1. Heapsort O(n lg n) worst case Another design paradigm –Use of a data structure (heap) to manage information during execution of algorithm Comparision-based Sorting Algorithm Analysis of Algorithms1

2. Heap Data Structure Analysis of Algorithms2

3. Heap Property Analysis of Algorithms3

4. A Heap Example Analysis of Algorithms4

5. Heap Data Structure Analysis of Algorithms5

6. Heap Operations Analysis of Algorithms6

7. Heap Operations Analysis of Algorithms7

8. Maintaining Heap Analysis of Algorithms8

9. Runtime Analysis of HEAPIFY Analysis of Algorithms9

10. Formal Runtime Analysis of HEAPIFY Analysis of Algorithms10

11. Formal Runtime Analysis of HEAPIFY Analysis of Algorithms11

12. Building Heap Analysis of Algorithms12

13. Proof of Lemma Analysis of Algorithms13

14. Building Heap Analysis of Algorithms14

15. Build Heap - Example Analysis of Algorithms15

16. Build Heap - Example Analysis of Algorithms16

17. BuildHeap – Tighter Running Time Analysis Analysis of Algorithms17

18. BuildHeap – Tighter Running Time Analysis Analysis of Algorithms18

19. BuildHeap – Tighter Running Time Analysis Analysis of Algorithms19

20. BuildHeap – Tighter Running Time Analysis Analysis of Algorithms20

21. HEAPSORT Analysis of Algorithms21

22. HEAPSORT Analysis of Algorithms22

23. Heapsort - Example Analysis of Algorithms23

24. Heapsort - Example Analysis of Algorithms24

25. Heapsort - Example Analysis of Algorithms25

26. Heapsort – Running Time Analysis Analysis of Algorithms26

27. How fast can we sort? The comparison sorting algorithms such as insertion sort, mergesort, quicksort and heapsort only use comparisons to determine the relative order of elements. The best worst-case running time that we’ve seen for comparison sorting is O(n lg n). Is O(n lg n) the best we can do? Decision trees can help us answer this question. Analysis of Algorithms27

28. Decision-tree example Analysis of Algorithms28

29. Decision-tree example Analysis of Algorithms29

30. Decision-tree model Analysis of Algorithms30

31. Lower bound for decision-tree sorting Analysis of Algorithms31 Corollary. Heapsort and merge sort are asymptotically optimal comparison sorting algorithms.