chapter - 6.ppt
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The heap data structure is a nearly complete binary tree implemented as an array. There are two types of heaps: max-heaps and min-heaps. The MAX-HEAPIFY algorithm maintains the heap property by allowing a value to "float down" the tree. BUILD-MAX-HEAP builds a max-heap by calling MAX-HEAPIFY on each node, and HEAPSORT sorts an array using the heap structure.
![Maintaining the Heap Property The function of MAX-HEAPIFY is to let the value at A[i] “float down” in the max-heap so that the sub-tree rooted at index i becomes a max-heap](https://image.slidesharecdn.com/chapter-6ppt3761/85/chapter-6-ppt-5-320.jpg)
![Maintaining the Heap Property MAX-HEAPIFY(A, i) l LEFT(i) r RIGHT(i) if l <= heap-size[A] and A[l] >A [i] then largest l else largest i](https://image.slidesharecdn.com/chapter-6ppt3761/85/chapter-6-ppt-6-320.jpg)
![Maintaining the Heap Property If r<= heap-size[A] and A[r] > A[largest] then largest r If largest ≠ i then exchange A[i] A[largest] MAX-HEAPIFY(A, largest)](https://image.slidesharecdn.com/chapter-6ppt3761/85/chapter-6-ppt-7-320.jpg)
![Building a Heap BUILD-MAX-HEAP(A) heap-size[A] length[A] for i length[A]/2 downto 1 do MAX-HEAPIFY(A, i)](https://image.slidesharecdn.com/chapter-6ppt3761/85/chapter-6-ppt-11-320.jpg)
![Heapsort Algorithm HEAPSORT(A) BUILD-MAX-HEAP(A) for I length[A] downto 2 do exchange A[1] A[i] heap-size[A] heap-size[A] -1 MAX-HEAPIFY(A,1)](https://image.slidesharecdn.com/chapter-6ppt3761/85/chapter-6-ppt-19-320.jpg)
chapter - 6.ppt
- 1. Heap Data Structure The heap data structure is an array object that can be viewed as a nearly complete binary tree.
- 2. Heap 1 2 3 4 5 6 7 8 9 10 (a) 16 2 9 8 3 10 7 14 4 1
- 3. Heap (b) 1 4 2 3 9 7 8 10 14 16 5 4 3 2 1 10 9 8 7 6
- 4. Heap There are two kinds of binary heaps max-heaps min heaps
- 5. Maintaining the Heap Property The function of MAX-HEAPIFY is to let the value at A[i] “float down” in the max-heap so that the sub-tree rooted at index i becomes a max-heap
- 6. Maintaining the Heap Property MAX-HEAPIFY(A, i) l LEFT(i) r RIGHT(i) if l <= heap-size[A] and A[l] >A [i] then largest l else largest i
- 7. Maintaining the Heap Property If r<= heap-size[A] and A[r] > A[largest] then largest r If largest ≠ i then exchange A[i] A[largest] MAX-HEAPIFY(A, largest)
- 8. Maintaining the Heap Property 1 2 3 4 5 6 7 8 9 10 i (a) 16 2 9 14 3 10 7 4 8 1
- 9. Maintaining the Heap Property 1 2 3 4 5 6 7 8 9 10 i (b) 16 2 9 4 3 10 7 14 8 1
- 10. Maintaining the Heap Property 1 2 3 4 5 6 7 8 9 10 i (c) 16 2 9 6 3 10 7 14 4 1
- 11. Building a Heap BUILD-MAX-HEAP(A) heap-size[A] length[A] for i length[A]/2 downto 1 do MAX-HEAPIFY(A, i)
- 12. Building a Heap 3 2 16 1 10 4 7 8 14 9 A
- 13. Building a Heap 1 2 3 4 5 6 7 8 9 10 i (a) 4 14 9 2 10 3 16 1 8 7
- 14. Building a Heap 1 2 3 4 5 6 7 8 9 10 i (b) 4 14 9 2 10 3 16 1 8 7
- 15. Building a Heap 1 2 3 4 5 6 7 8 9 10 i (c) 4 2 9 14 10 3 16 1 8 7
- 16. Building a Heap 1 2 3 4 5 6 7 8 9 10 i (d) 4 2 9 14 3 10 16 1 8 7
- 17. Building a Heap 1 2 3 4 5 6 7 8 9 10 i (e) 4 2 9 14 3 10 7 16 8 1
- 18. Building a Heap 1 2 3 4 5 6 7 8 9 10 (f) 16 2 9 8 3 10 7 14 4 1
- 19. Heapsort Algorithm HEAPSORT(A) BUILD-MAX-HEAP(A) for I length[A] downto 2 do exchange A[1] A[i] heap-size[A] heap-size[A] -1 MAX-HEAPIFY(A,1)
- 20. Heapsort Algorithm (a) 16 2 9 8 3 10 7 14 4 1
- 21. Heapsort Algorithm (b) i 14 2 9 4 3 10 7 8 1 16
- 22. Heapsort Algorithm (c) i 10 2 1 4 3 9 7 8 16 14
- 23. Heapsort Algorithm (d) i 9 1 4 2 3 7 8 16 14 10
- 24. Heapsort Algorithm (e) i 8 1 4 3 2 7 16 14 10 19
- 25. Heapsort Algorithm (f) i 7 1 3 2 4 16 14 10 9 8
- 26. Heapsort Algorithm (g) i 4 1 3 2 16 14 10 9 8 7
- 27. Heapsort Algorithm (h) i 3 1 2 16 14 10 9 8 7 4
- 28. Heapsort Algorithm (i) i 2 1 16 14 10 9 8 7 4 3
- 29. Heapsort Algorithm (j) i 1 2 16 14 10 9 8 7 4 3
- 30. Heapsort Algorithm k 3 4 7 2 9 1 16 14 10 8 A