1. Analysis of Algorithms
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 Algorithms

2. Analysis of Algorithms
Heap Data Structure
Analysis of Algorithms

3. Analysis of Algorithms
Heap Property
Analysis of Algorithms

4. Analysis of Algorithms
A Heap Example
Analysis of Algorithms

5. Analysis of Algorithms
Heap Data Structure
Analysis of Algorithms

6. Analysis of Algorithms
Heap Operations
Analysis of Algorithms

7. Analysis of Algorithms
Heap Operations
Analysis of Algorithms

8. Analysis of Algorithms
Maintaining Heap
Analysis of Algorithms

9. Runtime Analysis of HEAPIFY
Analysis of Algorithms

10. Formal Runtime Analysis of HEAPIFY
Analysis of Algorithms

11. Formal Runtime Analysis of HEAPIFY
Analysis of Algorithms

12. Analysis of Algorithms
Building Heap
Analysis of Algorithms

13. Analysis of Algorithms
Proof of Lemma
Analysis of Algorithms

14. Analysis of Algorithms
Building Heap
Analysis of Algorithms

15. Analysis of Algorithms
Build Heap - Example
Analysis of Algorithms

16. Analysis of Algorithms
Build Heap - Example
Analysis of Algorithms

17. BuildHeap – Tighter Running Time Analysis
Analysis of Algorithms

18. BuildHeap – Tighter Running Time Analysis
Analysis of Algorithms

19. BuildHeap – Tighter Running Time Analysis
Analysis of Algorithms

20. BuildHeap – Tighter Running Time Analysis
Analysis of Algorithms

21. Analysis of Algorithms
HEAPSORT
Analysis of Algorithms

22. Analysis of Algorithms
HEAPSORT
Analysis of Algorithms

23. Analysis of Algorithms
Heapsort - Example
Analysis of Algorithms

24. Analysis of Algorithms
Heapsort - Example
Analysis of Algorithms

25. Analysis of Algorithms
Heapsort - Example
Analysis of Algorithms

26. Heapsort – Running Time Analysis
Analysis of Algorithms

27. Analysis of Algorithms
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 Algorithms

28. Decision-tree example
Analysis of Algorithms

29. Decision-tree example
Analysis of Algorithms

30. Analysis of Algorithms
Decision-tree model
Analysis of Algorithms

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