1. Dr.Surasak Mungsing E-mail: Surasak.mu@spu.ac.th
CSE 221/ICT221 Analysis and Design of Algorithms Lecture 05-2: Analysis of time Complexity of Priority Queue
Dr.Surasak Mungsing
Feb-19

2. Priority Queue Collection of elements.
Each element has a priority or key.
Supports following operations:
isEmpty
size
add/put an element into the priority queue
get element with min/max priority
remove element with min/max priority
Feb-19

3. Min Tree Example 2 4 9 3 8 7
Root is the minimum element
Feb-19

4. Max Tree Example 9 4 8 2 7 3 1
Root is the maximum element
Feb-19

5. Min Heap Definition complete binary tree min tree2 4 6 7 9 3 8
Complete binary tree with 9 nodes that is also a min tree.
Feb-19

6. Max Heap With 9 Nodes 9 8 6 7 2 5 1
Complete binary tree with 9 nodes that is also a max tree.
Feb-19

7. Heap Height What is the height of an n node heap ?
Since a heap is a complete binary tree, the height of an n node heap is log2 (n+1).
Feb-19

8. A Heap Is Efficiently Represented As An Array9 8 6 7 2 5 1 9 8 7 6 2 5 1 3 4 10

9. Moving Up And Down A Heap9 8 6 7 2 5 1 3 4

10. Putting An Element Into A Max Heap9 8 6 7 2 5 1 7
Complete binary tree with 10 nodes.

11. Putting An Element Into A Max Heap9 8 7 6 7 2 6 5 1 5 7
New element is 5.

12. Putting An Element Into A Max Heap9 8 7 6 7 2 6 5 1 7 7
New element is 20.

13. Putting An Element Into A Max Heap9 8 7 6 2 6 5 1 7 7 7
New element is 20.

14. Putting An Element Into A Max Heap9 7 6 8 2 6 5 1 7 7 7
New element is 20.

15. Putting An Element Into A Max Heap20 9 7 6 8 2 6 5 1 7 7 7
New element is 20.

16. Putting An Element Into A Max Heap20 9 7 6 8 2 6 5 1 7 7 7
Complete binary tree with 11 nodes.

17. Putting An Element Into A Max Heap20 9 7 6 8 2 6 5 1 7 7 7
New element is 15.

18. Putting An Element Into A Max Heap20 9 7 6 2 6 5 1 7 7 8 7 8
New element is 15.

19. Putting An Element Into A Max Heap20 15 7 6 9 2 6 5 1 7 7 8 7 8
New element is 15.

20. Complexity is O(log n), where n is heap size.
Complexity Of Put 8 6 7 2 5 1 20 9 15
Complexity is O(log n), where n is heap size.

21. Removing The Max Element8 6 7 2 5 1 20 9 15
Max element is in the root.

22. Removing The Max Element15 7 6 9 2 6 5 1 7 7 8 7 8
After max element is removed.

23. Removing The Max Element15 7 6 9 2 6 5 1 7 7 8 7 8
Heap with 10 nodes.
Reinsert 8 into the heap.

24. Removing The Max Element15 7 6 9 2 6 5 1 7 7 7
Reinsert 8 into the heap.

25. Removing The Max Element15 7 6 9 2 6 5 1 7 7 7
Reinsert 8 into the heap.

26. Removing The Max Element15 9 7 6 8 2 6 5 1 7 7 7
Reinsert 8 into the heap.

27. Removing The Max Element15 9 7 6 8 2 6 5 1 7 7 7
Max element is 15.

28. Removing The Max Element9 7 6 8 2 6 5 1 7 7 7
After max element is removed.

29. Removing The Max Element9 7 6 8 2 6 5 1 7 7 7
Heap with 9 nodes.

30. Removing The Max Element9 7 6 8 2 6 5 1
Reinsert 7.

31. Removing The Max Element9 7 6 8 2 6 5 1
Reinsert 7.

32. Removing The Max Element9 8 7 6 7 2 6 5 1
Reinsert 7.

33. Complexity Of Remove Max Element6 2 5 1 7 9 8
Complexity is O(log n).

34. Complexity of Operations
Two good implementations are heaps and leftist trees.
isEmpty, size, and get => O(1) time
put and remove => O(log n) time where n is the size of the priority queue

35. Practical Complexities
109 instructions/second
Teraflop computer the 32yr time becomes approx 10 days.
2/24/2019

36. Impractical Complexities
109 instructions/second
2/24/2019

37. Summary O(n2) O(n7/6) O(n log n) 10 0.00044 0.00041 0.00057 0.00052
Insertion Sort
O(n2)
Shellsort
O(n7/6)
Heapsort
O(n log n)
Quicksort 10
100
1000
10000
58864
100000 NA
5.7298
8.8591
4.2298
71.164
104.68
47.065
2/24/2019

38. Faster Computer Vs Better Algorithm
Algorithmic improvement more useful
than hardware improvement.
E.g. 2n to n3
2/24/2019

39. 24-Feb-19