1. 1 HeapSort CS 3358 Data Structures

2. 2 Heapsort: Basic Idea Problem: Arrange an array of items into sorted order. 1) Transform the array of items into a heap. 2) Invoke the “retrieve & delete” operation repeatedly, to extract the largest item remaining in the heap, until the heap is empty. Store each item retrieved from the heap into the array from back to front. Note: We will refer to the version of heapRebuild used by Heapsort as rebuildHeap, to distinguish it from the version implemented for the class PriorityQ.

3. 3 Transform an Array Into a Heap: Basic Idea We have seen how the consecutive items in an array can be considered as the nodes of a complete binary tree. Note that every leaf is a heap, since a leaf has two empty subtrees. (Note that the last node in the array is a leaf.) It follows that if each child of a node is either a leaf or empty, then that node is the root of a semiheap. We can transform an array of items into a heap by repetitively invoking rebuildHeap, first on the parent of the last node in the array (which is the root of a semiheap), followed by each preceding node in the array (each of which becomes the root of a semiheap).

4. 4 Transform an Array Into a Heap: Example The items in the array, above, can be considered to be stored in the complete binary tree shown at right. Note that leaves 2, 4, 9 & 10 are heaps; nodes 5 & 7 are roots of semiheaps. rebuildHeap is invoked on the parent of the last node in the array (= 9). 72410 9 35 6 543210 427536 76 9 rebuildHeap

5. 5 Transform an Array Into a Heap: Example Note that nodes 2, 4, 7, 9 & 10 are roots of heaps; nodes 3 & 5 are roots of semiheaps. rebuildHeap is invoked on the node in the array preceding node 9. 92410 7 35 6 543210 429536 76 7 rebuildHeap

6. 6 Transform an Array Into a Heap: Example Note that nodes 2, 4, 5, 7, 9 & 10 are roots of heaps; node 3 is the root of a semiheap. rebuildHeap is invoked on the node in the array preceding node 10. 9245 7 310 6 543210 429 36 76 75 rebuildHeap

7. 7 Transform an Array Into a Heap: Example Note that nodes 2, 4, 5, 7 & 10 are roots of heaps; node 3 is the root of a semiheap. rebuildHeap is invoked recursively on node 3 to complete the transformation of the semiheap rooted at 9 into a heap. 3245 7 910 6 543210 423 96 76 75 rebuildHeap

8. 8 Transform an Array Into a Heap: Example Note that nodes 2, 3, 4, 5, 7, 9 & 10 are roots of heaps; node 6 is the root of a semiheap. The recursive call to rebuildHeap returns to node 9. rebuildHeap is invoked on the node in the array preceding node 9. 7245 3 910 6 543210 427 96 76 35 rebuildHeap

9. 9 Transform an Array Into a Heap: Example Note that node 10 is now the root of a heap. The transformation of the array into a heap is complete. 7245 3 96 10 543210 42769 76 35

10. Transform an Array Into a Heap (Cont’d.) Transforming an array into a heap begins by invoking rebuildHeap on the parent of the last node in the array. Recall that in an array-based representation of a complete binary tree, the parent of any node at array position, i, is  (i – 1) / 2  Since the last node in the array is at position n – 1, it follows that transforming an array into a heap begins with the node at position  (n – 2) / 2  =  n / 2  – 1 and continues with each preceding node in the array.

11. 11 Transform an Array Into a Heap: C++ // transform array a[ ], containing n items, into a heap for( int root = n/2 – 1; root >= 0; root – – ) { // transform a semiheap with the given root into a heap rebuildHeap( a, root, n ); }

12. 12 Rebuild a Heap: C++ // transform a semiheap with the given root into a heap void rebuildHeap( ItemType a[ ], int root, int n ) { int child = 2 * root + 1;// set child to root’s left child, if any if( child < n )// if root’s left child exists... { int rightChild = child + 1; if( rightChild a[ child ] ) child = rightChild;// child indicates the larger item if( a[ root ] < a[ child ] ) { swap( a[ root ], a[ child ] ); rebuildHeap( a, child, n ); }

13. 13 Transform a Heap Into a Sorted Array After transforming the array of items into a heap, the next step in Heapsort is to: –invoke the “retrieve & delete” operation repeatedly, to extract the largest item remaining in the heap, until the heap is empty. Store each item retrieved from the heap into the array from back to front. If we want to perform the preceding step without using additional memory, we need to be careful about how we delete an item from the heap and how we store it back into the array.

14. 14 Transform a Heap Into a Sorted Array: Basic Idea Problem: Transform array a[ ] from a heap of n items into a sequence of n items in sorted order. Let last represent the position of the last node in the heap. Initially, the heap is in a[ 0.. last ], where last = n – 1. 1) Move the largest item in the heap to the beginning of an (initially empty) sorted region of a[ ] by swapping a[0] with a[ last ]. 2) Decrement last. a[0] now represents the root of a semiheap in a[ 0.. last ], and the sorted region is in a[ last + 1.. n – 1 ]. 3) Invoke rebuildHeap on the semiheap rooted at a[0] to transform the semiheap into a heap. 4) Repeat steps 1 - 3 until last = -1. When done, the items in array a[ ] will be arranged in sorted order.

15. 15 Transform a Heap Into a Sorted Array: Example We start with the heap that we formed from an unsorted array. The heap is in a[0..7] and the sorted region is empty. We move the largest item in the heap to the beginning of the sorted region by swapping a[0] with a[7]. 7245 3 96 10 42769 54321076 35a[ ]: Heap

16. 16 Transform a Heap Into a Sorted Array: Example a[0..6] now represents a semiheap. a[7] is the sorted region. Invoke rebuildHeap on the semiheap rooted at a[0]. 7245 96 3 427693 54321076 105a[ ]: SemiheapSorted rebuildHeap

17. 17 Transform a Heap Into a Sorted Array: Example rebuildHeap is invoked recursively on a[1] to complete the transformation of the semiheap rooted at a[0] into a heap. 7245 36 9 427639 54321076 105a[ ]: Becoming a HeapSorted rebuildHeap

18. 18 Transform a Heap Into a Sorted Array: Example a[0] is now the root of a heap in a[0..6]. We move the largest item in the heap to the beginning of the sorted region by swapping a[0] with a[6]. 3245 76 9 423679 54321076 105a[ ]: HeapSorted

19. 19 Transform a Heap Into a Sorted Array: Example a[0..5] now represents a semiheap. a[6..7] is the sorted region. Invoke rebuildHeap on the semiheap rooted at a[0]. 324 76 5 423675 54321076 109a[ ]: SemiheapSorted rebuildHeap

20. 20 Transform a Heap Into a Sorted Array: Example Since a[1] is the root of a heap, a recursive call to rebuildHeap does nothing. a[0] is now the root of a heap in a[0..5]. We move the largest item in the heap to the beginning of the sorted region by swapping a[0] with a[5]. 324 56 7 423657 54321076 109a[ ]: HeapSorted rebuildHeap

21. 21 Transform a Heap Into a Sorted Array: Example a[0..4] now represents a semiheap. a[5..7] is the sorted region. Invoke rebuildHeap on the semiheap rooted at a[0]. 32 56 4 723654 54321076 109a[ ]: SemiheapSorted rebuildHeap

22. 22 Transform a Heap Into a Sorted Array: Example a[0] is now the root of a heap in a[0..4]. We move the largest item in the heap to the beginning of the sorted region by swapping a[0] with a[4]. 32 54 6 723456 54321076 109a[ ]: HeapSorted

23. 23 Transform a Heap Into a Sorted Array: Example a[0..3] now represents a semiheap. a[4..7] is the sorted region. Invoke rebuildHeap on the semiheap rooted at a[0]. 3 54 2 763452 54321076 109a[ ]: SemiheapSorted rebuildHeap

24. 24 Transform a Heap Into a Sorted Array: Example rebuildHeap is invoked recursively on a[1] to complete the transformation of the semiheap rooted at a[0] into a heap. 3 24 5 Becoming a Heap 763425 54321076 109a[ ]: Sorted rebuildHeap

25. 25 Transform a Heap Into a Sorted Array: Example a[0] is now the root of a heap in a[0..3]. We move the largest item in the heap to the beginning of the sorted region by swapping a[0] with a[3]. 2 34 5 762435 54321076 109a[ ]: HeapSorted

26. 26 Transform a Heap Into a Sorted Array: Example a[0..2] now represents a semiheap. a[3..7] is the sorted region. Invoke rebuildHeap on the semiheap rooted at a[0]. 34 2 765432 54321076 109a[ ]: SemiheapSorted rebuildHeap

27. 27 Transform a Heap Into a Sorted Array: Example a[0] is now the root of a heap in a[0..2]. We move the largest item in the heap to the beginning of the sorted region by swapping a[0] with a[2]. 32 4 765234 54321076 109a[ ]: HeapSorted

28. 28 Transform a Heap Into a Sorted Array: Example a[0..1] now represents a semiheap. a[2..7] is the sorted region. Invoke rebuildHeap on the semiheap rooted at a[0]. 3 2 765432 54321076 109a[ ]: SemiheapSorted rebuildHeap

29. 29 Transform a Heap Into a Sorted Array: Example a[0] is now the root of a heap in a[0..1]. We move the largest item in the heap to the beginning of the sorted region by swapping a[0] with a[1]. 2 3 765423 54321076 109a[ ]: HeapSorted

30. 30 Transform a Heap Into a Sorted Array: Example a[1..7] is the sorted region. Since a[0] is a heap, a recursive call to rebuildHeap does nothing. We move the only item in the heap to the beginning of the sorted region. 2 765432 54321076 109a[ ]: HeapSorted

31. 31 Transform a Heap Into a Sorted Array: Example Since the sorted region contains all the items in the array, we are done. 765432 54321076 109a[ ]: Sorted

32. 32 Heapsort: C++ void heapsort( ItemType a[ ], int n ) { // transform array a[ ] into a heap for( int root = n/2 – 1; root >= 0; root – – ) rebuildHeap( a, root, n ); for( int last = n – 1; last > 0; ) { // move the largest item in the heap, a[ 0.. last ], to the // beginning of the sorted region, a[ last+1.. n–1 ], and // increase the sorted region swap( a[0], a[ last ] ); last – – ; // transform the semiheap in a[ 0.. last ] into a heap rebuildHeap( a, 0, last ); }

33. 33 Heapsort: Efficiency rebuildHeap( ) is invoked  n / 2  times to transform an array of n items into a heap. rebuildHeap( ) is then called n – 1 more times to transform the heap into a sorted array. From our analysis of the heap-based, Priority Queue, we saw that rebuilding a heap takes O( log n ) time in the best, average, and worst cases. Therefore, Heapsort requires O( [  n / 2  + (n – 1) ] * log n ) = O( n log n ) time in the best, average and worst cases. This is the same growth rate, in all cases, as Mergesort, and the same best and average cases as Quicksort. Knuth’s analysis shows that, in the average case, Heapsort requires about twice as much time as Quicksort, and 1.5 times as much time as Mergesort (without requiring additional storage).

34. 34 Growth Rates for Selected Sorting Algorithms † According to Knuth, the average growth rate of Insertion sort is about 0.9 times that of Selection sort and about 0.4 times that of Bubble Sort. Also, the average growth rate of Quicksort is about 0.74 times that of Mergesort and about 0.5 times that of Heapsort.