lecture24.pdf

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1. Session 24 ICS 211 Heaps
2. Memory Upload • Heaps • Array implementation • Insertion into a heap • Removal from a heap
3. Terminology • A heap is a complete binary tree, in which each node has a greater (or equal) priority value than its children • A more accurate name for this is max heap • If each node has a priority value less than its children, it is called a min heap
4. Example Heap • Here is a max heap of Integers 99 20 37 2 24 4 16 8 3
5. Example Heap • Here is a min heap of Integers 9 20 37 52 44 24 21 47 81 33 25
6. What is a Tree? • A tree is set of nodes with the following restrictions: 1. Each node has one parent (except for the root node) 2. Each node has zero or more child nodes
7. Terminology • A binary tree is a tree in which each node has at most 2 children
8. Terminology • The depth is the number of edges from root node to current node • For example, the root is at depth 0, its children at depth 1, their children are at depth 2, etc. • Nodes on the same level have the same depth
9. Complete Binary Trees • A complete binary tree is a tree in which every level is filled, except for the bottom level, where all of the nodes are pushed to the left
10. Example Max Heap Level 0 Level 1 Level 2 Level 3 50 14 77 53 82 11 71 58 66 5 20 4 18
11. Heap Array Implementation • Since a heap is a complete binary tree, it is convenient to implement with an array  Next node that is added is put at end of array • If the parent node has array index i  Then left child is at index 2*i+1  And right child is at index 2*i+2
12. Heap Array Implementation • If left or right child has array index i  Then parent is at index (i-1)/2
13. Convert to Array • Let's convert this tree representation of a max heap into the corresponding array representation 99 20 37 2 24 4 16 8 3
14. Convert to Array • Take each node in the heap, from the top level to the bottom level, from left to right and insert into an array starting from index zero 99 20 37 2 24 4 16 8 3
15. Convert to Array • Node 99 is at index 0 • Node 20 is at index 1 • Node 37 is at index 2 • Node 16 is at index 3 • Etc. 99 20 37 2 24 4 16 8 3
16. Convert to Array • Index: 0 1 2 3 4 5 6 7 8 9 • Element: |99|20|37|16|5|24| 2| 3|12|3| 99 20 37 2 24 5 16 12 3 3
17. Max Heap Array Example • Index: 0 1 2 3 4 5 6 7 8 9 • Element: | 9| 8| 7| 6| 5| 4| 3| 2| 1| 0| 9 8 7 3 4 5 6 1 2 0
18. Max Heap Array Example • Index: 0 1 2 3 4 5 6 7 8 9 • Element: | 9| 3| 9| 2| 1| 8| 8| 2| 1| 1| 9 3 9 8 8 1 2 1 2 1
19. Not a Max Heap!! • Index: 0 1 2 3 4 5 6 7 8 9 • Element: | 9|17| 9| 2| 1| 8| 8| 2| 1| 2| 9 17 9 8 8 1 2 1 2 2
20. Max Heap Insertion Algorithm • Step 1: To keep the tree complete, the new node is inserted at bottom left position in tree • This corresponds to the end of the array
21. Max Heap Insertion Algorithm • Step 2: To keep the node relationship of the max heap intact, the new node is swapped with its parent, until the parent node is larger than the child node (“trickles up” the tree)  Each node in a max heap has a larger value than its two child nodes
22. Max Heap Insertion • Let’s insert the following values into a max heap: 5, 4, 6, 3, 3, 8, 8, 4, 5, 9
23. Max Heap Insertion Example • Insert value: 5 (empty heap, so this is the root node) • Index: 0 1 2 3 4 5 6 7 8 9 • Element: | 5| | | | | | | | | | 5
24. Max Heap Insertion Example • Insert value: 4 (at the bottom left position) • Index: 0 1 2 3 4 5 6 7 8 9 • Element: | 5| 4| | | | | | | | | 5 4
25. Max Heap Insertion Example • Insert value: 6 (at the bottom left position) • Index: 0 1 2 3 4 5 6 7 8 9 • Element: | 5| 4| 6| | | | | | | | 5 4 6
26. Max Heap Insertion Example • Insert value: 6 (greater than parent 5, so swap) • Index: 0 1 2 3 4 5 6 7 8 9 • Element: | 6| 4| 5| | | | | | | | 6 4 5
27. Max Heap Insertion Example • Insert value: 3 (at the bottom left position) • Index: 0 1 2 3 4 5 6 7 8 9 • Element: | 6| 4| 5| 3| | | | | | | 6 4 5 3
28. Max Heap Insertion Example • Insert value: 3 (at the bottom left position) • Index: 0 1 2 3 4 5 6 7 8 9 • Element: | 6| 4| 5| 3| 3| | | | | | 6 4 5 3 3
29. Max Heap Insertion Example • Insert value: 8 (at the bottom left position) • Index: 0 1 2 3 4 5 6 7 8 9 • Element: | 6| 4| 5| 3| 3| 8| | | | | 6 4 5 8 3 3
30. Max Heap Insertion Example • Insert value: 8 (greater than parent 5, so swap) • Index: 0 1 2 3 4 5 6 7 8 9 • Element: | 6| 4| 8| 3| 3| 5| | | | | 6 4 8 5 3 3
31. Max Heap Insertion Example • Insert value: 8 (greater than parent 6, so swap) • Index: 0 1 2 3 4 5 6 7 8 9 • Element: | 8| 4| 6| 3| 3| 5| | | | | 8 4 6 5 3 3
32. Max Heap Insertion Example • Insert value: 8 (at bottom left position position) • Index: 0 1 2 3 4 5 6 7 8 9 • Element: | 8| 4| 6| 3| 3| 5| 8| | | | 8 4 6 5 3 3 8
33. Max Heap Insertion Example • Insert value: 8 (greater than parent 6, so swap) • Index: 0 1 2 3 4 5 6 7 8 9 • Element: | 8| 4| 8| 3| 3| 5| 6| | | | 8 4 8 5 3 3 6
34. Max Heap Insertion Example • Insert value: 4 (at bottom left position) • Index: 0 1 2 3 4 5 6 7 8 9 • Element: | 8| 4| 8| 3| 3| 5| 6| 4| | | 8 4 8 5 3 3 6 4
35. Max Heap Insertion Example • Insert value: 4 (greater than parent 3, so swap) • Index: 0 1 2 3 4 5 6 7 8 9 • Element: | 8| 4| 8| 4| 3| 5| 6| 3| | | 8 4 8 5 3 4 6 3
36. Max Heap Insertion Example • Insert value: 5 (at bottom left position) • Index: 0 1 2 3 4 5 6 7 8 9 • Element: | 8| 4| 8| 4| 3| 5| 6| 3| 5| | 8 4 8 5 3 4 6 3 5
37. Max Heap Insertion Example • Insert value: 5 (greater than parent 4, so swap) • Index: 0 1 2 3 4 5 6 7 8 9 • Element: | 8| 4| 8| 5| 3| 5| 6| 3| 4| | 8 4 8 5 3 5 6 3 4
38. Max Heap Insertion Example • Insert value: 5 (greater than parent 4, so swap) • Index: 0 1 2 3 4 5 6 7 8 9 • Element: | 8| 5| 8| 4| 3| 5| 6| 3| 4| | 8 5 8 5 3 4 6 3 4
39. Max Heap Insertion Example • Insert value: 9 (at bottom left position) • Index: 0 1 2 3 4 5 6 7 8 9 • Element: | 8| 5| 8| 4| 3| 5| 6| 3| 4| 9| 8 5 8 5 3 4 6 3 4 9
40. Max Heap Insertion Example • Insert value: 9 (greater than parent 3, so swap) • Index: 0 1 2 3 4 5 6 7 8 9 • Element: | 8| 5| 8| 4| 9| 5| 6| 3| 4| 3| 8 5 8 5 9 4 6 3 4 3
41. Max Heap Insertion Example • Insert value: 9 (greater than parent 5, so swap) • Index: 0 1 2 3 4 5 6 7 8 9 • Element: | 8| 9| 8| 4| 5| 5| 6| 3| 4| 3| 8 9 8 5 5 4 6 3 4 3
42. Max Heap Insertion Example • Insert value: 9 (greater than parent 8, so swap) • Index: 0 1 2 3 4 5 6 7 8 9 • Element: | 9| 8| 8| 4| 5| 5| 6| 3| 4| 3| 9 8 8 5 5 4 6 3 4 3
43. Max Heap Insertion Big-O • Inserting a node into a heap follows a path from the leaf to the root  A maximum of h comparisons have to be made, where h is the height of the tree  The height of a complete binary tree (heap) is log2 n (n is number of nodes)  Therefore, the Big-O is O(log n)
44. Max Heap Removal Algorithm • Removal from a heap is always from the top • Step 1: The top node is deleted and its value returned  The deleted node is replaced with the rightmost, bottommost node  This maintains the heap as a complete binary tree
45. Max Heap Removal Algorithm • Step 2: To keep largest value at top of tree, the new value must be compared to its children  If it is smaller than its largest child node, it is swapped with its largest child node  So the node “trickles down” the tree
46. Max Heap Removal Example • Starting max heap • Index: 0 1 2 3 4 5 6 7 8 9 • Element: | 9| 8| 8| 4| 5| 5| 6| 3| 4| 3| 9 8 8 5 5 4 6 3 4 3
47. Max Heap Removal Example • Remove value: 9 (replace with last node 3) • Index: 0 1 2 3 4 5 6 7 8 9 • Element: | 3| 8| 8| 4| 5| 5| 6| 3| 4| | 3 8 8 5 5 4 6 3 4
48. Max Heap Removal Example • Remove value: 9 (swap 3 with largest child 8) • Index: 0 1 2 3 4 5 6 7 8 9 • Element: | 8| 3| 8| 4| 5| 5| 6| 3| 4| | 8 3 8 5 5 4 6 3 4
49. Max Heap Removal Example • Remove value: 9 (swap 3 with largest child 5) • Index: 0 1 2 3 4 5 6 7 8 9 • Element: | 8| 5| 8| 4| 3| 5| 6| 3| 4| | 8 5 8 5 3 4 6 3 4
50. Max Heap Removal Example • Remove value: 8 (replace with last node 4) • Index: 0 1 2 3 4 5 6 7 8 9 • Element: | 4| 5| 8| 4| 3| 5| 6| 3| | | 4 5 8 5 3 4 6 3
51. Max Heap Removal Example • Remove value: 8 (swap 4 with largest child 8) • Index: 0 1 2 3 4 5 6 7 8 9 • Element: | 8| 5| 4| 4| 3| 5| 6| 3| | | 8 5 4 5 3 4 6 3
52. Max Heap Removal Example • Remove value: 8 (swap 4 with largest child 6) • Index: 0 1 2 3 4 5 6 7 8 9 • Element: | 8| 5| 6| 4| 3| 5| 4| 3| | | 8 5 6 5 3 4 4 3
53. Max Heap Removal Example • Remove value: 8 (replace with last node 3) • Index: 0 1 2 3 4 5 6 7 8 9 • Element: | 3| 5| 6| 4| 3| 5| 4| | | | 3 5 6 5 3 4 4
54. Max Heap Removal Example • Remove value: 8 (swap 3 with largest child 6) • Index: 0 1 2 3 4 5 6 7 8 9 • Element: | 6| 5| 3| 4| 3| 5| 4| | | | 6 5 3 5 3 4 4
55. Max Heap Removal Example • Remove value: 8 (swap 3 with largest child 5) • Index: 0 1 2 3 4 5 6 7 8 9 • Element: | 6| 5| 5| 4| 3| 3| 4| | | | 6 5 5 3 3 4 4
56. Max Heap Removal Example • Remove value: 6 (replace with last node 4) • Index: 0 1 2 3 4 5 6 7 8 9 • Element: | 4| 5| 5| 4| 3| 3| | | | | 4 5 5 3 3 4
57. Max Heap Removal Example • Remove value: 6 (swap 4 with largest child 5) • Index: 0 1 2 3 4 5 6 7 8 9 • Element: | 5| 4| 5| 4| 3| 3| | | | | 5 4 5 3 3 4
58. Max Heap Removal Example • Remove value: 5 (replace with last node 3) • Index: 0 1 2 3 4 5 6 7 8 9 • Element: | 3| 4| 5| 4| 3| | | | | | 3 4 5 3 4
59. Max Heap Removal Example • Remove value: 5 (swap with largest child 5) • Index: 0 1 2 3 4 5 6 7 8 9 • Element: | 5| 4| 3| 4| 3| | | | | | 5 4 3 3 4
60. Max Heap Removal Big-O • Removing a node from a heap follows a path from the root to the leaf  A maximum of h comparisons have to be made, where h is the height of the tree  The height of a complete binary tree (heap) is log2 n (n is number of nodes)  Therefore, the Big-O is O(log n)
61. Memory Defragmenter • Heaps • Array implementation • Insertion into a heap • Removal from a heap
62. Task Manager • Before the next class, you need to: 1.Do the assignment corresponding to this lecture 2.Email me any questions you may have about the material 3.Turn in the assignment before the due date 4.Stop playing video games!