CS 2468: Assignment 2 (Due Week 9, Tuesday. Drop a hard copy in Mail Box 75 or hand in during the lecture)

Use array representation (double a[]) to implement BinaryTree. Each cell in the array is a real number. You can just set the size of the array as 1000.

Hint: (1) Please read the file BinaryTree.java on Week 5’s website, which uses BNode to implement BinaryTree. (2) You can also look the java code for heap.

Trees 1

Array-Based Representation of Binary Trees

• nodes are stored in an array

Trees 2

…

let rank(node) be defined as follows: rank(root) = 1 if node is the left child of parent(node),

rank(node) = 2*rank(parent(node)) if node is the right child of parent(node),

rank(node) = 2*rank(parent(node))+1

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6 74 5

10 11

A

HG

FE

D

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J

Array-Based Representations

• How many cells can be wasted with array based representation?

Trees 3

Linked Structure for Binary Trees

• A node is represented by an object storing– Element– Parent node– Left child node– Right child node

• Node objects implement the Position ADT

Trees 4

B

DA

C E

B

A D

C E

Linked Structure for Binary Trees

• A node is represented by an object storing– Element– Parent node– Left child node– Right child node

• Node objects implement the Position ADT

Trees 5

B

DA

C

G

B

A D

CE

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D

F G

Full Binary Tree

• A full binary tree: – All the leaves are at the bottom level– All nodes which are not at the bottom level have two children. – A full binary tree of height h has 2h leaves and 2h-1 internal

nodes.

Trees 6A full binary tree of

height 2

This is not a full binary tree.

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23

4

5 6 7

Properties of Proper Binary Trees

• Notationn number of nodese number of external

nodesi number of internal

nodesh height

Trees 7

Properties for proper binary tree: e i 1 n 2e 1 h i e 2h

h log2 e

No need to

remember.

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1514

Depth(v): no. of ancestors of v

Trees 8

Make Money Fast!

1. Motivations References2. Methods

2.1 StockFraud

2.2 PonziScheme

1.1 Greed 1.2 Avidity2.3 BankRobbery

0

1

2

1

2 2 2 2

1

Algorithm depth(T,v)If T.isRoot(v) then return 0;else return 1+depth(T, T.parent(v))

Height(T): the height of T is max depth of an external node

Trees 9

Algorithm height1(T) h=0;

for each vT.positions() do if T.isExternal(v) then h=max(h, depth(T, v)) return h

T.positions() holds all nodes in T. See the method in LinkedBinaryTree.java. The max

is 2.Make Money Fast!

1. Motivations2.

Methods

2.1 StockFraud

2.2 PonziScheme1.1 Greed 1.2 Avidity

2.3 BankRobbery

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2

0

2 2 2 2

1

Height(T,v):

Trees 10

Algorithm height2(T,v)if T.isExternal(v) then return 0else h=0 for each wT.children(v) do h=max(h, height2(T, w)) return 1+h

1. If v is an external node, then height of v is 0.

2. Otherwise, the height of v is one +max height of a child of v.

Height(T,v):

Trees 11

Make Money Fast!

1. Motivations References2. Methods

2.1 StockFraud

2.2 PonziScheme

1.1 Greed 1.2 Avidity2.3 BankRobbery

2

1

0

1

0 0 0 0

1

Algorithm height2(T,v)if T.isExternal(v) then return 0else h=0 for each wT.children(v) do h=max(h, height2(T, w)) return 1+h

Priority Queues 12

Part-D1

Priority Queues

Priority Queues 13

Priority Queue ADT • A priority queue stores a

collection of entries• Each entry is a pair

(key, value)• Main methods of the Priority

Queue ADT– insert(k, x)

inserts an entry with key k and value x

– removeMin()removes and returns the entry with smallest key

• Additional methods– min()

returns, but does not remove, an entry with smallest key

– size(), isEmpty()

• Applications:– Standby flyers– Auctions– Stock market

Priority Queues 14

Entry ADT

• An entry in a priority queue is simply a key-value pair

• Priority queues store entries to allow for efficient insertion and removal based on keys

• Methods:– key(): returns the key for

this entry– value(): returns the value

associated with this entry

• As a Java interface:/** * Interface for a key-value * pair entry **/public interface Entry { public Object key(); public Object value();}

Priority Queues 15

Comparator ADT

• A comparator encapsulates the action of comparing two objects according to a given total order relation

• A generic priority queue uses an auxiliary comparator

• The comparator is external to the keys being compared

• When the priority queue needs to compare two keys, it uses its comparator

• The primary method of the Comparator ADT:– compare(a, b): Returns an

integer i such that i < 0 if a < b, i = 0 if a = b, and i > 0 if a > b; an error occurs if a and b cannot be compared.

Priority Queues 16

Priority Queue Sorting

• We can use a priority queue to sort a set of comparable elements

1. Insert the elements one by one with a series of insert operations

2. Remove the elements in sorted order with a series of removeMin operations

• The running time of this sorting method depends on the priority queue implementation

Algorithm PQ-Sort(S, C)Input sequence S, comparator C for the elements of SOutput sequence S sorted in increasing order according to CP priority queue with

comparator Cwhile S.isEmpty ()

e S.removeFirst ()P.insert (e, 0)

while P.isEmpty()e

P.removeMin().key()S.insertLast(e)

Priority Queues 17

Sequence-based Priority Queue

• Implementation with an unsorted list

• Performance:– insert takes O(1) time since

we can insert the item at the beginning or end of the sequence

– removeMin and min take O(n) time since we have to traverse the entire sequence to find the smallest key

• Implementation with a sorted list

• Performance:– insert takes O(n) time since

we have to find the place where to insert the item

– removeMin and min take O(1) time, since the smallest key is at the beginning

4 5 2 3 1 1 2 3 4 5

Priority Queues 18

Selection-Sort

• Selection-sort is the variation of PQ-sort where the priority queue is implemented with an unsorted list

• Running time of Selection-sort:1. Inserting the elements into the priority queue with n insert

operations takes O(n) time2. Removing the elements in sorted order from the priority queue

with n removeMin operations takes time proportional to 1 2 …n-1

• Selection-sort runs in O(n2) time

Priority Queues 19

Selection-Sort Example Sequence S Priority Queue PInput: (7,4,8,2,5,3,9) ()

Phase 1(a) (4,8,2,5,3,9) (7)(b) (8,2,5,3,9) (7,4).. .. ... . .(g) () (7,4,8,2,5,3,9)

Phase 2(a) (2) (7,4,8,5,3,9)(b) (2,3) (7,4,8,5,9)(c) (2,3,4) (7,8,5,9)(d) (2,3,4,5) (7,8,9)(e) (2,3,4,5,7) (8,9)(f) (2,3,4,5,7,8) (9)(g) (2,3,4,5,7,8,9) ()

Running time:

1+2+3+…n-1=O(n2).

No advantage is

shown by using

unsorted list.

Priority Queues 20

Complete Binary Trees

1 5 62 43

A binary tree of height 3 has the first p leaves at the bottom for p=1, 2, …, 8=23.

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.

Priority Queues 21

Complete Binary Trees

1 5 6 7 82 43

A binary tree of height 3 has the first p leaves at the bottom for p=1, 2, …, 8=23.

Once the number of nodes in the

complete binary tree is fixed, the

tree is fixed.

For example, a complete binary

tree of 15 node is shown in the slide. A CBT of 14 nodes is the one without

node 8.

A CBT of 13 node is the one without

nodes 7 and 8.

Priority Queues 22

Heaps • A heap is a binary tree storing

keys at its nodes and satisfying the following properties:– Heap-Order: for every node v

other than the root,key(v) key(parent(v))

– Complete Binary Tree: let h be the height of the heap• for i 0, … , h 1, there are

2i nodes of depth i• at depth h 1, the internal

nodes are to the left of the external nodes

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• The last node of a heap is the rightmost node of depth h

• Root has the smallest key

last nodeA full binary without the last few nodes at the bottom on the right.

Priority Queues 23

Height of a Heap • Theorem: A heap storing n keys has height O(log n)

Proof: (we apply the complete binary tree property)– Let h be the height of a heap storing n keys– Since there are 2i keys at depth i 0, … , h 1 and at least one key at

depth h, we have n 1 2 4 … 2h1 1=2h.

– Thus, n 2h , i.e., h log n

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2h1

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keys

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h1

h

depth

Priority Queues 24

Heaps and Priority Queues

• We can use a heap to implement a priority queue• We store a (key, element) item at each internal node• We keep track of the position of the last node• For simplicity, we show only the keys in the pictures

(2, Sue)

(6, Mark)(5, Pat)

(9, Jeff) (7, Anna)

Priority Queues 25

Insertion into a Heap • Method insertItem of the

priority queue ADT corresponds to the insertion of a key k to the heap

• The insertion algorithm consists of three steps– Create the node z (the new

last node). Store k at z– Put z as the last node in the

complete binary tree.– Restore the heap-order

property, i.e., upheap (discussed next)

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insertion node

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79 1

z

z

Priority Queues 26

Upheap• After the insertion of a new key k, the heap-order property may be

violated• Algorithm upheap restores the heap-order property by swapping k along

an upward path from the insertion node• Upheap terminates when the key k reaches the root or a node whose

parent has a key smaller than or equal to k • Since a heap has height O(log n), upheap runs in O(log n) time

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79 6z

Priority Queues 27

An example of upheap

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Different entries may have the same key. Thus, a key may appear more than once.

Priority Queues 28

Removal from a Heap

• Method removeMin of the priority queue ADT corresponds to the removal of the root key from the heap

• The removal algorithm consists of three steps– Replace the root key with

the key of the last node w– Remove w – Restore the heap-order

property .e., downheap(discussed next)

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last node

w

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9w

new last node

Priority Queues 29

Downheap• After replacing the root key with the key k of the last node, the heap-

order property may be violated• Algorithm downheap restores the heap-order property by swapping key

k along a downward path (always use the child with smaller key) from the root

• Downheap terminates when key k reaches a leaf or a node whose children have keys greater than or equal to k

• Since a heap has height O(log n), downheap runs in O(log n) time

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Priority Queues 30

An example of downheap

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9

Priority Queues 31

Priority Queue ADT using a heap

• A priority queue stores a collection of entries

• Each entry is a pair(key, value)

• Main methods of the Priority Queue ADT– insert(k, x)

inserts an entry with key k and value x O(log n)

– removeMin()removes and returns the entry with smallest key O(log n)

• Additional methods– min()

returns, but does not remove, an entry with smallest key O(1)

– size(), isEmpty() O(1)

Running time of Size(): when constructing the heap, we keep the size in a variable. When inserting or removing a node, we update the value of the variable in O(1) time.

isEmpty(): takes O(1) time using size(). (if size==0 then …)

Priority Queues 32

Heap-Sort

• Consider a priority queue with n items implemented by means of a heap– the space used is O(n)

– methods insert and removeMin take O(log n) time

– methods size, isEmpty, and min take time O(1) time

• Using a heap-based priority queue, we can sort a sequence of n elements in O(n log n) time

• The resulting algorithm is called heap-sort

• Heap-sort is much faster than quadratic sorting algorithms, such as insertion-sort and selection-sort

Priority Queues 33

Vector-based Heap Implementation

• We can represent a heap with n keys by means of an array of length n 1

• For the node at rank i– the left child is at rank 2i– the right child is at rank 2i 1

• Links between nodes are not explicitly stored

• The cell of at rank 0 is not used• Operation insert corresponds to

inserting at rank n 1• Operation removeMin corresponds

to removing at rank n• Yields in-place heap-sort• The parent of node at rank i is i/2

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1 2 3 4 50

Priority Queues 34

Merging Two Heaps

• We are given two two heaps and a key k

• We create a new heap with the root node storing k and with the two heaps as subtrees

• We perform downheap to restore the heap-order property

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Priority Queues 35

• We can construct a heap storing n given keys in using a bottom-up construction with log n phases

• In phase i, pairs of heaps with 2i 1 keys are merged into heaps with 2i11 keys

Bottom-up Heap Construction (§2.4.3)

2i 1 2i 1

2i11

Priority Queues 36

Example

1516 124 76 2023

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1516

5

124

11

76

27

2023

Priority Queues 37

Example (contd.)

25

1516

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124

11

96

27

2023

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2516

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125

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911

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2723

Priority Queues 38

Example (contd.)

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2516

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911

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2723

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2516

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911

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2723

Priority Queues 39

Example (end)

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Priority Queues 40

Analysis• We visualize the worst-case time of a downheap with a proxy path that

goes first right and then repeatedly goes left until the bottom of the heap (this path may differ from the actual downheap path)

• Since each node is traversed by at most two proxy paths, the total number of nodes of the proxy paths is O(n)

• Thus, bottom-up heap construction runs in O(n) time • Bottom-up heap construction is faster than n successive insertions and

speeds up the first phase of heap-sort