algorithm questions solution

8/10/2019 Algorithm Questions Solution

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Rayan DankarID:201302670Computer Algorithms Assig.

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Question 1

Solve the following recurrence relations (show every step in your solution)

T (1) = 1

T (n) = T (n-1) + 2* n / 5

T (n) = T (n-1) + 2n / 5

T(n) = T(n-2) + 2(n-1) / 5 + 2n / 5= T(n-2) + 4n/52/5

T(n) = T(n-3) + 2(n-2) / 5 + 4n/52/5

= T(n-3) + 6n/56/5

T(n) = T(n-4) + 2(n-3) / 5 + 6n/56/5

= T(n-4) + 8n / 512 / 5

.

.

.

T(n) = T(n-i) + 2*i*n / 52(i1) / 5

T(n) = T(1) + 2(n-1)(n) / 5 2(n-2) / 5T(n) = T(1) + 2n

2/ 54n / 5 + 4 / 5

O( n2)

T(n-1) = T(n-2) + 2(n-1) / 5

T(n-2) = T(n-3) + 2(n-2) / 5

T(n-3) = T(n-4) + 2(n-3) / 5

n-i = 1i = n-1


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T (1) = 1

T (n) = 3 * T (n / 2) + 2

T(n) = 3T(n / 2) + 2

T(n) = 3[ 3T(n / 4) + 2 ] + 2= 9T(n / 4) + 3*2 + 2

T(n) = 9[ 3T(n / 8) + 2 ] + 3*2 + 2

= 27T(n / 8) + 9*2 + 3*2 + 2

T(n) = 27[ 3T(n / 16 ) +2 ] + 9*2 +3*2 + 2

= 81T(n / 16) + 27*2 + 9*2 +3*2 + 2

= 81T(n / 16) + 2[ 27 + 9 +3 + 1 ].

..

T(n) = 3i

* T(n / 2i) + 2[3

i-1+ 3

i-2+ . 3

0]

T(n) = 3i

* T(n / 2i) + 2[3

i1 ]

T(n) = 3log

2n * T(1) + 2 [3

log2

n1 ]

O(3log

2n)

T(n / 2) = 3T(n / 4) + 2

T(n / 4) = 3T(n / 8) + 2

T(n / 8) = 3T(n / 16 ) +2

n / 2i= 1n = 2

ii = log2

n


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T (n) = n +

T (1) = 1

T (n) = n + T(i)

T (n-1) = n1 + T(i)

T(n)T(n-1) = 1 + T(n-1) T(n) = 2T(n-1) + 1

T(n) = 2T(n-1) + 1

T(n) = 2[ 2T(n-2) + 1 ] + 1

= 4T(n-2) + 2 +1

T(n) = 4[ 2T(n-3) +1 ] + 2 + 1

= 8T(n-3) + 4 + 2 +1

.

.

.

T(n) = 2i* T(n-i) + 2

i+11

T(n) = 2n-1

* T(1) + 2n1

O( 2n)

T(n-1) = 2T(n-2) + 1

T(n-2) = 2T(n-3) +1

n-i = 1i= n -1


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Question 2

Convert the following postfi x expression to i nf ix form:

5 9 8 + 4 6 * * 7 + *

Expression Stack

5 9 8 + 4 6 * * 7 + * NULL

9 8 + 4 6 * * 7 + * 5

8 + 4 6 * * 7 + * 95

+ 4 6 * * 7 + * 8

95

4 6 * * 7 + * 9+8

56 * * 7 + * 4

9+8

5

* * 7 + * 6

49+8

5

* 7 + * 4*69+8

5

7 + * (9+8)*4*65

+ * 7

(9+8)*4*65

* (9+8)*4*6+75

NULL 5+((9+8)*4*6+7)

5 * ((9 + 8) * 4 * 6 + 7)


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Question 3:

Given the following Binary search tree, Show the order of the visited nodes when we traversethis tree using:

Preorder traversal:

rootleftright :

FBADCEGIH

Inorder traversal:

leftrootright :

ABCDEFGHI

Postorder traversal:

leftrightroot :

ACEDBHIGF


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Question 4

Draw the binary search tree that results from inserting the keys 12, 17, 20, 13, 14, 10, 11, 8 intoan initially empty tree.

Draw the binary tree that result from

deletingthe

node

withkey

valu

e 16

fromthe

follo

wing

tree:


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Question 5

What is a heap, what are the conditions that should be specified inside a heap, and what is therelation between the indexes of parents and children inside of a heap.

A heap is a tree where the value of the node is less than or equal to that of its descendants (minheap) or vice versa (max heap).

Parent:

Left child: 2*+ 1

Right child: 2*1

Given the following max heap:

Illustrate the

effect of:

a. Inserting a

nodewhos

e key

is Pinto

the

follo

wing

heap


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b. Removing the largest node from the resulting heap

Question 6

1)Show the detailed workings for the fir st two parti tioning of the foll owing array duri ng

Quicksort.

The answer should show the behavior of vari ables as the parti tioning proceeds.

Quick Sort


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publi cvoid quicksort(I TEM [] a, intL , int R)

{

if (R left& & a[R] >= p) R--;

if (L < R) { int temp = a[L] ; a[L ] = a[R]; a[R] = temp; }

}

a[left] = a[R];

a[R] = p;

return R; }

0 1 2 3 4 5 6 7

A 5 3 1 9 8 2 4 7

left=

0

right=7 p=5 L=1

L=2L=3

R=7

R=6

Swap a[3] & a[6]

new a 5 3 1 4 8 2 9 7

left=0

right=7 p=5 L=3L=4

R=6R=5

Swap a[4] & a[5]

new a 5 3 1 4 2 8 9 7


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left=0

right=7 p=5 L=4L=5

R=5R=4

a[0]=a[4]a[4]=5

new a 2 3 1 4 5 8 9 7

I STOPPED BECAUSE THE GIVEN IS WRONG!

PARTITION TAKES ONE PARAMETER HOW COME IT IS

GIVEN 3???

2) When does the best case of Quicksort occurs, when does the worst caseoccurs (explain your answer showing the complexity in each case)

3) Using tree diagrams, show how the above array would be sorted using themax Heapsortalgorithm.

4) What is the complexity of Heapsort? Compare it with that of Quicksort.

Question 7

How many comparisons and swaps are needed to sort 10 numbers in ascendi ng order using?

1. Selection sort:

Select ion Sort

public void selectionSor t(int[ ] arr )

{

int i , j, minI ndex, tmp;

int n = arr .length;for (i = 0; i < n - 1; i++)

{

minI ndex = i;

for (j = i + 1; j < n; j++)

if (arr[j ] < arr [minI ndex])

minI ndex = j;

if (minI ndex != i )


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{

tmp = arr [i] ;

arr[i ] = arr[minI ndex];

arr[minI ndex] = tmp;

}

}}

a. I f the data is originall y sorted in ascending order

i = 0

minIndex = 0

i = 1

minIndex = 1

i = 2

minIndex = 2

i = 3

minIndex = 3

i = 4

minIndex =4

Comparison 9 9 9 9 9

minIndex = i minIndex = i minIndex = i minIndex = i minIndex = i

Swap 0 0 0 0 0

i = 5

minIndex = 5

i = 6

minIndex = 6

i = 7

minIndex = 7

i = 8

minIndex = 8

Total


minIndex = i minIndex = i minIndex = i minIndex = i

Swap 0 0 0 0 0

b. I f the data is originall y sorted in descending order

i = 0minIndex = 0

j = 9minIndex = 9

i = 1minIndex = 1

j = 9minIndex = 9

i = 2minIndex = 2

j = 9minIndex = 9

i = 3minIndex = 3

j = 9minIndex = 9

i = 4minIndex = 4

j = 9minIndex = 9


minIndex =! i minIndex! = i minIndex != i minIndex! = i minIndex != i

Swap 1 1 1 1 1

i = 5

minIndex = 5

j = 9

minIndex = 9

i = 6

minIndex = 6

j = 9

minIndex = 9

i = 7

minIndex = 7

j = 9

minIndex = 9

i = 8

minIndex = 8

j = 9

minIndex = 9

Total


minIndex! = i minIndex != i minIndex! = i minIndex != iSwap 1 1 1 1 9

2. I nserti on sort:


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Ins ert ion Sort

void inser tionSort(i nt[ ] arr )

{

in t i, j, newValue;

for (i = 1; i < arr .length; i++){

newValue = arr[ i] ;

j = i;

while (j > 0 && arr[ j - 1] > newValue)

{

arr[ j] = arr[j - 1];

j--;

}

arr[ j] = newValue;

}

}


i = 1 i = 2 i = 3 i = 4 i = 5


Swap 0 0 0 0 0

i = 6 i = 7 i = 8 i = 9 Total


Swap 0 0 0 0 0

b. If the data is originally sorted in descending order

i = 1 i = 2 i = 3 i = 4 i = 5

Comparison 9 9 9 9 9Swap 1 1 1 1 1

i = 6 i = 7 i = 8 i = 9 Total


Swap 1 1 1 1 9


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3. Bubble sort:Bubble Sort

public void bubbleSort(int[] arr)

{

int j = 0;

int tmp;

for (int i = 0; i < arr.length-1; i++)

for (int j = arr.length-1; j > i; j--)

if (arr[j-1] > arr[j] )

{

tmp = arr[j-1];

arr[j-1] = arr[j];

arr[j] = tmp;

}

}


i = 0

j = 9

i = 1

j = 9

i = 2

j = 9

i = 3

j = 9

i = 4

j = 9


arr[j-

1]!>arr[j]

arr[j-

1]!>arr[j]

arr[j-

1]!>arr[j]

arr[j-1]!>arr[j] arr[j-1]!>arr[j]

Swap 0 0 0 0 0

i = 5

j = 9

i = 6

j = 9

i = 7

j = 9

i = 8

j = 9

Total


arr[j-1]!>arr[j] arr[j-1]!>arr[j] arr[j-1]!>arr[j] arr[j-1]!>arr[j]

Swap 0 0 0 0 0

b. I f the data is originall y sorted in descending order

i = 0j = 9

i = 1j = 9

i = 2j = 9

i = 3j = 9

i = 4j = 9


arr[j-1] >arr[j] arr[j-1] >arr[j] arr[j-1] >arr[j] arr[j-1] >arr[j] arr[j-1] >arr[j]

Swap 1 1 1 1 1


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i = 5

j = 9

i = 6

j = 9

i = 7

j = 9

i = 8

j = 9

Total


arr[j-1] >arr[j] arr[j-1] >arr[j] arr[j-1] >arr[j] arr[j-1] >arr[j]

Swap 1 1 1 1 9


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Question 8

The following functions are given:

(en)

240 * n

2n 14logn

427 * n

2+ 3 * n + 1 14(logn)

4100nlogn

a) Ar range these functions according to O notation.

(en)2

>

> 40 * n2n > 27 * n

2+ 3 * n + 1 > 100nlogn > 14(logn)

4>

14logn

4

b) Assume these functions are runtimes for 7 algori thms that all solve the same problem. For

which values of n (nearest in teger) whi ch algor ithm has the shor test runtime?

(e

n

)

2

: for n=0

40 * n2n : for n=0

14logn4: for n=1

27 * n2+ 3 * n + 1 : for n=0

14(logn)4

: for n=1

100nlogn : for n=1

: for n=0

c) Assume you have an algor ithm whose runtime is T(n) = n2log(n). For what ranges of n

(order of magni tude estimate) thi s algor ithm wil l r un in a second, an hour and a week on a

200MHz Processor, assuming one instructi on is executed in one clock cycle.

in one second: n2log(n) = 200 * 10

6

in one hour: n2log(n) = 200 * 10

6* 3600

in one hour: n2log(n) = 200 * 10

6* 3600 * 24 * 7


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Question 9

A divide-and-conquer algorithm for multi plying two N x N matrices reduces the calculation to

7 products of

matr ices and 18 matr ix additi ons of N x N matri ces.

A matr ix addition can be implemented by a double for -loop li ke this:

for i = 1 to N

for j =1 to N

a(i,j) = b(i,j) + c(i,j)

Give a recur rence soluti on for the runtime T (N) required to multi ply two N x N matri ces and

using Masters theorem, give an O-estimate for T (N).

T ( n ) = 7T (

) + 18n

2

Using Masters theorem:

a= 7 b=2 c=2

log27 > 2

( n log27)


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b) Suppose now that the set S is given in sorted order. Design an algor i thm to

solve the pro blem in t im e O (n) .

i = 1;

j = n;

for( i = 1 j )

{ O ( n )

for ( j =n1 i )

{

if (S[i] + S[j] < x)

i++;

else if (S[i] + S[j] >x)

j;

else

break;

}

}

O ( n )


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Consider the weighted undirected graph below.

1. In what order would the vertices be traversed, using breadth-first search starting atvertex E, assuming that adjacent vertices are visited in alphabetical order?

ECBADFG

2. In what order would the vertices be traversed, using depth-first search starting at vertexB, assuming that adjacent vertices are visited in alphabetical order?

BADCEGF

3. Illustrate the execution of Kruskals algorithm applied to the graph. Clearly show theresult after each step of the algorithm by listing the edges added in the order in which

they are added.


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2 + 3 + 7 + 7 + 9 + 12 +12

= 52

4.Illustrate the execution of Prims algorithmapplied to the graph, starting at vertex C. Clearly show the result after each step of the

algorithm by listing the edges added in the order in which they are added.


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3 + 7 + 9 + 12 + 2 + 12 + 7

= 52

5.Illustrate the execution of Dijkstras algorithm applied to the graph, starting at vertex

C. Clearly show the resul t after each step of the algori thm by li sting the edges added

in the order in which they are added.

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