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Chapter 21 The Binary Heap

Chapter 21 The Binary Heap. Bernard Chen Spring 2006. What’s priority queue. Problem: find MIN and Delete . Possible methods: unsorted array sorted array Binary Search tree. What’s Binary Heap.

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Chapter 21 The Binary Heap

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  1. Chapter 21The Binary Heap Bernard Chen Spring 2006

  2. What’s priority queue

  3. Problem: find MIN and Delete • Possible methods: • unsorted array • sorted array • Binary Search tree

  4. What’s Binary Heap • The Binary Heap supports the insertion of new items and delete of MIN item in logarithmic worst-case time. • It uses only an array to implement. (Instead of linked list) • It is the classic method used to implement priority queues

  5. Structure Property • A COMPLETE BINARY TREE is a tree that complete filled.

  6. Complete Binary Tree

  7. Advantages of Complete Tree • The height of a complete binary tree is at most logN • Left and right pointers are not needed

  8. Advantages of Complete Tree • The parent is in position i/2 • The left child is in position 2i • The right child is in position 2i+1

  9. Heap-Order property • In a Heap, for every node X with parent P, the key in P is smaller than or equal to the key in X

  10. 21.2 Implementation of the Basic Operations • Insert operation • Delete operation

  11. 21.2.1 The INSERT operation • Insertion is implemented by creating a hole at the next available location and then percolating it up until the new item can be placed in it without violating a heap order. • Insert takes constant time on average but logarithmic time in worst case.

  12. Insert step 1

  13. Insert step 2

  14. Insert step 3

  15. Insert step 4

  16. Insert step 5

  17. 21.2.2 The DeleteMIN Operation • Deletion of the min involves placing the former last item in a hole that is created at the root. • The hole is percolated down the tree through min children until the item can be placed without violating the heap order property.

  18. DeleteMIN step 1

  19. DeleteMIN step 2

  20. DeleteMIN step 3

  21. DeleteMIN step 4

  22. DeleteMIN step 5

  23. 21.3 the buildHeap Operation:Linear-Time Heap construction • The buildHeap operation can be done in linear time by applying a percolate down routine to nodes in reverse order

  24. BuildHeap step 1

  25. BuildHeap step 2

  26. BuildHeap step 3

  27. BuildHeap step 4

  28. BuildHeap step 5

  29. BuildHeap step 6

  30. BuildHeap step 7

  31. BuildHeap step 8

  32. BuildHeap step 8

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