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Lec 6 Feb 17, 2011

Lec 6 Feb 17, 2011 Section 2.5 of text (review of heap) Chapter 3. Review of heap (Sec 2.5). Heap is a data structure that supports a priority queue. Two versions (Max-heap, min-heap) Max-heap operations (can do in O(log n) time.

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Lec 6 Feb 17, 2011

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  1. Lec 6 Feb 17, 2011 • Section 2.5 of text (review of heap) • Chapter 3

  2. Review of heap (Sec 2.5) Heap is a data structure that supports a priority queue. Two versions (Max-heap, min-heap) Max-heap operations (can do in O(log n) time. • Insert(H, x) – add x to H. • Delete-max(H) – remove the max elt. from H. • Other operations: increase-key, decrease-key, delete(j) – delete the key stored in index j of heap etc. • Operations that take O(n) time: search(x), delete(x) etc.

  3. Min Heap with 9 Nodes Complete binary tree with 9 nodes.

  4. 2 4 3 6 7 9 3 8 6 Min Heap With 9 Nodes Min-heap property: A[k] <= A[2*k] (if 2*k <= n) and A[k] <= A[2*k+1] (if 2*k+1 <= n).

  5. 9 8 7 6 7 2 6 5 1 Max Heap With 9 Nodes Example of a Max-heap

  6. Heap Height Since a heap is a complete binary tree, the height of an n node heap is log2 (n+1).

  7. 9 8 7 6 7 2 6 5 1 A Heap Is Efficiently Represented As An Array 9 8 7 6 7 2 6 5 1 0 1 2 3 4 5 6 7 8 9 10

  8. 1 9 2 3 8 7 4 7 5 6 6 7 2 6 5 1 8 9 Moving Up And Down A Heap Parent of node with index k is k/2 Left child of a node with index j is 2*j Right child of a node with index j is 2*j + 1

  9. 9 8 7 6 7 2 6 5 1 7 Putting An Element Into A Max Heap Place to add the new key

  10. Putting An Element Into A Max Heap 9 8 7 6 7 2 6 5 1 7 5 Example: New element is 5.

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

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

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

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

  15. Putting An Element Into A Max Heap 20 9 7 6 8 2 6 5 1 7 7 7 Complete binary tree with 11 nodes.

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

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

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

  19. 20 7 15 6 9 2 6 8 5 1 7 7 8 7 Complexity of insert Complexity is O(log n), where n is heap size.

  20. 20 7 15 6 9 2 6 8 5 1 7 7 8 7 DeleteMax operation Max element is in the root.

  21. DeleteMax 7 15 6 9 2 6 8 5 1 7 7 8 7 After max element is removed.

  22. DeleteMax 7 15 6 9 2 6 8 5 1 7 7 8 7 Heap with 10 nodes. Location needs to be vacated. Find the right place to reinsert 8.

  23. DeleteMax 7 15 6 9 2 6 5 1 7 7 7 Reinsert 8 into the heap.

  24. DeleteMax 15 7 6 9 2 6 5 1 7 7 7 Reinsert 8 into the heap.

  25. DeleteMax 15 9 7 6 2 6 8 5 1 7 7 7 Reinsert 8 into the heap.

  26. DeleteMax – Another example 15 9 7 6 2 6 8 5 1 7 7 7 Max element is 15.

  27. DeleteMax – Ex 2 9 7 6 2 6 8 5 1 7 7 7 After max element is removed.

  28. DeleteMax – Ex 2 9 7 6 2 6 8 5 1 7 7 7 Heap with 9 nodes.

  29. DeleteMax – Ex 2 9 7 6 2 6 8 5 1 Reinsert 7.

  30. DeleteMax – Ex 2 9 7 6 2 6 8 5 1 Reinsert 7.

  31. DeleteMax – Ex 2 9 8 7 6 7 2 6 5 1 Reinsert 7.

  32. 9 8 7 6 7 2 6 5 1 Complexity of DeleteMax • Complexity is O(log n). • Involves working down the heap, two comparisons and 1 assignment per level. There are at most log2 (n+1) levels. • Total complexity <= 3 log2 (n+1) = O(log n).

  33. 9 8 7 6 7 2 6 5 1 Delete a key at a given index Delete (2) • Want an algorithm of complexity O(log n).

  34. 9 8 7 6 7 2 6 5 1 Delete a key at a given index Delete (2) similar to DeleteMax To perform Delete(j): A[j] = A[size]; size--; adjust the heap at position j; How to adjust?

  35. 19 18 7 17 16 2 6 15 Delete a key at a given index : Ex – 2 Delete (6) Adjustment may require percolate_up or percolate_down

  36. Augmenting a heap • Suggest a data structure that acts like both a min-heap and max-heap. • i.e., it should support all three operations in O(log n) time: • Insert • DeleteMin • DeleteMax • Any suggestion?

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