Lecture 8 COMPSCI 220 - AP G. Gimel'farb 1

Algorithm HeapSort

• J. W. J. Williams (1964): a special binary tree calledheap to obtain an O(n log n) worst-case sorting

• Basic steps:– Convert an array into a heap in linear time O(n)

– Sort the heap in O(n log n) time by deleting n timesthe maximum item because each deletion takes thelogarithmic time O(log n)

Lecture 8 COMPSCI 220 - AP G. Gimel'farb 2

Complete Binary Tree: linear array representation

Lecture 8 COMPSCI 220 - AP G. Gimel'farb 3

Complete Binary Tree

• A complete binary tree of the height h contains between2h and 2h+1−1 nodes

• A complete binary tree with the n nodes has the heightlog2n

• Node positions are specified by the level-order traversal(the root position is 1)

• If the node is in the position p then:– the parent node is in the position p/2– the left child is in the position 2p– the right child is in the position 2p + 1

Lecture 8 COMPSCI 220 - AP G. Gimel'farb 4

Binary Heap

• A heap consists of a complete binary tree ofheight h with numerical keys in the nodes

• The defining feature of a heap:

the key of each parent node is greater thanor equal to the key of any child node

• The root of the heap has the maximum key

Lecture 8 COMPSCI 220 - AP G. Gimel'farb 5

Binary Heap: linear array representation

Lecture 8 COMPSCI 220 - AP G. Gimel'farb 6

Binary Heap: insert a new key

• Heap of k keys into a heap of k + 1 keys

• Logarithmic time O( log k ) to insert a new key:

– Create a new leaf position k + 1 in the heap

– Bubble (or percolate) the new key up byswapping it with the parent if the parent key issmaller than the new key

Lecture 8 COMPSCI 220 - AP G. Gimel'farb 7

Binary Heap:an example ofinserting a key

Lecture 8 COMPSCI 220 - AP G. Gimel'farb 8

Binary Heap:delete the maximum key

• Heap of k keys into a heap of k − 1 keys

• Logarithmic time O( log k ) to delete the root(or maximum) key:– Remove the root key– Delete the leaf position k and move its key into the

root

– Bubble (percolate) the root key down by swappingit with the largest child if that child is greater

Lecture 8 COMPSCI 220 - AP G. Gimel'farb 9

Binary Heap:an example ofdeleting themaximum key

Lecture 8 COMPSCI 220 - AP G. Gimel'farb 10

Linear Time Heap Construction

• Do not use n insertions → O(n log n) time!• Alternative O(n) procedure uses a recursively

defined heap structure:

– form recursively the left and right subheaps– percolate the root down to establish the heap order

everywhere

Lecture 8 COMPSCI 220 - AP G. Gimel'farb 11

Non-recursive Heap Building

• Nodes percolate down in reverse level order– Each node p is processed after its descendants

have been already processed– Leaves need not be percolated down

• Worst-case time T(h) to build a heap of height h:T(h) = 2T(h−1) + ch → T(h) = O(2h)

– Form two subheaps of height at most h − 1– Percolate the root down a path of length at most h

Lecture 8 COMPSCI 220 - AP G. Gimel'farb 12

Time to Build a Heap

!

T(h) = 2T(h "1) + ch

2T(h "1) = 22T(h " 2) + 2c(h "1)

... ... ...

2h"2

T(2) = 2h"1

T(1) + 2h"2

c # 2

2h"1

T(1) = 2hT(0) + 2

h"1c #1= 2

h"1c #1

!

T(h) = c " 1" 2h#1 + 2 " 2h#2 + ...+ (h # 2) " 22 + (h #1) " 21 + h " 20( )= c " 2h+1

# h #1( )

Lecture 8 COMPSCI 220 - AP G. Gimel'farb 13

Worst-case Time Complexity

• A heap of n nodes is of height h = log2n sothat 2h ≤ n ≤ 2h+1−1

• Therefore, the time for converting an array into aheap is linear: T(h) = O(2h), or T(n) = O(n)

• To sort a heap, the maximum element is deletedn times, so that the worst-case time complexityof HeapSort is O( n log n )– Each deletion takes logarithmic time O( log n)

Lecture 8 COMPSCI 220 - AP G. Gimel'farb 14

Steps of HeapSort

215202550831706591h

28

10/9

70915091

20318H

EAPIFY

15312591220506570a

9/88/7

7/66/5

5/44/3

3/22/1

1/0p/i

Lecture 8 COMPSCI 220 - AP G. Gimel'farb 15

Steps of HeapSort

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15201531

91

91

1565R.h.7020252831506515a2

1520252831506570H9

250270Restore

the heap(R.h.)

1520255083170652a1

Lecture 8 COMPSCI 220 - AP G. Gimel'farb 16

Steps of HeapSort

2815252031h6

1520

91

91

1531R.h.7065502820253115a4

152820253150h7

15251550R.h.

7065252820503115a3

Lecture 8 COMPSCI 220 - AP G. Gimel'farb 17

Steps of HeapSort

821520h4

815

91

91

820R. h.7065503125152208a6

81522025h5

225R. h.7065503181525202a5

Lecture 8 COMPSCI 220 - AP G. Gimel'farb 18

Steps of HeapSort

917065503125201582a9

28h2

91

91

28R. h.7065503125201582a8

2815h3

815R. h.7065503125202158a7

s o r t e d a r r a y