Linked list

(c) Stack

(d) Queue

(e) Graph.

84. The following are the statements regarding the NP problems. Chose the right option from the following options:

I.     All NP-complete problems are not NP-hard.

II.     Some NP-hard problems are not known to be NP-complete.

(a) Both (I) and (II) are true

(b) Both (I) and (II) are false

(c) Only (I) is true

(d) Only (II) is true

(e) (I) is true but (II) is not always true.

85. Let G be a graph with ‘n’ nodes and let ‘m’ be the chromatic number of the graph. Then the time taken by the backtracking algorithm to color it is

(a) O(nm)

(b) O(n+m)

(c) O(mnm)

(d) O(nm)

(e) O(nmn).

86. The time complexity of the shortest path algorithm can be bounded by

(a) O(n2)

(b) O(n4)

(c) O(n3)      

(d) O(n)

(e) O(n log n ).

87. Read the following statements carefully and pick the correct option:

I.     The worst time complexity of the Floyd’s algorithm is O(n3).

II.     The worst time complexity of the Warshall’s algorithm is O(n3).

(a) (I) is false but (II) is true

(b) (I) is true but (II) is false

(c) Both (I) and (II) are true

(d) (I) is true and (II) is not true always

(e) Both (I) and (II) are false.

88. The asymptotic notation for defining the average time complexity is

(a) Equivalence

(b) Symmetric

(c) Reflexive

(d) Transitive

(e) Both (c) and (d) above.

89. The suitable data structure to represent the IDs of employees is

(a) Stack

(b) Queue

(c) Linked List

(d) Array

(e) Tree.

90. The number of nodes a null tree can have is

(a) One

(b) Two

(c) Three

(d) Zero

(e) Null tree is invalid tree.

Answers

81. Answer : (a)

Reason : As the Stack behaves in Last In First Out manner.

82. Answer : (e)

Reason : There should be a self referential structure in linked list so as to point to the other node in the link list. The other options are wrong.

83. Answer : (b)

Reason : If a linked list is used then a node can hold the address of both the left child and right child. Stack, Queue and Graph can never be used. Arrays can also be used but careful indexing is required to be done and hence is not a valid data structure.

84. Answer : (d)

Reason : According to the theory of NP-complete and NP-hard problems.

85. Answer : (e)

Reason : As the number of internal nodes in the state space tree are mn, and O(mn) is the time spent by the NextValue algorithm to determine the children corresponding to legal colorings. Hence the total time is bounded by O(nmn).

86. Answer : (c)

Reason : As the algorithm contains three ‘for loops’ nested together running from 1 to ‘n’ and hence the total time is O(n3).

87. Answer : (c)

Reason : As the both the algorithms contains three ‘for loops’ nested together running from 1 to ‘n’ and hence the total time is O(n3) for both the algorithms.

88. Answer : (a)

Reason : This is a theta notation which is Reflexive, Symmetric and Transitive and hence it is an Equivalence relation.

89. Answer : (d)

Reason : As the type of all Id for all the employees is same then a simple array is enough to store the values.

90. Answer : (d)

Reason : As a null tree is a valid tree with zero number of nodes.

91. For the bubble sort algorithm, what is the time complexity of the best/worst case? (assume that the computation stops as soon as no more swaps in one pass)

(a) best case: O(n)      worst case: O(n*n)

(b) best case: O(n)      worst case: O(n*log(n))

(c) best case: O(n*log(n)) worst case: O(n*log(n))

(d) best case: O(n*log(n)) worst case: O(n*n)

(e) best case: O( 1)     worst case: O( n ).

92. For the quick sort algorithm, what is the time complexity of the best/worst case?

(a) best case: O(n)      worst case: O(n*n)

(b) best case: O(n)      worst case: O(n*log(n))

(c) best case: O(n*log(n)) worst case: O(n*log(n))

(d) best case: O(n*log(n)) worst case: O(n*n)

(e) best case: O(n*log(n)) worst case: O(n*n*n).

93. In an arbitrary tree ( not a search tree) of order M. Its size is N, and its height is K. The computation time needed to find a data item on T is

(a) O(K*K)

(b) O(M*M)

(c) O(N)

(d) O(K)

(e) O( 1 ).

94. When we organize our data as an ordered list, what is the time complexity of inserting/deleting a data item to/from the list?

(a) O(length_of_list*length_of_list)

(b) O(length_of_list)

(c) O(log(length_of_list * length_of_list))

(d) O(1)

(e) O( log(length_of_list)*length_of_list ).

95. Five statements about B-trees are below. Four of them are correct. Which one is INCORRECT?

(a) All B-trees are also search trees

(b) The word B-tree stands for balanced tree

(c) The word B-tree also stands for binary tree

(d) All leaves of a B-tree must be on the same level

(e) B-tree is not same as B+-tree.

96. For any B-tree of height H (H>1), after inserting a new key, is it possible for a key, K, which was located in a leaf-node to move up to the root in this regard which of the following is correct?

(a) Can’t be defined without data

(b) Never

(c) Yes, only if H=2

(d) Yes, only when the half of the keys in the root are less than K and the other half of the keys in the root are greater than K

(e) Yes.

97. When we say the order of a tree is M, we mean

(a) Every non-leaf node must have M subtrees

(b) Every non-leaf node must have M keys

(c) Every non-leaf node can have at most M subtrees

(d) Every non-leaf node can have at most M keys

(e) The Height of the tree is M.

98. T is a search tree of order M, its size is N, and its height is K. The computation time needed to INSERT/DELETE a data item on T is

(a) O( 1 )

(b) O( M )

(c) O( Log K )

(d) O( N )

(e) O( K ).

99. Suppose that we have a data file containing records of famous people, and we need to build a hash table to find a record from the person's birthday. The size of the hash table is 4096. The following are hash functions which convert a birthday to an integer. Which of the following function is the best?

(a)  h1( day/month/year ) = day + month + year

(b)  h2( day/month/year ) = day + month*31 + year

(c)  h3( day/month/year ) = (day + month*31 + year*365) mod 4096

(d)  h4( day/month/year ) = (day + month*31 + year*365) mod 4093

(e)  h5(day/month/year ) = (day + month*31 + year*365).

100.What is collision resolution with open addressing?

(a) When collision happens, we create a new memory location outside of the existing table, and use a chain to link to the new memory location

(b) When collision happens, we enlarge the hash table

(c) When collision happens, we look for an unoccupied memory location in the existing table

(d) We use an extra table to collect all collided data

(e) We use indexed hash table to resolve collision.

Answers

91. Answer : (a)

Reason:  Best Case is the one in which it is already sorted, and hence it takes O(n) time, otherwise it takes O(n2) time.

92. Answer : (d)

Reason:  The worst case here is, if the list is already sorted in descending order and we want to sort it in ascending order. In this case it takes O(n2) time. Otherwise it takes O( n * log(n) ) time.

93. Answer : (c)

Reason:  As the size is N, we have to traverse almost entire tree to look for a node ( because it is not a binary search tree). And hence it takes O(N) time.

94. Answer : (b)

Reason:  Because we have to search for deletion/insertion both in the ordered list. And the algorithm takes O(n) in this case[ best option in the given options].

95. Answer : (c)

Reason:  As B-tree is not a binary tree, but a balanced m-way tree

96. Answer : (e)

Reason:  Yes, this is the best option given because, if the insertion in a node causes the node to contain the keys equal to order of the tree, then the middle key of the node will be passed to one level up (it can be a root also ). A common mistake is to tick D. But, the condition stated in D is too strong. We were only asking whether a key can move up to the root, and we were not asking whether a key can become a new root!

97. Answer : (c)

Reason:  Here in the answer ‘can’ is most important. That is “every non-leaf node ‘can’ have at most M subtrees”

98. Answer : (e)

Reason:  In this case it is a search tree ( BST ). And hence the search time is proportional to the height of the tree. This answer assumes that M < K If M >> K, you could answer B However, M is a fixed number, but K grows when the size of data grows. Therefore, e is a better answer.

99. Answer : (d)

Reason:  With the answer ‘a’ some of the entries in the hash table, say 3000-4000, are never used. A similar problem also applies to ‘b’, ‘e’. In the answer ‘c’, 4096 is a bad choice. It should a prime number.

100. Answer : (c)

Reason:  The meaning of the open addressing it self is to look and probe for unoccupied and open location to place the incoming key.

DATA STRUCTURES AND ALGORITHM ANALYSIS SET 11

Data Structures and Algorithm AnalysisQuestions 101 To 110

101.What of the following cases is a so-called "collision"?

(a) A hash function produces the same address for two different keys:

            h( key1 ) = h( key2 ) where key1 =\= key2

(b) Two different hash functions produce the same address for a given key:

            h1( key ) = h2( key )

(c) Two different hash functions produce the same address for two different keys:

            h1( key1 ) = h2( key2 ) where key1 =\= key2

(d) A hash function produces the same address for two different keys with

different lengths: h( key1 ) = h( key2 ) where length(key1) =\= length (key2)

(e) A hash function produces the same address for two keys of the same length:

            h( key1 ) = h( key2 ) where length(key1) = length(key2).

102.Which of the following statements is INCORRECT when the differences/similarities between binary search trees and heaps?

(a) Both binary search trees and heaps are binary trees

(b) Both binary search trees and heaps require ascending order between sibling nodes

(c) With heaps the smallest value is stored at the root, while with binary trees the smallest value is on the leftmost leaf

(d) In heaps no gaps are allowed apart from the right side of the leaves' level, but in binary search trees gaps are allowed

(e) The order of root, left, right nodes are different for both the trees.

103.Find the odd one out from the following categories of algorithms.

(a) TVSP

(b) OBST

(c) N-Queens

(d) 15-Puzzle

(e) Bin-Packing.

104.For the ease of searching in a linked list, the linked list be created in

(a) Descending order

(b) Ascending order

(c) Without underflow condition

(d) Any order

(e) With underflow condition.

105.The time complexity of binary search in best, worst cases for an array of size N is

(a) N, N2

(b) 1, Log N

(c) Log N, N2

(d) 1, N log N

(e) N, N.

106.Folding is a method of generating

(a) A hash function

(b) Index function for a triangular matrix

(c) Linking the tail node to the head node in linked list

(d) Linear probing

(e) Chaining.

107.Which of following algorithm scans the list by swapping the entries whenever pair of adjacent keys are out of desired order?

(a) Insertion sort

(b) Quick sort

(c) Shell sort

(d) Bubble sort

(e) Radix sort.

108.What is the linked list that contains a variable whose value is the address of the location of the other node in the list?

(a) Integer

(b) Float

(c) Char

(d) Void

(e) Pointer.

109.The mathematical definition for Omega can be defined as, provided f,g:NR+ and c is a positive constant and n > n0,

(a) f(n) ≥ c. g(n)  n

(b) f(n) = c. g(n)  n

(c) f(n) ≥ c + g(n)  n

(d) f(n) = c + g(n)  n

(e) f(n) =  g(n)  n.

110.The  notation is

(a) Symmetric

(b) Reflexive

(c) Transitive

(d) Both (b) and (c) above

(e) (a), (b) and (c) above.

Answers

101. Answer : (a)

Reason:  Collision happens if the hash function produces the same address for two different keys in the prime area.

102. Answer : (b)

Reason:  As this statement is totally wrong as Binary search trees require ascending order between siblings node, but heaps don't.

103. Answer : (e)

Reason:  This one belongs to NP-Class category.

104. Answer : (d)

Reason:  What ever the order it may be, always we have to search for a node in the entire linked list, which is always organized in discrete locations. Hence no any particular order is needed.

105. Answer : (b)

Reason:  Best case is; if the element to search is there at mid of he list. Worst case is; if the element we are searching is either not there in the list or at the beginning of the list or end of the list.

106. Answer : (a)

Reason:  Folding is one of the methods in hashing functions. It can be shift folding or rounded folding.

107. Answer : (d)

Reason:  Bubble sort only is the algorithm from the given options which compares the adjacent keys.

108. Answer : (e)

Reason:  A self referential pointer variable is used to hold the address of the next node in the list.

109. Answer : (a)

Reason:  According to the mathematical definition of the Asymptotic notations.

110. Answer : (e)

Reason:  Because  notation is equivalence relationship in nature.

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DATA STRUCTURES AND ALGORITHM ANALYSIS SET 12

Data Structures and Algorithm AnalysisQuestions 111 To 120

111.Which of the following belongs to the algorithm paradigm?

(a) Minimum & Maximum problem

(b) Knapsack problem

(c) Selection problem

(d) Merge sort

(e) Quick sort.

112.If  f, t: N→ R+, then t (n)  Ω (f (n)), iff  f(n)  O (t (n)) is known as

(a) Limit rule

(b) Rule of inference

(c) Duality rule

(d) Rule of consequences

(e) Rule of symmetricity.

113.How many children do an external node of a binary tree of order N contains?

(a) N exactly

(b) N-1 exactly

(c) One exactly

(d) 0 exactly

(e) N/2 exactly.

114.To calculate c(i, j )’s, w( i, j)’s and r(i, j)’s; Howmuch time does the OBST algorithm in worst case takes?

(a) O(log n)

(b) O (n4)

(c) O (n3)

(d) O (n log n)

(e) O (n2).

115.Which of the following in breadth first search uses data structures as an auxiliary structure to hold nodes for future processing?

(a) Queue

(b) Linked list

(c) Graph

(d) B-Tree

(e) Stack.

116.The number of loop(s) of a node in a simple graph of ‘N’ nodes is

(a) One

(b) N

(c) Zero

(d) Exactly two

(e) N-1.

117.The time taken by NP-class sorting algorithm is

(a) O(1)

(b) O(log n)

(c) O(n2)

(d) O(n log n)

(e) O(n).

118.Pick the correct one from the following by reading carefully the below statements

I.     Floyd’s algorithm takes n3 among of time.

  Warshall’s algorithm takes n3 among of time.

(a) I is true but II is false

(b) I is false but II is true

(c) Both are false

(d) Both are true

(e) I is true and II takes n4 amount of time.

119.What defines the average case for the functions f, g : N –> R  for all n ,  

n –> f(n)/g(n) =

(a) f(n) ( g(n)), and f(n) ( g(n))

(b) f(n) ( g(n)), but f(n)  ( g(n))

(c) f(n) ( g(n)), but f(n) ( g(n))

(d) f(n) ( g(n)), but g(n)   ( f(n))

(e) f(n) ( g(n)), but f(n)  ( g(n)).

120.For the functions f, g : N –> R  for all n,   f(n) = n and g(n) = n log n then

(a) f(n) ( g(n)), but g(n) ( f(n))

(b) f(n) ( g(n)), but g(n) ( f(n))

(c) g(n) ( f(n)), but f(n) ( g(n))

(d) f(n) ( g(n)), and g(n) ( f(n))

(e) f(n) ( g(n)), but f(n) ( g(n)).

Answers

111. Answer : (b)

Reason:  Remaining all belongs to divide and conquer paradigm.

112. Answer : (c)

Reason:  According to the asymptotic notation rules. And this rule is used to apply the limit rule for the omega notated values.

113. Answer : (d)

Reason:  Because, the leaf nodes can’t have the children.

114. Answer : (c)

Reason:  Because, we have to calculate all c(i, j )’s, w( i, j)’s and r(i, j)’s for the tree of ‘n’ identifiers.

115. Answer : (a)

Reason:  As in BFS, all the adjacent nodes are traversed first, before traversing the descendent nodes. And hence a queue is used.

116. Answer : (c)

Reason:  Because, the simple graph should not contain any loop in the entire graph.

117. Answer : (e)

Reason:  Here only one loop from one to n is there. And hence it runs for ‘n’ times only.

118. Answer : (d)

Reason:  As there are three nested loops running from 1 to n in both of the algorithms. And hence it is O(n3)

119. Answer : (b)

Reason:  According to the limit rules for big Oh notation.

120. Answer : (a)

Reason:  According to the limit rules for big Oh notation.

121. Consider that a heap is implemented using a partially filled array called data, and the array contains n elements (n > 0), where is the entry with the greatest value?

(a) data[0]   (b) data[n-1]   (c) data[n]   (d) data[2*n + 1]   (e) data[2*n + 2].  122. Which of the following are used to represent the sparse matrices?

(a) Singly Linked List where each node contains non- zero elements of the matrix 

(b) Circular Linked List for each row, column with each node corresponding to non-zero element  (c) Doubly Linked List  (d) Can’t be represented using Linked List, but with arrays only  (e) Can be represented only by queues.  123. Which of the following has a desired key is searched, starting itself from hash address, sequentially in a table?

(a) Quadratic probing  (b) Random probing  (c) Reverse probing  (d) Chaining  (e) Linear probing.

 124. Consider A is an array of order m*n stored in a Row Major order. If the address of the first element in the array is K which is there at A[0, 0], then What is the address of A[i, j]?

(a) K + (i-1) * m + j – 1  (b) K + (i-1) * n + j – 1  (c) K + i * m + j  (d) K + (j-1) * m + j – 1  (e) K + j * n +  i.  125. Which of the following is used in conversion of an infix expression to a postfix expression?

(a) Circular list  (b) Array  (c) Stack  (d) Queue  (e) Dequeue.  126. What is the output of the following function when it is called?void disp( node *first ) if( first->next ){ {   printf("%4d", first->data);  disp( first->next ); }}

(a) Display the data of next node always in the list  (b) Display the data of all nodes of the list  (c) Does not execute  (d) Display the data of the first node only  (e) Display the data of the last node only.  127. With the “wrap around” implementation of a queue, which of the following code should be used to work out the next location of deletion?

(a) front++   (b) front--   (c) front = (front % max_queue_size) + 1   (d) front = (front ++) % max_queue_size   (e) front = 0.

 128. Consider the order of a tree is represented by M. Then, which of the following is true?

(a) Every non-leaf node must have M subtrees   (b) Every non-leaf node must have M keys   (c) Every non-leaf node can have atmost M subtrees   (d) Every non-leaf node can have atmost M keys   (e) The Height of the tree is M.  129. What are the time complexities of binary search for an array of size N in best and worst cases respectively?

(a) N, N2  (b) 1, Log N  (c) Log N, N2  (d) 1, N Log N  (e) N, N.  130. What is the algorithm, which scans the list by swapping the entries whenever pair of adjacent keys is out of desired order?

(a) Insertion sort  (b) Quick sort  (c) Shell sort  (d) Bubble sort  (e) Radix sort.

Answers

121. Answer : (a)

Reason:  if  a heap is implemented using a partially filled array called data, and the array contains n elements (n > 0), then the entry with the greatest value is  data[0] .

122. Answer : (a)

Reason:  To represent the sparse matrices, we have Singly Linked List where each node contains non- zero elements of the matrix

123. Answer : (e)

Reason:  Linear Probing has a desired key is searched, starting itself from hash address, sequentially in a table.

124. Answer : (b)

Reason:  If A is an array of order m*n stored in a Row Major order. If the address of the first element in the array is K which is there at A[0, 0], then the address of A[i, j] is  K + (i-1) * n + j – 1

125. Answer : (c)

Reason:  Stack is used in conversion of an infix expression to a postfix expression.

126. Answer : (b)

Reason:  if the  following function  is called.

void disp( node *first )

              if( first->next )

{

              {

                     printf("%4d", first->data);

               disp( first->next );

              }

}             then  it Display the data of all nodes of the list.

127. Answer : (d)

Reason:  With the ``wrap around'' implementation of a queue, front = (front ++) % max_queue_size should be used to work out the next location of deletion.

128. Answer : (c)

Reason:  if the order of a tree is represented by M. Then , Every non-leaf node can have at most M subtrees is true.

129. Answer : (b)

Reason:  1, Log N are the time complexities of binary search for an array of size N in best and worst cases respectively

130. Answer : (d)

Reason:  Bubble sort is the algorithm, which scans the list by swapping the entries whenever pair of adjacent keys is out of desired order.

DATA STRUCTURES AND ALGORITHM ANALYSIS SET 14

Data Structures and Algorithm AnalysisQuestions 131 To 140

131.The in-order traversal of some binary tree produced the sequence HFIEJGZ, and the post-order traversal of the same tree produced the sequence HIFJZGE. What will be the total number of nodes in the left sub tree of the given tree?

(a) 1

(b) 2

(c) 3

(d) 4

(e) 5.

132.Which of the following shows the difference between a queue and a stack?

(a) Queues require linked lists, but stacks do not

(b) Stacks require linked lists, but queues do not

(c) Queue is used for complex programs and stack for simple programs

(d) Stacks use two ends of the structure, queues use only one

(e) Queues use two ends of the structure; stacks use only one.

133.Which of the following is generated by Folding method?

(a) A hash function

(b) Index function for a triangular matrix

(c) Linking the tail node to the head node in linked list

(d) Linear probing

(e) Chaining.

134.From the following choose the one which belongs to the algorithm paradigm other than to which others from the following belongs to.

(a) Minimum & Maximum problem

(b) Knapsack problem

(c) Selection problem

(d) Merge sort

(e) Quick sort.

135.Which of the following data structures is used by breadth first search as an auxiliary structure to hold nodes for future processing?

(a) Queue

(b) Linked list

(c) Graph

(d) B-Tree

(e) Stack.

136.Pick the correct statement(s) from the following set of statements.

  In the Kruskal’s algorithm, for the construction of minimal spanning tree for a graph, the selected edges always form a forest.

   In Prim’s algorithm, for the construction of minimal spanning tree for a graph, the selected edges always form an orchard.

III.    DFS, BFS algorithms always make use of a queue, and stack respectively.

(a) Only (I) above

(b) Only (II) above

(c) Only (III) above

(d) Both (I) and (III) above

(e) Both (II) and (III) above.

137.Which of the following methods of collision processing has some of their addresses may remain unchecked?

(a) Linked collision processing

(b) Quadratic collision processing

(c) Linear collision processing

(d) Random collision processing

(e) Clustering collision processing.

138.Consider a tree ‘t’ has two subtrees t1 and t2. Identify the tree where the subtree t1 has minimum height and the subtree t2 has maximum height.

(a) Fibonacci      

(b) B+   

(c) Sparse   

(d) Complete binary   

(e) B.

139.Which of the following is used to select the first entry which has the free block equal to or more than required one in memory allocation?

(a) Best fit method

(b) Average fit method

(c) Worst fit method

(d) Big fit method

(e) First fit method.

140.What is the total number of nodes at every level (L) of a complete binary tree?

(a) 2L

(b) 2L–1

(c) 2L–1 

(d) 2L–1

(e) 2L.

Answers

131. Answer : (c)

Reason:  The in-order traversal of some binary tree produced the sequence HFIEJGZ, and the post-order traversal of the same tree produced the sequence HIFJZGE.  Then the total number of nodes in the left sub tree of the given tree is 3.

132. Answer : (e)

Reason:  Queues use two ends of the structure; stacks use only one show the difference between them.

133. Answer : (a)

Reason:  a hashing function is generated by Folding method.

134. Answer : (b)

Reason:  Knapsack problem is  the one which belongs to the algorithm paradigm other than to which others from the following belongs to

135. Answer : (a)

Reason:  Queue is the data structure used by breadth first search as an auxiliary structure to hold nodes for future processing.

136. Answer : (a)

Reason:  In the Kruskal’s algorithm, for the construction of minimal spanning tree for a graph, the selected edges always form a forest.

137. Answer : (b)

Reason:  Quadratic collision processing has some of their addresses may remain unchecked.

138. Answer : (a)

Reason:  Consider a tree‘t’ has two sub trees t1 and t2. Fibonacci is the tree where the sub tree t1 has minimum height and the sub tree t2 has maximum height.

139. Answer : (e)

Reason:  First fit method is used to select the first entry which has the free block equal to or more than required one in memory allocation.

140. Answer : (e)

Reason:  The total number of nodes at every level ( L ) of a complete binary tree is 2L.

141.

Which of the following statements is true related to a B-tree?

(a) All entries of a node are greater than or equal to the entries in the node's children

(b) All leaves are at the different depths

(c) All the leaf nodes appear at same level

(d) All non-leaf nodes have the exact same number of children

(e) All nodes contain the exact same number of entries.

142.

A simple graph has no loops. Which of the following forms another property of a simple graph?

(a) Contain un-weighted edges

(b) Undirected

(c) Have at least one vertex

(d) Have no multiple edges

(e) Must be directed.

143.

Which of the following information is not saved in the activation record, when ever a function call is executed?

(a) Static and dynamic address links

(b) Current depth of recursion

(c) Formal parameters

(d) Location where the function should return when done

(e) Local variables.

144.

Use the following code and the linked list pictured below to answer the questions 24 - 26.

First

Then, First nextnextdata is

(a) Null

(b) Syntax Error

(c) 30

(d) 20

(e) 45.

145.

Pick the right option from the following:

I.     Firstnext == Temp1.

II.     Temp1nextdata == 60.

III.    Temp2next next == NULL.

(a) Only (I) is TRUE

(b) Only (II) is TRUE

(c) Both (I) and (II) are TRUE

(d) Both (I) and  (III) are TRUE

(e) Both (II) and (III) are TRUE.

146.

Decide whether the syntax of each of the following statements is valid or invalid.

I.     Firstnext = Temp1next;

II.     First next = *( Temp2next);

III.    * First = Temp2;

(a) Only (I) is valid

(b) Only (II) is valid

(c) Only (III) is valid

(d) Both (I) and (III) are valid

(e) Both (II) and (III) are valid.

147.

Which of the following is not possible as a balance factor of any node of an AVL tree?

I.     -2.

II.     ±1.

III.    0.

IV.   2.

(a) Only (I) above

(b) Only (II) above

(c) Only (III) above

(d) Both (I) and (IV) above

(e) Both (III) and (IV) above.

148.

Which of the following statement(s) is/are TRUE in view of a multi-way search tree? If a node has

I.     4 sub trees it contains 3 keys.

II.     5 keys, it has 7 sub trees.

III.    6 sub trees, it contains 6 keys.

IV.   10 keys if it contains 11 children.

(a) Only (I) above

(b) Only (II) above

(c) Only (III) above

(d) Both (I) and (IV) above

(e) Both (II) and (III) above.

149.

Identify the name of the sorting in which time is not proportional to n2.

(a) Selection sort

(b) Bubble sort

(c) Qucik sort

(d) Merge sort

(e) Insertion sort.

150.

How many fields are required by a node to represent the polynomials in computer memory using linked list?

(a) Two fields

(b) Four fields

(c) More than three fields

(d) One is enough

(e) Three fields.

Answers

141. Answer : (c)

Reason:  All the leaf nodes appear at same level  is true related to a B-tree.

142. Answer : (d)

Reason:  A simple graph has no loops. Other property is to have no multiple edges.

143. Answer : (b)

Reason:  Current depth of recursion  is not saved in the activation record, when ever a function call is executed.

144. Answer : (e)

Reason:  the following code and the linked list pictured

Then, First nextnextdata = 45

145. Answer : (b)

Reason:  Firstnext == Temp1 and Temp2next next == NULL are true.

146. Answer : (a)

Reason:  The syntax of Firstnext = Temp1next is valid.

147. Answer : (e)

Reason:  -2 and 2 is not possible as a balance factor of any node of an AVL tree.

148. Answer : (e)

Reason:  If a node has 4 sub trees it contains 3 keys and 10 keys if it contains 11 children. is TRUE in view of a multi-way search tree.

149. Answer : (d)

Reason:  Merge Sort is a sort  in which time is not proportional to n2.

150. Answer : (e)

Reason:  Three fields are required by a not to represent the polynomials in computer memory using linked list.

161.The optimal solution to a problem is a combination of optimal solutions to its sub-problems. This is known as

(a) Principle of Duality

(b) Principle of Equality

(c) Principle of Feasibility

(d) Principle of Optimality

(e) Principle of Dynamicity.

162.Which of the following versions of merge sort algorithm does uses space efficiently?

(a) Contiguous version

(b) Array version

(c) Linked version

(d) Structure version

(e) Heap version.

163. Identify the correct problem for multistage graph from the list given below.

(a) Resource allocation problem

(b) Traveling salesperson problem

(c) Producer consumer problem

(d) Barber’s problem

(e) Dining philosopher problem.

164.How many edges are there in a Hamiltonian cycle if the edge cost is ‘c’ and the cost of cycle is ‘cn’

(a) c

(b) 2n

(c) cn

(d) 2c

(e) n.

165.A problem L is NP-complete iff L is NP-hard and

(a) L ≈ NP

(b) L α NP

(c) L ε NP

(d) L = NP

(e) L ≠ NP.

166.What would be the cost value for any answering node of a sub tree with root ‘r’ using branch-bound algorithm?

(a) Maximum

(b) Minimum

(c) Optimal

(d) Average

(e) Negative.

167.Name the process which executes a correct program on data sets measuring the time and space of a program, in order to compute the results

(a) Partitioning

(b) Hashing

(c) Validating

(d) Profiling

(e) Verifying.

168.What would be the hash address for the following function by using the folding method Hk = H(4326)?

(a) 69

(b) 32

(c) 46

(d) 105

(e) 42.

169.Name the node which has been generated but none of its children nodes have been generated in state space tree of backtracking method.

(a) Dead node

(b) Live node

(c) E-Node

(d) State Node

(e) Bounded Node.

170.How many nodes are there in a full state space tree with n = 6?

(a) 65

(b) 64

(c) 63

(d) 32

(e) 31.

Answers

161. Answer : (d)

Reason:  According to the principle of optimality, the optimal solution to a problem is a combination of optimal solutions to its sub-problems which are optimum.

162. Answer : (c)

Reason:  Because, in linked version auxiliary memory is not required, as needed in array version.

163. Answer : (a)

Reason:  Because, the resource allocation problem also same as allocating the right resource to appropriate person. The remaining options are not suitable.

164. Answer : (e)

Reason:  Because. Simple in Hamiltonian graph there should be exactly ‘n’ edges to connect all the nodes exactly once to form a Hamiltonian cycle.

165. Answer : (c)

Reason:  According to the definitions of NP-class, NP-complete and NP-hard class problems.

166. Answer : (b)

Reason:  Because the objective in branch-bound algorithms is to minimize the objective function.

167. Answer : (d)

Reason:  Because, in order to compute the results the profiling process executes the correct program on data sets measuring the time and space of a program. The remaining options are invalid.

168. Answer : (a)

Reason:  Because, according to folding method this is 43+26 which is 69.

169. Answer : (b)

Reason:  Because in state space tree organization live node is a node which has been generated but none of its children nodes have been generated, but are being generated.

170. Answer : (c)

Reason:  As this is 2n-1 ( i.e. 26-1 = 63 ).

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171.Consider the following Traveling salesperson problem instance matrix. What would be the value of g(1, {2, 3})?

∞ 8 5

1 ∞ 7

2 3 ∞

(a) 2

(b) 12

(c) 14

(d) 17

(e) 9.

172.The running time complexity of graph coloring problem is

(a) O(nm)

(b) O(mn)

(c) O(mnm)

(d) O(nmn)

(e) O(mnnm).

173.What is the time taken by the binary search algorithm to search a key ‘k’ in a sorted array of ‘n’ elements?

(a) O(log2 n)

(b) O(n)

(c) (n log2 n)

(d) O(n2)

(e) O(1).

174.Name the function which is used to kill live nodes without generating all their children in state space tree using backtracking method

(a) Solution function

(b) Bounding function

(c) Optimal function

(d) State Space function

(e) Randomized function.

175.Which of the following search method will always find a goal node nearest to the root of the tree?

(a) Depth first search

(b) Binary search

(c) Breadth first search

(d) Heuristic search

(e) Randomized search.

176.What do we call the selected keys in quick sort?

(a) Recombine key

(b) Inner key

(c) Outer key

(d) Pivot key

(e) Median key.

177.This sort finds location ‘pos’ of smallest elements in a(i), …. , a(n) and then interchange a(pos) with a(i) for i = 1, …., n – 1.

(a) Selection sort

(b) Quick sort

(c) Heap sort

(d) Bubble sort

(e) Insertion sort.

178.Adjacency matrix of a digraph is

(a) Identity matrix

(b) Symmetric matrix

(c) Asymmetric matrix

(d) Sparse matrix

(e) Zero matrix.

179.We can efficiently reverse a string using

(a) Linear queue

(b) Stack

(c) Circular queue

(d) Doubly linked list

(e) Circular linked list.

180.The technique of linear probing for collision resolution can lead to

(a) Underflow

(b) Radix ordering

(c) Efficient storage utilization

(d) Overflow

(e) Clustering.

Answers

171. Answer  :        (e)

Reason : This is  g( 1, {2,3}) = min( C12 + g( 2, {3}), C13 + g( 3, {2})

= min( 8+9, 5+4 ) = 9.

172. Answer : (d)

Reason:  According to the complexity analysis of the algorithm.

173. Answer : (a)

Reason:  According to the mathematical analysis binary search takes ( log2 n ) time.

174. Answer : (b)

Reason:  Because, in state space tree generation of backtracking approach a bounding function is used to kill live nodes without generating all their children, if that node is not promising the solution.

175. Answer : (c)

Reason:  Because in this method nodes are generated level by level and hence the goal node is found nearest to root node.

176. Answer : (d)

Reason:  Because, in quick sort division is made along the pivotal element.

177. Answer : (a)

Reason:  As this process is the key process for the selection sort algorithm.

178. Answer : (c)

Reason:  Because, in a diagraph for directed edge <a, b>; b is adjacent to a, but reverse is not true. Hence the adjacency matrix is asymmetric matrix.

179. Answer : (b)

Reason:  This is nothing but pushing the characters of the string in a stack and then popping the stack to get the reversed order of the string.

180. Answer : (e)

Reason:  Because, if we follow the linear probing method to resolve the collision then it may lead to the clustering.

181.

For the smart bubble sort algorithm, what is the time complexity of the best and worst case?

(a) best case: O(n)                   worst case: O(n2)

(b) best case: O(n)                   worst case: O(n*log n)

(c) best case: O(n*log n)         worst case: O(n*log n)

(d) best case: O(n*log n)         worst case: O(n2)

(e) best case: O(1 )            worst case: O(n).

182.

There are two sorted lists L1 and L2. Their lengths are n1 and n2 respectively. You are asked to design an algorithm to generate a sorted list L3 from L1 and L2. That is, for example, if L1=[1,5,30] andL2=[3,6,12,24,43] then L3=[1,3,5,6,12,24,30,43]. Which of the following methods is the most efficient?

(a) Taking elements from L1 one by one, and inserting them into L2such that the order of L2 remains unchanged

(b) Appending two lists (i.e. join one list's beginning to the other one's end), then applying quick sort to the appended list.

(c) Using a merge method that compares two elements at the head of both lists. The smaller one is then taken out and inserted at the end of L3 while the larger one is kept its location unchanged. Repeat this until either L1 or L2 is empty.

(d) Append two lists and apply Shell sort.

(e) Append two lists and apply Selection sort.

183.

What is the time complexity of the algorithm ‘a’ in question 2?

(a) O(log(n1)*n2)

(b) O(n1*log(n2))

(c) O(n1+n2)

(d) O(max(n1,n2))

(e) O(n1*n2).

184.

What is the best choice for the time complexity of the algorithm C in question 2?

(a) O(n1*n2)

(b) O(n1*log(n2))

(c) O(n1+n2)

(d) O(max(n1,n2))

(e) O(log(n1)*n2).

185.

What is the Postfix form of the following in fix expression? 

(2+3)*(4+(5*6)/(7*8/9*1))

(a) 23+456*78*9/1*/+*

(b) 23+456*78*91/*/+*

(c) 23+456*78*91*//+*

(d) 23+456*78*91*//*+

(e) 23+456*+78*91*//*.

186.

The direct or random access of element is not possible in

(a) Array

(b) String

(c) Linked List

(d) Both (a) and (b) above

(e) Both (b) and (c) above.

187.

If for a given ‘Queue’ initially f=0 and r=-1, then f = r refers to

(a) Queue is empty

(b) Queue is full

(c) Queue has two elements

(d) Exactly one element is there

(e) Not possible at all.

188.

Any string of bits of length ‘n’ represents a unique non-negative integer between

(a) 0 and 2n-1-1

(b) 0 and 2n –1

(c) 2n-1

(d) 0 and 2n

(e) 0 and 2n-1.

189.

Pick the statement(s) which is(are) not true

(a) ADT is useful tool for specifying the logical properties of data type

(b) While defining the ADT as a mathematical concept, the space efficiency isn’t a major concern

(c) Every ADT can be implemented using any programming language

(d) ADT helps us to hide the organization of the data

(e) A & B.

190.

Using arrays the most efficient implementation of the QUEUE is as

(a) Linear queue

(b) Circular queue

(c) Linked queue

(d) Can’t be implemented

(e) Priority queue.

Answers

181.

Answer : (a)

Reason  :       The smart bubble sort assumes that the computation stops as soon as no more swaps in pass one.

182.

Answer : (c)

Reason  :       As the time complexity of merge sort is small when compared with all the remaining algorithms. Also the lists L1 and L2 already sorted.

183.

Answer : (e)

Reason  :       Each ordered insertion costs O(n2). There is a total of n1 insertions. Therefore, the answer is O(n1*n2).

184.

Answer : (c)

Reason  :       If the size of each list is n1 and n2, then in merge sort the final list contains n1+n2 elements.

185.

Answer : (a)

Reason  :       According to precedence of operators.

186.

Answer : (c)

Reason  :       Because, in linked list the elements are always not stored at contiguous memory locations, but are stored at discrete locations. And hence random access is not possible.

187.

Answer : (d)

Reason  :       If front pointer ‘f’ and rear pointer ‘r’ points to same location then there is only one element in the queue.

188.

Answer : (e)

Reason  :       As all the bits are used to hold the data bits, and there is no sign bit in the array.

189.

Answer : (c)

Reason  :       Though ADT is independent of any programming language, but still every ADT can’t be implemented using any programming language.

190.

Answer : (b)

Reason  :       Because in Circular queue the memory can be used efficiently.

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191.One has implemented the queue with a circular array, keeping track offirst, last, and count (the number of items in the array). Suppose firstis zero, and last is SIZE-1. What can you tell me about count?

(a) count must be zero

(b) count must be SIZE

(c) count could be zero or SIZE, but no other values could occur

(d) count must be SIZE+1

(e) count must be SIZE-2.

192.When we say the order of a tree is M, we mean

(a) every non-leaf node must have M subtrees

(b) every non-leaf node must have M keys

(c) every non-leaf node can have at most M keys

(d) every non-leaf node can have at most M subtrees

(e) the Height of the tree is M.

193.Find the odd one out from the following categories of algorithms

(a) Bin-packing

(b) OBST

(c) N-Queens

(d) 15-Puzzle

(e) TVSP.

194.This algorithm scans the list by swapping the entries whenever pair of adjacent keys are out of desired order.

(a) Insertion sort.

(b) Bubble sort.

(c) Shell sort.

(d) Quick sort.

(e) Radix sort.

195.The mathematical definition for Omega can be defined as, provided f,g:NR+ and c is a positive constant and n > n0,

(a) f(n) =   g(n)   n

(b) f(n) = c. g(n)   n

(c) f(n) ≥ c + g(n)   n

(d) f(n) = c + g(n)   n

(e) f(n) ≥ c. g(n)   n.

196.Let f, t: N→ R+, then t (n)  Ω (f (n)), iff  f(n)  O (t (n)) is known as what?

(a) Limit rule

(b) Rule of inference

(c) Duality rule

(d) Rule of consequences

(e) Rule of symmetricity.

197.The  notation is

(a) Symmetric

(b) Reflexive

(c) Transitive

(d) B & C only

(e) A, B, & C.

198.How many children do an external node of a binary tree of order N contains.

(a) N exactly

(b) N-1 exactly

(c) One exactly

(d) 0 exactly

(e) N/2 exactly.

199.From the following chose the one which belongs to the algorithm paradigm other than to which others from the following belongs to.

(a) Minimum & Maximum problem.

(b) Knapsack problem.

(c) Selection problem.

(d) Merge sort.

(e) Quick sort.

200.To calculate c(i, j )’s, w( i, j)’s and r(i, j)’s; the OBST algorithm in worst case takes the following time.

(a) O(log n)

(b) O (n4)

(c) O (n3)

(d) O (n log n)

(e) O (n2).

Answers

191.

Answer : (c)

Reason  :       This is the case where either the list is full or the list the made empty after it was full.

192.

Answer : (d)

Reason  :       Here in the answer ‘can’ is most important. That is “every non-leaf node ‘can’ have at most M subtrees.

193.

Answer : (a)

Reason  :       This one belongs to NP-Class category.

194.

Answer : (b)

Reason  :       Bubble sort only is the algorithm from the given options which compares the adjacent keys

195.

Answer : (e)

Reason  :       According to the mathematical definition of the Asymptotic notations.

196.

Answer : (c)

Reason  :       According to the asymptotic notation rules. And this rule is used to apply the limit rule for the omega notated values

197.

Answer : (e)

Reason  :       Because  notation is equivalence relationship in nature.

198.

Answer : (d)

Reason  :       Because, the leaf nodes can’t have the children.

199.

Answer : (b)

Reason  :       Remaining all belongs to divide and conquer paradigm.

200.

Answer : (c)

Reason  :       Because, we have to calculate all c(i, j )’s, w( i, j)’s and r(i, j)’s for the tree of ‘n’ identifiers

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201.The time taken by NP-class sorting algorithm is

(a) O(1)

(b) O(log n)

(c) O(n2)

(d) O(n log n)

(e) O(n).

202.For the functions f, g : N –> R  for all n,   f(n) = n and g(n) = n log n then

(a) f(n) ( g(n)), but g(n) ( f(n))

(b) f(n) ( g(n)), but g(n) ( f(n))

(c) g(n) ( f(n)), but f(n) ( g(n))

(d) f(n) ( g(n)), and g(n) ( f(n))

(e) f(n) ( g(n)), but f(n) ( g(n)).

203.What is the type of the algorithm used in solving the 8 Queens problem?

(a) Greedy

(b) Dynamic

(c) Branch and Bound

(d) Divide and Conquer

(e) Backtracking.

204.Let G be a graph with ‘n’ nodes and let ‘m’ be the chromatic number of the graph. Then the time taken by the backtracking algorithm to color it is

(a) O(nmn)

(b) O(n+m)

(c) O(mnm)

(d) O(nm)

(e) O(nm).

205.What is the major data structure used in the Network data model?

(a) Stack

(b) Array

(c) Graph

(d) Tree

(e) Queue.

206.Minimum number of queue(s) needed to implement the priority queue is(are)

(a) Four

(b) Three

(c) Two

(d) One

(e) Depends upon the application.

207.Sorting is not possible by using which of the following methods?

(a) Insertion    

(b) Selection    

(c) Exchange

(d) Deletion

(e) Partitioning.    

208.The number of null branches for a binary tree with 20 nodes is

(a) 21

(b) 20

(c) 22

(d) 19

(e) 18.

209.The time complexity of binary search in best, worst cases for the array of size N is

(a) N, N2

(b) N, N

(c) 1, logN

(d) 1, NlogN

(e) logN, N2.

210.A desired key in a table is searched, starting itself from hash address, sequentially for the following.

(a) Quadratic probing

(b) Linear probing

(c) Random probing

(d) Chaining

(e) Reverse probing.

Answers

201.

Answer : (e)

Reason  :       Here only one loop from one to n is there. And hence it runs for ‘n’ times only

202.

Answer : (a)

Reason  :       According to the limit rules for big Oh notation

203.

Answer : (e)

Reason  :       As the other types of algorithms are not suitable to find the solution for the given problem.

204.

Answer : (a)

Reason  :       As the number of internal nodes in the state space tree are mn, and O(mn) is the time spent by the NextValue algorithm to determine the children corresponding to legal colorings. Hence the total time is bounded by O(nmn).

205.

Answer : (c)

Reason  :       As the graph is the suitable data structure to represent the connectivity between the nodes as edges.

206.

Answer : (c)

Reason  :       Two : One queue is used for actual storing of data and another for storing priorities.

207.

Answer : (d)

Reason  :       There is no sorting algorithm available where we delete an element. Using insertion we can perform insertion sort, using selection we can perform selection sort, using exchange we can perform the bubble sort (and other similar sorting methods) and using the portioning we can perform merge sort(and other similar sorting methods) But no sorting method can be done just using deletion.

208.

Answer : (a)

Reason  :       For ‘n’ nodes there are always ‘n+1’ null branches in a binary tree.

209.

Answer : (c)

Reason  :       In best case if the required element is at the middle then it takes O( 1 ) time other wise it takes O( log n ) time in worst case.

210.

Answer : (b)

Reason  :       According to opening addressing techniques, the linear probing starts probing from the hash address and it continues sequentially in a linear manner.

211.

Find the correct sequence for the Big Oh values for all n > 2

(a) O(1) < O( n) < O( log n) < O(n log n) < O(n2) < O(n3) < O(2n)

(b) O(1) < O(log n) < O(n) < O(n log n) < O(n2) < O(2n) < O(n3)

(c) O(1) < O(log n) < O(n) < O(n log n) < O(n2) < O(n3) < O(2n)

(d) O(1) < O(n) < O(log n) < O(n log n) < O(n2) < O(2n) < O(n3)

(e) O(1) < O(log n) < O(n) < O(n2) < O(n log n) < O(n3) < O(2n).

212.

The   notation is

I.     Reflexive.

II.    Symmetric.

III.   Transitive.

(a) Only (I) above

(b) Only (III) above

(c) Both (I) and (II) above

(d) Both (II) and (III) above

(e) All (I), (II) and (III) above.

213.

Data structure

(a) May be helpful to develop efficient algorithms in different phases of data processing

(b) Need not give relationship between data items

(c) Is programming language dependent

(d) Is procedural language dependent

(e) Need to describe the relationship among data items.

214.

Modular programming uses

(a) Only top-down method

(b) Only bottom-up method

(c) Both (a) and (b) above

(d) Procedural approach

(e) Object oriented approach.

215.

In Knapsack problem, the best strategy to get the optimal solution, where Pi, Wi is the Profit, Weight associated with each of the Xi

th object respectively is to

(a) Arrange the values Pi/Wi in ascending order

(b) Arrange the values Pi/Xi in ascending order

(c) Arrange the values Pi/Wi in descending order

(d) Arrange the values Pi/Xi in descending order

(e) Arrange the values Wi/ Xi in decreasing order.

216.

In the following weighted graph, the length of the minimum spanning tree is

(a) 39

(b) 41

(c) 43

(d) 52

(e) depends on the starting vertex.

217.

The time complexities of Max Min recursive and non-recursive algorithms are

(a)  – 1,  – 2    

(b) ,  – 2

(c)  – 2,  – 1

(d) ,

(e) 3n – 2, 3n – 1.

218.

Greedy job scheduling with deadlines algorithms’ complexity is defined as 

(a) O(N)

(b) Ω( n log n)

(c) O (n2 log n)

(d) O ( n log n)

(e) Ω (N2).

219.

The divide and conquer merge sort algorithm’s time complexity can be defined as

(a)  (long n)

(b)  (n)

(c) Ω (n log n)

(d)  (n log n)

(e)  (n2 log n).

220.

In analysis of algorithm, approximate relationship between the size of the job and the amount of work required to do it is expressed by using

(a) Order of magnitude or Big - O

(b) Central tendency

(c) Differential equation

(d) Polynomial equation

(e) Space complexity.

Answers

211.

Answer : (c)

Reason  :       O(1) < O(log n) < O(n) < O(n log n) < O(n2) < O(n3) < O(2n)

212.

Answer : (e)

Reason  :       (I), (II), & (III).

213.

Answer : (a)

Reason  :       may be helpful to develop efficient algorithms in different phases of data processing

214.

Answer : (c)

Reason  :       both a & b

215.

Answer : (d)

Reason  :       Arrange the values Pi/Xi in descending order

216.

Answer : (e)

Reason  :       depends on the starting vertex

217.

Answer : (a)

Reason  :       – 1,  – 2

218.

Answer : (a)

Reason  :       O(N)

219.

Answer : (d)

Reason  :        (n log n)

220.

Answer : (a)

Reason  :       order of magnitude or Big - O

221.

In linked list, a node contains atleast

(a) Node address field, data field

(b) Node number, data field

(c) Next address field, information field

(d) Node number, information field

(e) Node address field, information field.

222.

Which of the following is the correct big-O expression for 1 + 2 + 3 + ... + n?

(a) O(log n)

(b) O(n)

(c) O(n log n)

(d) O(n²)

(e) O(2n).

223.

A B-tree of order n is also called

 (a) (n–n) –1 tree

(b) n–(n–2) tree

(c) (n–1)–n tree

(d) n–(n–1) tree

(e) n–1 tree.

224.

The minimum number of keys contained in each non root node of a B-tree of order 11 are

(a) 4

(b) 5

(c) 3

(d) 2

(e) 6.

225.

Row-major order in two dimensional array refers to

(a) All elements of a row are stored in memory in sequence followed by next row in sequence and so on

(b) All elements in the sorted order according to their column sequence

(c) All elements of a column in sequence and so on

(d) All elements in the sorted order according to their row sequence

(e) All elements of a row are stored in memory in sequence followed by next column in sequence and so on.

226.

Null pointer is used to define

I.     End of linked list.

II.    Empty pointer field of a structure.

III.   The linked list is empty.

(a) Only (I) above

(b) Only (II) above

(c) Only (III) above

(d) Both (I) and (II) above

(e) All (I), (II) and (III) above.

227.

Direct or random access of element is not possible in

(a) Linked list

(b) Array

(c) String

(d) Queue

(e) Double linked list.

228.

“Get a node, store new element and insert the new node at the top” refers to insert operation in non empty

(a) Stack

(b) Queue

(c) Array

(d) Linked list

(e) Tree.

229.

A technique which collects all deleted spaces onto free storage list is called

(a) Static memory allocation

(b) Garbage collection

(c) Calloc

(d) Dynamic memory allocation

(e) Malloc.

230.

datatype  *s;

datatype  *p;

datatype  *i;

p = malloc(100); t=p; s=p+ (100/size of (datatype))-sizeof (datatype);

Func(datatype j )

{ if (p > s) {printf (“overflow/n”); return:}

*p= j; p++;}

is referring to

(a) Push operation of stack

(b) Pop operation of stack

(c) Push operation of queue

(d) Pop operation of queue           

(e) Push and pop of the stack.

Answers

221.

Answer : (c)

Reason  :       next address field, information field.

222.

Answer : (d)

Reason  :       O(n²)

223.

Answer : (c)

Reason  :       (n-1)-n tree

224.

Answer : (b)

Reason  :       5

225.

Answer : (a)

Reason  :       all elements of a row are stored in memory in sequence followed by next row in sequence and so on.

226.

Answer : (e)

Reason  :       (I), (II), (III) are correct

227.

Answer : (c)

Reason  :       string

228.

Answer : (a)

Reason  :       stack

229.

Answer : (b)

Reason  :       Garbage collection

230.

Answer : (a)

Reason  :       push operation of stack

231.

malloc() returns pointer to

(a) Integer

(b) Structure

(c) Character

(d) Boolean

(e) Float.

232.

If a binary tree is threaded for an in order traversal order, a NULL right link of any node is replaced by the address of its

(a) Successor

(b) Predecessor

(c) Root

(d) Postdecessor

(e) None of these.

233.

If graph G has no edges then corresponding adjacency matrix is

(a) Unit matrix

(b) Zero matrix

(c) Matrix with all 1’s

(d) Diagonal matrix

(e) Upper diagonal matrix.

234.

Worst case efficiency of binary search is

(a) log2 n + 1

(b) n

(c) N2

(d) 2n

(e) log n.

235.

The property that is not expected from good hashing technique should

(a) Produce keys uniformly distributed over the range

(b) Easy to program

(c) Produce no collisions

(d) Produces collisions

(e) All above properties are expected.

236.

Graph structure is available in

(a) C

(b) C++

(c) Pascal

(d) Java

(e) Data structures.

237.

The element at the root of heap is

(a) Largest

(b) Smallest

(c) Depending on type of heap it may be smallest or largest

(d) Fixed

(e) Unlimited.

238.

Worst case efficiency of which search is O(n)?

(a) Sequential search

(b) Binary search

(c) Indexed search

(d) Hashing

(e) Depth first search.

239.

Breadth first search

(a) Scans all incident edges before moving to other vertex

(b) Scans adjacent unvisited vertex as soon as possible

(c) Is same as backtracking

(d) Computes a path between two vertices of graph or equivalently reporting that no such path exists

(e) Scans for cycle in graph, that no such cycle exists.

240.

Which of the following searching methods requires that all keys must reside in internal memory?

(a) Binary search

(b) Sequential search

(c) Hashing

(d) Depth first search

(e) Breadth first search.

Answers

231.

Answer : (c)

Reason  :       character

232.

Answer : (a)

Reason  :       successor

233.

Answer : (b)

Reason  :       zero matrix

234.

Answer : (a)

Reason  :       log2 n+1

235.

Answer : (c)

Reason  :       produce no collisions.

236.

Answer : (e)

Reason  :       data structures

237.

Answer : (c)

Reason  :       depending on type of heap it may be smallest or largest

238.

Answer : (a)

Reason  :       Sequential search

239.

Answer : (a)

Reason  :       scans all incident edges before moving to other vertex

240.

Answer : (a)

Reason  :       Binary search

241. Standard queue operations are

(a)  empty(), fill(), place(), remove()                        (b)  enque(), dequeue(), isempty(), isfull()

(c)  init(), delete(), add()                                         (d)  isempty(), isfull(), fill(), remove()

(e)  isempty(), isfull(), init(), delete(), add().

242. Assume that variable A is a sorted array of integers of length L. Consider the following code segment:

  int k=1, flag=0;

for (   ;  k<L;   k++ )

{

                if (A[k-l] == A[k] ) flag = 1;

  }

Which of the following best describes when variable flag is 1 after this code segment executes?

  Always

  Never

  If and only if array A really is sorted

  If and only if array A contains adjacent duplicate values                

  If and only if all values in array A are the same.

243. Consider using binary search to look for a given value in an array of integers. Which of the following must be true in order for the search to work?

  The values in the array are stored in sorted order.

   The array does not contain any duplicate values.

   The array does not contain any negative values.

(a)   Only (I) above                                 (b)  Only (II) above   (c)  Both (I) and (II) above

(d)  Both (I) and (III) above                      (e)  Both (II) and (III) above.

244. A program is required to store and process information about the solar system. Which types of data structure would be most appropriate for storing the following respectively : (i) A list of the names of the planets, (ii) full details of a planet including distance from sun, diameter, atmospheric composition, etc, (iii) a predetermined list of scientific observations made of the planet.

  (i) linked list           (ii) structure       (iii) linked list

  (i) array                  (ii) structure       (iii) linked list

  (i) linked list           (ii) array            (iii) array

  (i) linked list           (ii) linked list    (iii) linked list

  (i) array                  (ii) structure       (iii) array.

245. If data is a circular array of CAPACITY elements, and R is an rear index into that array, what is the formula for the index after R?

(a)  (R + 1) % CAPACITY                        (b)  R % (1 + CAPACITY)                       

(c)  (R % 1) + CAPACITY                        (d)  R + (1 % CAPACITY)              (e)  R = R + 1.

246. In the linked list implementation of the queue, where does the insert method place the new entry on the linked list?

  At the head

  At the tail

  After all other entries that are greater than the new entry

  After all other entries that are smaller than the new entry

  This operation is not possible.

247. One difference between a queue and a stack is:

  Queues require linked lists, but stacks do not

  Stacks require linked lists, but queues do not

  Queue is used for complex programs and stack is used for simple programs

  Stacks use two ends of the structure, queues use only one

Queues use two ends of the structure; stacks use only one.

248. Suppose cursor refers to a node in a linked list. What boolean expression will be true when cursor refers to the tail node of the list?

(a)  cursor.next == null                                                          (b)  cursor == null    (c) cursor.data == null

(d)  cursor.data == 0                                                            (e)  cursor.next == cursor.

249. Five statements about lists and stacks are below. Four of them are correct. Which one isincorrect?

  Lists can be implemented by using arrays or linked lists

  Lists are the dynamic data structures

  A stack is a special kind of list in which all insertions and deletions take place at one end

  A list is a sequence of one or more data items

  Stacks are easier to implement than lists.

250. For questions 10-11, refer to the following:

Queue q;

int i=1;const int value = 10;for ( ; i < value; i++) insert(i, q);printf(“%d”, frontofqueue(q) ) ;

delete(q);

delete(q);

printf(“%d”, frontofqueue(q) ) ;

while (! isempty(q))

delete(q);

What is the output for the above code?

(a)  1, 3                   (b)  1, 4                   (c)  No output          (d)  9, 6                   (e)  9, 7.

Answers

241. Answer : (b)

Reason:  As fill( ), place( ), add( ),empty( )are not the standard operations on queue, while enque ( ), dequeue( ), isempty( ), isfull( ) are the standard one.

242. Answer : (d)

Reason:  Because this is an array, and k is the index. So k, k-1 are the adjacent index values.

243. Answer : (c)

Reason:  Because the precondition for binary search is, the array must be there in sorted order, should not contain duplicate values and may have positive or negative values.

244. Answer : (e)

Reason:  To keep track of the planets a fixed size array is enough as the number of plants is finite value. In addition, to keep track of complex information as a single logical unit structure is the best suitable data structure. For predetermined list array is suitable.

245. Answer : (a)

Reason:  As the circular queue is given, if the empty places are there at any place and rear pointer R is a rear index, then this is the best ever formula to increment R. All other are not valid or appropriate one.

246. Answer : (b)

Reason:  Because, in  queue always the new entries are inserted at the end of queue, no matter the queue is implemented as array or linked list.

247. Answer : (e)

Reason:  As all the other options are meaning less.

248. Answer : (a)

Reason:  As the ‘next’ field of the node is holding NULL address, means it is the last node       .

249. Answer : (d)

Reason:  As in lists only one data item of some type can only be stored. More data items can’t be stored in a list.

250. Answer : (a)

Reason:  As using insert we are inserting the elements in queue are 1, 2, 3, 4, 5, 6, 7, 8, 9. Then using printf statement we are displaying the first element, then we are deleting the front element twice (i.e. 1, 2) then again we are displaying front element.

251. At the end of this code segment the queue

(a)  Contains numbers 1 to 8                   (b)  Contain numbers 1 to 10 but we can't see them

(c)  Is empty                                          (d)  Contains numbers 3 to 10

(e)  Contains only 1.

252. What is the correct post fix form of the given infix expression a+b/(c+d)%e*(g-h/(i+j)*k)?

(a)  abcd+/e%ghij+/k**-+                        (b)  abcd+e/%ghij+/k**-+

(c)  ab+cd/e%ghij+/k**-                          (d)  ab+cd+/e%ghij+/k*-*

(e)  abcd+/e%ghij+/k*-*+.

253. With the “wrap around” implementation of a queue, which of the following code should be used to work out the next location of deletion?

  front++

  front--

  front = (front % max_queue_size) + 1

  front = (front ++) % max_queue_size

  front = 0.

254. Which boolean expression indicates whether the number in two nodes (p and q) are the same? Assume that neither p nor q is null.

(a)  p == q                                             (b)  p.data == q.data                      (c)  p.link == q.link

(d)  p.link = q.link                                   (e)  p.data = q.data.

255. Which of the following statements is not true?

  An array is a data structure containing a number of items of the same type

  An array is normally used to store two or more items of data

  The lower bound of an array can be declared as any unsigned value

  The elements of an array can be initialized when the array is declared

  The number of elements in an array must be declared at compile time.

256. Consider searching for a given value in a large, sorted array. Under which of the following  conditions is sequential search slower than binary search?

  Always

  Never

  When the value being searched for is the first element in the array

  When the value being searched for is the second element in the array

  When the value being searched for is the last element in the array.

257. The following data structure is used to store the values of the extreme ends of the freely hanging simple pendulum for every oscillation.

(a)  Linked List                                       (b)  Priority Queue   (c)  Deque

(d)  Double Stack                                  (e)  Array.

258. The linked list which can be processed in either of the direction is

(a)  Single linked list                                                             (b)  Circular linked list

(c)  Stack implemented as linked list       (d)  Queue implemented as linked list

(e)  Double linked list.

259. What does a run-time analysis usually count?

  The number of arithmetic and other operations required for the program to run

  The number of megabytes required for the program to run

  The number of seconds required for the program to run

  The number of seconds plus the number of megabytes

  The number of seconds times the number of megabytes.

260. Which of these is the correct big-O expression for 1+2+3+...+n?

(a)  O(log n)             (b)  O(n)                  (c)  O(n log n)          (d)  O(n²)                 (e)  O(1).

Answers

251. Answer : (c)

Reason:  As using ‘while loop’ we have removed all the elements. Queue becomes empty.

252. Answer : (e)

Reason:  Once we convert it all the other options are wrong except E

253. Answer : (d)

Reason:  As in circular queue all the other options are not correct to identify the location pointed by front pointer where the recent deletion is made.

254. Answer : (b)

Reason:  As we are comparing the data of both of the nodes, we have to check for the data, a member of nodes p, q. Also we have to use relational operator( ==).

255. Answer : (c)

Reason:  As there is no any syntax to declare the lower bound of the array except the size of the array which can be declared.

256. Answer : (a)

Reason:  As in all the cases the time taken by sequential( linear) search is O(n) which is greater than the time taken by binary search which is O(log n).

257. Answer : (d)

Reason:  As the other data structures are not suitable to store the elements from both of the ends simultaneously.

258. Answer : (e)

Reason:  As the Doubly linked list contains two pointers to keep track the address of both the previous and next nodes.

259. Answer : (c)

Reason:  All the other options are meaning less. Because for run-time analysis, the number of operations and memory is not considered, but the unit time(i.e., seconds)

260. Answer : (d)

Reason:  Because this the summation of the first ‘n’ numbers, that is n(n+1)/2   => ( n2+n)/2  => then by ignoring the lower order terms and the constants this becomes n2.   

261. Which of the following formulas in Omega notation best represent the expression n²+35n+6?

(a)  Ω (n³)                (b)  Ω (n²)                (c)  Ω (n)                 (d)  Ω (35)               (e)  Ω (6).

262. What term is used to describe an O(n) algorithm?

(a)  Constant                                          (b)  Non Polynomial Deterministic          

(c)  Logarithmic                                      (d)  Quadratic                                        (e)  Linear.

263. Express the formula (n - 2)*(n - 4) using θ notation:

(a)  θ (n2)                 (b)  θ (8)                  (c)  θ (log n)            (d)  θ (n)                  (e)  θ (1).

264. Read the following statements carefully and pick the right most option.

  A linear algorithm to solve a problem must perform faster than a quadratic algorithm to solve the same problem.

   An algorithm with worst case time behavior of 3n takes at least 30 operations for every input of size n=10.

  Both (I) and (II) are TRUE   

  Both (I) and (II) are FALSE

  Only (I) is true but (II) depends upon the instance size.

  (I) is TRUE but (II) is FALSE           

  (I) is FALSE and (II) is TRUE.

265. Which of the following are essential statement types for describing algorithms?

(a)  Sequence                                        (b)  Selection           (c)  Repetition

(d)  All the above                                   (e)  A and B only.

266. When we say an algorithm has a time complexity of O (n), what does it mean?

  The algorithm has ‘n’ nested loops

  The computation time taken by the algorithm is proportional to n

  The algorithm is ‘n’ times slower than a standard algorithm

  There are ‘n’ number of statements in the algorithm

  The computation time taken by the algorithm is less than ‘n’ seconds.

267. Can we read a data item at any location of a list within a constant time (i.e. O(1))?

  Yes

  Yes, only if the list is implemented by pointers (i.e. linked-list)

  Yes, only if the list is implemented by an array

  No, we need O(n) computation steps no matter what kind of implementation is used

  No, as constant time is not possible for algorithms.

268. Sequential search has a time complexity of O(n), and binary search has a time complexity ofO(log(n)). What difference will it make when the size n is 1000?

(a)   You would not notice much difference because computers run very fast anyway

(b)   As n is 1000, binary search is twice as fast as sequential search

(c)   As n is 1000, binary search is 10 times as fast as sequential search

(d)   As n is 1000, binary search is 10 times as slow as sequential search

(e)   As n is 1000, binary search is 100 times as fast as sequential search.

269. Read the following statements carefully, and choose the correct answer.

I.     The Ω notation is Anti Symmetric.

II.     The big Oh notation is Semi Equivalence.

(a)  (I) is FALSE but (II) is TRUE              (b)  Both (I), (II) are TRUE

(c)  (I) is TRUE but (II) is FALSE              (d)  Both (I), (II) are FALSE

(e)  (II) is TRUE and (I) cannot be defined.

270. Find the odd one out.

(a)  Bin-Packing Problem                        (b)  TVSP Problem                    (c)  Knap Sack Problem

(d)  OBST Problem                                (e)  Sum of Sub Sets Problem.

Answers:

261. Answer : (b)

Reason:  As the given expression is a quadratic equation of degree 2, then by ignoring the constants and lower order terms, the highest term is n2, and hence it is Ω(n2).

262. Answer : (e)

Reason:  Because it is a single term of degree one. And all the remaining options are wrong.

263. Answer : (a)

Reason:  Again this is a polynomial of degree 2 i.e. n2 – 6n + 8. And hence n2 according to above reason of Question (21).

264. Answer : (d)

Reason:  As the first statement is true according to the definition of linear algorithms, the second is false as 30 operations are not needed.

265. Answer : (d)

Reason:  Because all are essential for the algorithm to state the solution of a given problem.

266. Answer : (b)

Reason:  As all the other options are meaning less and are wrong.

267. Answer : (c)

Reason:  For an array implementation, we can find the given location straightway with the index value and hence it takes constant time. But for a linked list, we have to follow the pointers one by one. The cost of this is proportional to the size of the list.

268. Answer : (e)

Reason:  log(1024) = 10,  1000/log(1000) is roughly equal to 1000/10 = 100

269. Answer : (b)

Reason:  According to the definitions of Omega, Big Oh notations.

270. Answer : (a)

Reason:  As Bin-Packing problem belongs to NP ( Non Deterministic Polynomial) hard problems and the remaining all are Deterministic polynomial time algorithms.

Data Structure and Algorithm Analysis

Questions 271 To 280

271. The suitable data structure to represent the data of rainfall of week days in ten cities of three states is __________

  Stack                                              (b)  Queue                                             (c)  Array

  Multi way tree                                  (e)  Connected graph.

272. The data structure which allows the insertion at both the ends, but allows the deletion at only one end is ________

   Output restricted Deque

   Input restricted Deque

   Circular queue

   Linear queue

   Priority queue.

273. Searching the linked list requires linked list be created in _________

  Ascending order                              (b)  Descending order                           

  With underflow condition                  (d)  Any order                (e)  Without underflow condition.

274. The time complexity of binary search in best, worst cases for an array of size N is

  N, N2                (b)  N, N                  (c)  Log N, N2          (d)  1, N log N         (e)  1, log N.

275. How many minimum number of spanning trees, one can have from a given connected graph with N nodes is having different weights for the edges.

  N-1                   (b)  One                  (c)  1/(N+1)  2NCN           (d)  2NCN           (e)  N.

276. How many children do an external node of a binary tree of order N contains.

  N at least          (b)  2 exactly           (c)  More than two          (d)  0                  (e)  N at most.

277. In-order traversal of binary search tree implies visiting the nodes in __________

Post-order      

The order of increasing magnitude of their key         

Pre-order 

The order of decreasing magnitude of their key N    

Arbitrary order.

278. For a binary tree the In-order traversal was found to be as HIGCEFD. Once the post-order traversal was conducted the output is IHGFEDC. What is the per-order for the given tree?

  CGDHEIF                                        (b)  GHICDEF          (c)  CGHIDEF

  CDEFGHI                                        (e)  Data insufficient and hence can’t be answered.

279. Breadth first search uses __________ as an auxiliary structure to hold nodes for future processing.

  Stack               (b)  Linked list         (c)  Graph                (d)  B-Tree               (e)  Queue.

280. Folding is a method of generating ________

  A hash function

  Index function for a triangular matrix

  Header node for a circular linked list

  Linear probing

  Chaining.

Answers:

271.Answer : (c)

Reason:  As the array can keep the data for multiple dimension values which is not possible by using any other data structure given in the question.

272.Answer :  (a)

Reason:  Because in Deque insertion and deletion can take place at any end, but in the question it is given as deletion is allowed only at one end. Hence it is Output restricted Deque

273.Answer :  (d)

Reason:  Because the reference of the next node is stored in the node itself with which one can go to next node, and hence a particular order is not required.

274.Answer :  (e)

Reason:  Because if the list is either small or the element is present at the mid position for the first call to the function or for the first iteration then it takes one unit of time. In worst case it takes log n time,

275.Answer :  (b)

Reason:  Because one can have at most one tree which is minimum, also the weights are different.

276.Answer :  (d)

Reason:  Because external node is a leaf node, that can’t have the children

277.Answer :  (b)

Reason:  Because if we traverse a BST  in in-order, then it is nothing but traversing in the order of increasing magnitude of their key.

278.Answer :  (c)

Reason:  Draw the tree according to the asked in-order and post-order and traverse the tree in pre-order.

279.Answer :  (e)

Reason:  As this requires the traversing of nodes on the same level before traversing the nodes at next level of the tree, it requires queue.

280.Answer :  (a)

Reason:  Because folding is one of the method of hashing function to locate the key.

Data Structure and Algorithm Analysis

Questions 271 To 280

271. The suitable data structure to represent the data of rainfall of week days in ten cities of three states is __________

  Stack                                              (b)  Queue                                             (c)  Array

  Multi way tree                                  (e)  Connected graph.

272. The data structure which allows the insertion at both the ends, but allows the deletion at only one end is ________

   Output restricted Deque

   Input restricted Deque

   Circular queue

   Linear queue

   Priority queue.

273. Searching the linked list requires linked list be created in _________

  Ascending order                              (b)  Descending order                           

  With underflow condition                  (d)  Any order                (e)  Without underflow condition.

274. The time complexity of binary search in best, worst cases for an array of size N is

  N, N2                (b)  N, N                  (c)  Log N, N2          (d)  1, N log N         (e)  1, log N.

275. How many minimum number of spanning trees, one can have from a given connected graph with N nodes is having different weights for the edges.

  N-1                   (b)  One                  (c)  1/(N+1)  2NCN           (d)  2NCN           (e)  N.

276. How many children do an external node of a binary tree of order N contains.

  N at least          (b)  2 exactly           (c)  More than two          (d)  0                  (e)  N at most.

277. In-order traversal of binary search tree implies visiting the nodes in __________

Post-order      

The order of increasing magnitude of their key         

Pre-order 

The order of decreasing magnitude of their key N    

Arbitrary order.

278. For a binary tree the In-order traversal was found to be as HIGCEFD. Once the post-order traversal was conducted the output is IHGFEDC. What is the per-order for the given tree?

  CGDHEIF                                        (b)  GHICDEF          (c)  CGHIDEF

  CDEFGHI                                        (e)  Data insufficient and hence can’t be answered.

279. Breadth first search uses __________ as an auxiliary structure to hold nodes for future processing.

  Stack               (b)  Linked list         (c)  Graph                (d)  B-Tree               (e)  Queue.

280. Folding is a method of generating ________

  A hash function

  Index function for a triangular matrix

  Header node for a circular linked list

  Linear probing

  Chaining.

Answers:

271.Answer : (c)

Reason:  As the array can keep the data for multiple dimension values which is not possible by using any other data structure given in the question.

272.Answer :  (a)

Reason:  Because in Deque insertion and deletion can take place at any end, but in the question it is given as deletion is allowed only at one end. Hence it is Output restricted Deque

273.Answer :  (d)

Reason:  Because the reference of the next node is stored in the node itself with which one can go to next node, and hence a particular order is not required.

274.Answer :  (e)

Reason:  Because if the list is either small or the element is present at the mid position for the first call to the function or for the first iteration then it takes one unit of time. In worst case it takes log n time,

275.Answer :  (b)

Reason:  Because one can have at most one tree which is minimum, also the weights are different.

276.Answer :  (d)

Reason:  Because external node is a leaf node, that can’t have the children

277.Answer :  (b)

Reason:  Because if we traverse a BST  in in-order, then it is nothing but traversing in the order of increasing magnitude of their key.

278.Answer :  (c)

Reason:  Draw the tree according to the asked in-order and post-order and traverse the tree in pre-order.

279.Answer :  (e)

Reason:  As this requires the traversing of nodes on the same level before traversing the nodes at next level of the tree, it requires queue.

280.Answer :  (a)

Reason:  Because folding is one of the method of hashing function to locate the key.

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What you think about these Questions and 

281. This algorithm scans the list by swapping the entries whenever pair of adjacent keys are out of desired order.

  Insertion sort     (b)  Quick sort         (c)  Shell sort           (d)  Bubble sort       (e)  Radix sort.

282. At a given node of a simple graph, the number of loops are _________

  More than one                                 (b)  Not more than one                           (c)  Zero  Exactly two                                     (e)  Equal to the number of nodes of the graph.

283. In the linked list implementation of the stack, where does the push member function place the new entry on the linked list?

  At the head  At the tail  After all other entries those are greater than the new entry  After all other entries those are smaller than the new entry  At null node.

284. I have implemented the queue with a circular array, keeping track of first, last, and count (the number of items in the array). Suppose first is zero, and last is SIZE-1. What can you tell me about count?       

  count must be zero.                          count must be SIZE          count must be SIZE-2  count must be SIZE+1      count could be zero or SIZE, but no other values could occur.

285. Convert the following postfix expression to a fully-parenthesized infix expression:

A B C - D E + * F * +  (A + ((B - C) * (D + E) * F))  (A + (((B - C) * (D + E)) * F))  A + (((B - C) * (D + E)) * F)  (A + ((B - C) * (D + E)) * F)  A + ((B - C) * (D + E)) * F.

286. The linked list would be able to make use of a ------------- whose value is the address of the location which stores the node.

  integer              (b)  float                  (c)  char                  (d)  void                  (e)  pointer.

287. What kind of list is best to answer questions such as "What is the item at position n?"

  Lists implemented with an array  Doubly-linked lists  Singly-linked lists  Doubly-linked or singly-linked lists are equally best  Circular linked list implemented with priorities.

288. Consider the following pseudo code:

  declare a stack of characters  while (there are more characters in the word to read ) {           read a character  push the character on the stack

   while ( the stack is not empty)  {  pop a character off the stack write the character to the screen

What is written to the screen for the input "carpets"?

  serc                  (b)  steprac             (c)  carpets              (d)  ccaarrppeettss  (e)  stepr.

289. What is the value of the postfix expression 6 3 2 4 + - *

  Something between 5 and 15  Something between -5 and -15  Something between 5 and -5  Something between -15 and -100  Something between 15 and 100.

290. Suppose we have a circular array implementation of the queue type, with ten items in the queue stored at data[2] through data[11]. The current SIZE is 22. Where does the insert method place the new entry in the array?

  data[1]              (b)  data[22]            (c)  data[12]             (d)  data[11]            (e)  data[21].

Answers:

281.Answer :  (d)

Reason:  As in bubble sort always we swap the adjacent keys if they are not in desired order

282.Answer :  (c)

Reason:  Because in simple graphs the loops are not allowed.

283.Answer :  (a)

Reason:  Because in stack implemented using linked list, head always points to the top most node.

284.Answer :  (e)

Reason:  Because first = 0 and rear = size-1 means either the stack is full or it was made to empty by after filling it to full. (Since this is a circular queue).

285. Answer :  (b)

Reason:  As the fully parenthesized expression requires its sub parts to be in the parenthesis

286. Answer :  (e)

Reason:  Because the pointer is used to hold the address of the next node in the list.

287. Answer :  (a)

Reason:  Because index is required to locate a particular position, and in arrays this index can be used to locate an element.

288.Answer :  (b)

Reason:  This nothing but pushing the characters on the stack and popping them from the stack and hence it will print the input in the reverse because of LIFO manner.

289.Answer :  (d)

Reason:  Because the answer is -18.

290.Answer :  (c)

Reason:  Because still the space is there in the queue and the next available location is 12thindexed one, therefore new element is inserted at that location.

291. The mathematical definition for Omega can be defined as, provided f,g:NR+ and c is a positive constant and n > n0,

  f(n) ≥ c. g(n)  n

  f(n) £ c. g(n)  n

  f(n) ≥ c + g(n)  n

  f(n) £ c + g(n)  n

  f(n) £   g(n)  n.

292. The   notation is __________

  Symmetric.           

   Reflexive.             

  Transitive.

Only (I) above

Only (II) above

  Only (III) above

  Both (I) and (II) above

  All (I), (II) and (III) above.

293. From the following pick the one which does not belongs to the same paradigm to which others belongs to.

  Minimum & Maximum problem

  Knapsack problem

  Selection problem

  Merge sort

  Quick sort.

294. The OBST algorithm in worst case takes _______ time if all c(i, j )’s and r(i, j)’s are calculated.

  O(log n)            (b)  O(n4)                 (c)  O(n3)                  (d)  O(n log n)          (e)  O(n2).

295. Read the following statements carefully, and choose the correct answer

  For the Backtracking algorithms stack data structure is used.

   For the Branch-and-bound algorithms queue data structure is used.

  (I) is FALSE but (II) is TRUE

  (I) and (II) both are FALSE

  (I) is TRUE but (II) is FALSE

  (I) and (II) both are TRUE

  (II) is TRUE and (I) can’t be defined.

296. The weighted array used in TVS problems for the following binary tree is  __________

(a)   [1,2,3,0,0,4,0,5,6]

(b)   [1,2,3,0,0,4,0,5,0,0,0,0,6]

(c)   [1,2,3,4,5,6]

(d)   [1,2,3,0,0,4,5,6]

(e)   [1,3,5,2,4,6].

297. Let f, t: N→ R+, then t (n)   Ω (f (n)), iff  f(n)   O (t (n)) is known as __________

  Limit rule                                         (b)  Rule of inference                             (c)  Duality rule

  Rule of consequences                     (e)  Rule of symmetricity.

298. Let f, t: N→R ≥ 0, and t (n)  O (f (n)) iff  t(n) ≤  c.f (n) where c is positive real constant and n ≥ no, then no is ___________

  Upper bound    (b)  Lower bound                                   (c)  Duality value

  Threshold value (e)  Maximum value.

299. For the 15-puzzle problem if the initial arrangement is as follows, then the value of ‘x’ used to find the reachability is ________

1 3 5 7

9 11 14

2 4 8 6

10 12 15 13

  0                      (b)  1                       (c)  8                       (d)  14                     (e)  13.

300. The time taken by nondeterministic sorting algorithm is ______

  O(1)                  (b)  O(log n)            (c)  O(n2)                 (d)  O(n log n)          (e)  O(n).

Answers

291.Answer :  (a)

Reason:  According to the definition of the Omega notation.

292.Answer :  (e)

Reason:  According to the asymptotic notational rules,   notation is equivalence one

293.Answer :  (b)

Reason:  The remaining all belongs to divide and conquer paradigm.

294.Answer :  (c)

Reason:  Because of the memory functions in OBST we calculate c(i, j )’s, r(i, j)’s and w(i , j)’s.

295.Answer :  (d)

Reason:  According to the need of the traverse, Backtracking algorithm uses stack and Branch-and-bound algorithms uses queue.

296.Answer :  (b)

Reason:  For any null node in the full binary tree it stores zeros.

297.Answer :  (c)

Reason:  According to the asymptotic notational rules.

298.Answer :  (d)

Reason:  According to the definition of Big Oh ( O )

299.Answer :  (a)

Reason: 

Accordingly the EMPTY cell is there in non shaded position and hence value of ‘x’ is 0, otherwise value of ‘x’ is 1.

300.Answer :  (e)

Reason:  As there it sorts the entries in one loop. Because in nondeterministic sorting algorithms the abstractions are used which can’t be programmed.

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