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Lecture 2 Sorting

Lecture 2 Sorting. Sorting Problem. e.g.,. Insertion Sort, Merge Sort. Efficiency. Running time from receiving the input to producing the output. Running time. Insertion Sort. Merge Sort. Heapsort. Heap, a data structure Max-Heapify procedure Building a heap Heapsort.

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Lecture 2 Sorting

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  1. Lecture 2 Sorting

  2. Sorting Problem e.g., Insertion Sort, Merge Sort

  3. Efficiency • Running time from receiving the input to producing the output. Running time Insertion Sort Merge Sort

  4. Heapsort • Heap, a data structure • Max-Heapify procedure • Building a heap • Heapsort

  5. A Data Structure Heap • A heap is an array object that can be viewed as a nearly complete binary tree. 1 6 2 3 5 3 6 5 3 2 4 1 4 5 6 2 4 1

  6. Max-Heap

  7. Min-Heap

  8. Max-Heapify • Max-Heapify(A,i) is a subroutine. • When it is called, two subtrees rooted at Left(i) and Right(i) are max-heaps, but A[i] may not satisfy the max-heap property. • Max-Heapify(A,i) makes the subtree rooted at A[i] become a max-heap by letting A[i] “float down”.

  9. 4 14 7 4 14 7 2 8 1 2 8 1 14 8 7 2 1 4

  10. Running time

  11. Building a Heap e.g., 4, 1, 3, 2, 16, 9, 10, 14, 8, 7.

  12. 4 1 3 10 9 2 16 8 7 14 Proof.

  13. 4 1 3 10 9 2 16 8 7 7 14

  14. 4 1 3 10 9 2 16 8 7 14

  15. 4 1 3 10 9 2 16 8 7 14

  16. Building a Max-Heap e.g., 4, 1, 3, 2, 16, 9, 10, 14, 8, 7.

  17. 4 1 3 10 9 2 16 8 7 14

  18. 4 1 3 10 9 2 16 8 7 14

  19. 4 1 3 10 9 16 14 2 8 7

  20. 4 1 3 10 9 14 16 2 8 7

  21. 4 1 10 3 9 14 16 2 8 7

  22. 4 1 10 3 9 14 16 2 8 7

  23. 4 16 10 3 9 14 1 2 8 7

  24. 4 16 10 3 9 14 7 2 8 1

  25. 16 4 10 3 9 14 7 2 8 1

  26. 16 14 10 3 9 4 7 2 8 1

  27. 16 14 10 3 9 7 8 2 4 1

  28. Analysis

  29. 16 14 10 3 9 7 8 2 4 1

  30. Running time

  31. Heapsort

  32. Input: 4, 1, 3, 2, 16, 9, 10, 14, 8, 7. Build a max-heap 16 14 10 3 9 7 8 2 4 1 16, 14, 10, 8, 7, 9, 3, 2, 4, 1.

  33. 1 14 10 3 9 7 8 2 4 16

  34. 1 14 10 3 9 7 8 2 4 16 1, 14, 10, 8, 7, 9, 3, 2, 4, 16.

  35. 14 8 10 3 9 7 4 2 1 16

  36. 14 8 10 3 9 7 4 2 1 16 14, 8, 10, 4, 7, 9, 3, 2, 1, 16.

  37. 1 8 10 3 9 7 4 2 14 16

  38. 1 8 10 3 9 7 4 2 14 16 1, 8, 10, 4, 7, 9, 3, 2, 14, 16.

  39. 10 8 9 3 1 7 4 2 14 16

  40. 10 8 9 3 1 7 4 2 14 16 10, 8, 9, 4, 7, 1, 3, 2, 14, 16.

  41. 2 8 9 3 1 7 4 14 16 10

  42. 2 8 9 3 1 7 4 14 16 10 2, 8, 9, 4, 7, 1, 3, 10, 14, 16.

  43. 9 8 3 2 1 7 4 14 16 10

  44. 9 8 3 2 1 7 4 14 16 10 9, 8, 3, 4, 7, 1, 2, 10, 14, 16.

  45. 2 8 3 9 1 7 4 14 16 10

  46. 8 7 3 9 2 1 4 14 16 10

  47. 1 7 3 9 2 8 4 14 16 10

  48. 7 4 3 9 2 8 1 14 16 10

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