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Data Structures and Algorithms

Data Structures and Algorithms . Merge Sort Hairong Zhao http://web.njit.edu/~hz2 New Jersey Institute of Technology. This Lecture. Divide and Conquer Merge Sort. Divide and Conquer. Base case: the problem is small enough, solve directly

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Data Structures and Algorithms

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  1. Data Structures and Algorithms Merge Sort Hairong Zhao http://web.njit.edu/~hz2 New Jersey Institute of Technology

  2. This Lecture • Divide and Conquer • Merge Sort

  3. Divide and Conquer • Base case: the problem is small enough, solve directly • Divide the problem into two or more similar and smaller subproblems • Recursively solve the subproblems • Combine solutions to the subproblems

  4. Divide and Conquer - Sort Problem: • Input: A[n] – unsorted array of n ≥1 integers. • Output:A[n] – sorted in non-decreasing order

  5. Divide and Conquer - Sort • Base case • single element (n=1), return • DivideA into twosubarrays: FirstPart, SecondPart • Two Subproblems: • sort the FirstPart • sort the SecondPart • Recursively • sort FirstPart • sort SecondPart • Combinesorted FirstPart and sorted second part

  6. Merge Merge Sort: Idea Divide into two halves A: FirstPart SecondPart Recursively sort SecondPart FirstPart A is sorted!

  7. Merge Sort: Algorithm • Merge-Sort(A, n) • if n=1 return • else • n1← n2← n/2 • create array L[n1], R[n2] • for i ← 0 to n1-1 do L[i] ← A[i] • for j ← 0 to n2-1 do R[j] ← A[n1+j] • Merge-Sort(L, n1) • Merge-Sort(R, n2) • Merge(A, L, n1, R, n2 ) Space: n Time: n Recursive Call

  8. Merge-Sort: Merge Sorted A: merge Sorted Sorted R: L:

  9. 1 2 6 8 3 4 5 7 Merge-Sort: Merge Example A: L: R:

  10. Merge-Sort: Merge Example A: 3 1 5 15 28 10 14 k=0 R: L: 3 1 15 2 28 6 30 8 3 6 4 10 14 5 7 22 i=0 j=0

  11. Merge-Sort: Merge Example A: 2 1 5 15 28 30 6 10 14 k=1 R: L: 3 1 2 5 15 6 28 8 6 3 10 4 5 14 22 7 j=0 i=1

  12. Merge-Sort: Merge Example A: 3 1 2 15 28 30 6 10 14 k=2 R: L: 1 2 6 8 6 3 4 10 5 14 7 22 j=0 i=2

  13. Merge-Sort: Merge Example A: 1 2 3 6 10 14 4 k=3 R: L: 1 2 6 8 3 6 10 4 14 5 7 22 j=1 i=2

  14. Merge-Sort: Merge Example A: 1 2 3 4 6 10 14 5 k=4 R: L: 1 2 6 8 3 6 10 4 14 5 7 22 i=2 j=2

  15. Merge-Sort: Merge Example A: 6 1 2 3 4 5 6 10 14 k=5 R: L: 1 2 6 8 6 3 4 10 5 14 7 22 i=2 j=3

  16. Merge-Sort: Merge Example A: 7 1 2 3 4 5 6 14 k=6 R: L: 1 2 6 8 6 3 4 10 14 5 7 22 j=3 i=3

  17. Merge-Sort: Merge Example A: 8 1 2 3 4 5 5 7 14 k=7 R: L: 1 3 5 2 6 15 28 8 6 3 10 4 5 14 22 7 i=3 j=4

  18. Merge-Sort: Merge Example A: 1 2 3 4 5 6 7 8 k=8 R: L: 3 1 2 5 15 6 8 28 3 6 10 4 14 5 22 7 j=4 i=4

  19. merge(A,L,n1,R,n2) i ← j ← 0 for k ← 0 to n1+n2-1 if i < n1 if j = n2 or L[i] ≤ R[j] A[k] ← L[i] i ← i + 1 else if j < n2 A[k] ← R[j] j ← j + 1 Number of iterations: (n1+n2) Total time: c(n1+n2) for some c

  20. Merge-Sort Execution Example Divide 3 7 5 1 6 2 8 4 6 2 8 4 3 7 5 1

  21. Merge-Sort Execution Example , divide Recursive call 3 7 5 1 8 4 6 2 6 2 8 4

  22. Merge-Sort Execution Example , divide Recursive call 3 7 5 1 8 4 2 6 6 2

  23. Merge-Sort Execution Example , base case Recursive call 3 7 5 1 8 4 2 6

  24. Merge-Sort Execution Example Recursive call return 3 7 5 1 8 4 2 6

  25. Merge-Sort Execution Example , base case Recursive call 3 7 5 1 8 4 6 2

  26. Merge-Sort Execution Example Recursive call return 3 7 5 1 8 4 2 6

  27. Merge-Sort Execution Example Merge 3 7 5 1 8 4 2 6

  28. Merge-Sort Execution Example Recursive call return 3 7 5 1 8 4 2 6

  29. Merge-Sort Execution Example , divide Recursive call 3 7 5 1 2 6 4 8 8 4

  30. Merge-Sort Execution Example Recursive call, base case 3 7 5 1 2 6 4 8

  31. Merge-Sort Execution Example Recursive call return 3 7 5 1 2 6 4 8

  32. Merge-Sort Execution Example Recursive call, base case 2 6 8 4

  33. Merge-Sort Execution Example Recursive call return 3 7 5 1 2 6 4 8

  34. Merge-Sort Execution Example merge 3 7 5 1 2 6 4 8

  35. Merge-Sort Execution Example Recursive call return 3 7 5 1 4 8 2 6

  36. Merge-Sort Execution Example merge 3 7 5 1 2 4 6 8

  37. Merge-Sort Execution Example Recursive call return 2 4 6 8 3 7 5 1

  38. Merge-Sort Execution Example Recursive call 2 4 6 8 3 7 5 1

  39. Merge-Sort Execution Example 2 4 6 8 1 3 5 7

  40. Merge-Sort Execution Example Recursive call return 2 4 6 8 1 3 5 7

  41. Merge-Sort Execution Example merge 1 2 3 4 5 6 7 8

  42. Merge-Sort Analysis • Time, divide n n 2 × n/2 = n n/2 n/2 log n levels 4 × n/4 = n n/4 n/4 n/4 n/4 n/2 × 2 = n Total time for divide: n log n

  43. Merge-Sort Analysis • Time, merging cn n 2 × cn/2 = n n/2 n/2 log n levels 4 × cn/4 = n n/4 n/4 n/4 n/4 n/2 × 2c = n Total time for merging: cn log n • Total running time: order of nlogn • Total Space: order of n

  44. Homework • What is the running time of Merge-Sort if the array is already sorted? What is the best case running time of Merge-Sort? Justify the answer.

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