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Merge Sort

Merge Sort. Section 8.7. Merge. A merge is a common data processing operation performed on two sequences of data with the following characteristics Both sequences contain items with a common compareTo method

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Merge Sort

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  1. Merge Sort Section 8.7

  2. Merge • A merge is a common data processing operation performed on two sequences of data with the following characteristics • Both sequences contain items with a common compareTo method • The objects in both sequences are ordered in accordance with this compareTo method • The result is a third sequence containing all the data from the first two sequences

  3. Merge Algorithm Merge Algorithm 1. Access the first item from both sequences. 2. while not finished with either sequence 3. Compare the current items from the two sequences, copy the smaller current item to the output sequence, and access the next item from the input sequence whose item was copied. 4. Copy any remaining items from the first sequence to the output sequence. 5. Copy any remaining items from the second sequence to the output sequence.

  4. Analysis of Merge • For two input sequences each containing n elements, each element needs to move from its input sequence to the output sequence • Merge time is O(n) • Space requirements • The array cannot be merged in place • Additional space usage is O(n)

  5. Code for Merge • Listing 8.5 (MergeSort.java, page 442)

  6. Merge Sort • We can modify merging to sort a single, unsorted array • Split the array into two halves • Sort the left half • Sort the right half • Merge the two • This algorithm can be written with a recursive step

  7. (recursive) Algorithm for Merge Sort

  8. Trace of Merge Sort 50 60 45 30 90 20 80 15 50 90 20 60 45 80 30 15 50 20 60 80 45 15 30 90

  9. Trace of Merge Sort (cont.) 50 60 45 30 50 90 60 20 45 80 30 15 50 20 60 80 45 15 30 90 50 60 45 30

  10. Trace of Merge Sort (cont.) 50 90 60 20 45 80 30 15 50 20 60 80 45 15 30 90 50 60 45 30 50 60 50 60

  11. Trace of Merge Sort (cont.) 50 90 60 20 45 80 30 15 50 20 60 80 45 15 30 90 50 60 45 30 50 60 60

  12. Trace of Merge Sort (cont.) 50 90 60 20 45 80 30 15 50 20 60 80 45 15 30 90 50 60 45 30 45 30 45 30

  13. Trace of Merge Sort (cont.) 50 90 60 20 45 80 30 15 50 20 60 80 45 15 30 90 30 45 50 60 45 30 45 30 45

  14. Trace of Merge Sort (cont.) 30 90 50 20 45 60 50 45 80 60 30 15 50 20 60 80 45 15 30 90 50 60 45 30 30 45 45 30

  15. Trace of Merge Sort (cont.) 30 90 50 20 45 60 50 45 80 60 30 15 50 20 60 80 45 15 30 90 90 20 80 15

  16. Trace of Merge Sort (cont.) 30 50 90 20 60 45 45 50 80 60 30 15 50 20 60 80 45 15 30 90 90 20 80 15 90 20

  17. Trace of Merge Sort (cont.) 30 90 50 20 60 45 45 50 80 30 60 15 50 20 60 80 45 15 30 90 90 20 20 90 80 15 90 20

  18. Trace of Merge Sort (cont.) 30 50 90 20 60 45 45 50 80 60 30 15 50 20 60 80 45 15 30 90 20 90 80 15 80 15

  19. Trace of Merge Sort (cont.) 30 90 50 20 60 45 45 50 80 30 60 15 50 20 60 80 45 15 30 90 20 15 80 90 80 15 80 15

  20. Trace of Merge Sort (cont.) 15 30 90 50 20 20 60 45 45 50 80 80 60 30 90 15 50 20 60 80 45 15 30 90 20 90 15 80

  21. Trace of Merge Sort (cont.) 15 30 50 90 20 20 45 60 50 45 80 85 60 30 15 90 50 15 20 60 60 20 80 80 45 30 15 90 30 45 90 50

  22. Analysis of Merge Sort • Each backward step requires a movement of n elements from smaller-size arrays to larger arrays; the effort is O(n) • The number of lines which require merging at each step is log n because each recursive step splits the array in half • The total effort to reconstruct the sorted array through merging is O(n log n)

  23. Analysis of Merge Sort (cont.)

  24. Heapsort Section 8.8

  25. Heapsort • Merge sort time is O(n log n) but still requires, temporarily, n extra storage locations • Heapsort does not require any additional storage • As its name implies, heapsort uses a heap to store the array

  26. First Version of a Heapsort Algorithm • When used as a priority queue, a heap maintains a smallest value at the top • The following algorithm • places an array's data into a heap, • then removes each heap item (O(n log n)) and moves it back into the array • This version of the algorithm requires n extra storage locations Heapsort Algorithm: First Version 1. Insert each value from the array to be sorted into a priority queue (heap). 2. Set i to 0 3. while the priority queue is not empty 4. Remove an item from the queue and insert it back into the array at position i 5. Increment i

  27. Revising the Heapsort Algorithm • In heaps we've used so far, each parent node value was less than the values of its children • We can build a heap so that each parent node value is larger than its children • Then, • move the top item to the bottom of the heap • reheap, ignoring the item moved to the bottom

  28. Trace of Heapsort 89 74 76 37 32 39 66 18 20 29 26 6 28

  29. Trace of Heapsort (cont.) 89 74 76 37 32 39 66 18 20 29 26 6 28

  30. Trace of Heapsort (cont.) 89 74 76 37 32 39 66 18 20 29 26 6 28

  31. Trace of Heapsort (cont.) 6 74 76 37 32 39 66 18 20 29 26 89 28

  32. Trace of Heapsort (cont.) 76 74 6 37 32 39 66 18 20 29 26 89 28

  33. Trace of Heapsort (cont.) 76 74 37 6 32 39 66 18 20 29 26 89 28

  34. Trace of Heapsort (cont.) 76 74 37 26 32 39 66 18 20 29 6 89 28

  35. Trace of Heapsort (cont.) 76 74 37 26 32 39 66 18 20 29 6 89 28

  36. Trace of Heapsort (cont.) 76 74 37 26 32 39 66 18 20 29 6 89 28

  37. Trace of Heapsort (cont.) 76 74 37 26 32 39 66 18 20 29 6 89 28

  38. Trace of Heapsort (cont.) 29 74 37 26 32 39 66 18 20 76 6 89 28

  39. Trace of Heapsort (cont.) 74 29 37 26 32 39 66 18 20 76 6 89 28

  40. Trace of Heapsort (cont.) 74 66 37 26 32 39 29 18 20 76 6 89 28

  41. Trace of Heapsort (cont.) 74 66 37 26 32 39 29 18 20 76 6 89 28

  42. Trace of Heapsort (cont.) 74 66 37 26 32 39 29 18 20 76 6 89 28

  43. Trace of Heapsort (cont.) 74 66 37 26 32 39 29 18 20 76 6 89 28

  44. Trace of Heapsort (cont.) 28 66 37 26 32 39 29 18 20 76 6 89 74

  45. Trace of Heapsort (cont.) 66 28 37 26 32 39 29 18 20 76 6 89 74

  46. Trace of Heapsort (cont.) 66 39 37 26 32 28 29 18 20 76 6 89 74

  47. Trace of Heapsort (cont.) 66 39 37 26 32 28 29 18 20 76 6 89 74

  48. Trace of Heapsort (cont.) 66 39 37 26 32 28 29 18 20 76 6 89 74

  49. Trace of Heapsort (cont.) 66 39 37 26 32 28 29 18 20 76 6 89 74

  50. Trace of Heapsort (cont.) 18 39 37 26 32 28 29 66 20 76 6 89 74

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