05b-heapsorting

8/13/2019 05b-HeapSorting

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Dr. Ahmad R. Hadaegh

A.R. Hadaegh Nati onal University Page 1

Heap

Sorting


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Heap Sort

Heap sort uses a heap as described in the earlier lectures

As we said before, a heap is a binary tree with the following two

properties:

Value of each node is not less than the values stored in each

of its children

The tree is perfectly balanced and the leaves in the level are

all in the leftmost positions


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The procedure is:

The data are transformed into a heap first

Doing this, the data are not necessarily sorted; however, we

know that the largest element is at the root

Thus, start with a heap tree,

Swap the root with the last element

Restore all elements except the last element into a heap

again

Repeat the process for all elements until you are done


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template

void HeapSort(T data[ ], int size){

for (int i = (size/2)-1; i>=0; i--)

MoveDown(data, i, size-1); // creates the heap

for (i=size-1; i>=1; --i)

{

Swap (data[0], data[i]); // move the largest item to data[i]

MoveDown(data, 0, i-1); // restores the heap

}

}

Algorithm and Code for Heap sort

HeapSort(data[ ] ,n)

transform data into a heap

for (i=n-1; i>1; i --)

swap the root with the element in position i ;

restore the heap property for the tree data[0] data[i-1]


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Example of Heap Sort

We first transform thedata into heap

2

8

6

1

10

15

3

12

11

2

101

68

1112

315

The initial tree is formedas follows


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2

1012

68

111

315

We turn the array into a heap first

2

101

68

1112

315

2

1012

158

111

36

2

1012

68

111

315


7/18Dr. Ahmad R. HadaeghA.R. Hadaegh Nati onal University Page 7

2

108

1512

111

36

2

1012

158

111

36

2

1011

1512

81

36

2

108

1512

111

36



15

1011

612

81

32

15

1011

212

81

36

15

1011

212

81

36

2

1011

1512

81

36



Now we start to sort the elements

15

1011

612

81

32

8

1011

612

151

32

12

108

611

151

32

Swap the root

with the last

element

Restore the heap



Swap the root

with the last

element

Restore the heap

12

108

611

151

32

1

108

611

1512

32

11

18

610

1512

32



Swap the root

with the last

element

Restore the heap

3

18

610

1512

112

10

13

68

1512

112

11

18

610

1512

32



Swap the root

with the last

element

Restore the heap

8

12

63

1512

1110

10

13

68

1512

112

2

13

68

1512

1110



Swap the root

with the last

element

Restore the heap

8

12

63

1512

1110

1

82

63

1512

1110

6

82

13

1512

1110



Swap the root

with the last

element

Restore the heap

6

82

13

1512

1110

2

86

13

1512

1110

3

86

12

1512

1110


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Swap the root

with the last

element

Restore the heap

2

86

31

1512

1110

1

86

32

1512

1110

1

86

32

1512

1110



Place the elements into array using

breadth first traversal

1

86

32

1512

1110

1

2

3

6

8

10

11

12

15


18/18D Ah d R H d hA R Hadaegh Nati onal Un iversity Page 18

Complexity of heap sort

The heap sort requires a lot of movement which can be inefficientfor large objects

In the second phase when we start to sort the elements whilekeeping the heap, we exchange n-1times the root with theelement in position iand also restore the heap n-1times whichtakes O(nlogn)

In general:

The first phase, where we turn the array into heap, requiresO(n)steps

And the second phase when we start to sort the elementsrequires

O(n-1)swap + O(nlogn)operations to restore the heap

Total = O(n) + O(nlogn) + O(n-1) = O(nlogn)

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