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Merge and Count

This algorithm counts the number of inversions where ai and aj are in different halves and merges two sorted halves into a sorted whole. Example input: 6 3 7 10 14 18 19 2 11 16 17 23 25.

dpeter
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Merge and Count

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  1. Merge and Count • Merge and count step. • Given two sorted halves, count number of inversions where ai and aj are in different halves. • Combine two sorted halves into sorted whole. i = 6 3 7 10 14 18 19 2 11 16 17 23 25 two sorted halves auxiliary array Total:

  2. Merge and Count • Merge and count step. • Given two sorted halves, count number of inversions where ai and aj are in different halves. • Combine two sorted halves into sorted whole. i = 6 3 7 10 14 18 19 2 11 16 17 23 25 two sorted halves 6 2 auxiliary array Total: 6

  3. Merge and Count • Merge and count step. • Given two sorted halves, count number of inversions where ai and aj are in different halves. • Combine two sorted halves into sorted whole. i = 6 3 7 10 14 18 19 2 11 16 17 23 25 two sorted halves 6 2 auxiliary array Total: 6

  4. Merge and Count • Merge and count step. • Given two sorted halves, count number of inversions where ai and aj are in different halves. • Combine two sorted halves into sorted whole. i = 6 3 7 10 14 18 19 2 11 16 17 23 25 two sorted halves 6 2 3 auxiliary array Total: 6

  5. Merge and Count • Merge and count step. • Given two sorted halves, count number of inversions where ai and aj are in different halves. • Combine two sorted halves into sorted whole. i = 5 3 7 10 14 18 19 2 11 16 17 23 25 two sorted halves 6 2 3 auxiliary array Total: 6

  6. Merge and Count • Merge and count step. • Given two sorted halves, count number of inversions where ai and aj are in different halves. • Combine two sorted halves into sorted whole. i = 5 3 7 10 14 18 19 2 11 16 17 23 25 two sorted halves 6 2 3 7 auxiliary array Total: 6

  7. Merge and Count • Merge and count step. • Given two sorted halves, count number of inversions where ai and aj are in different halves. • Combine two sorted halves into sorted whole. i = 4 3 7 10 14 18 19 2 11 16 17 23 25 two sorted halves 6 2 3 7 auxiliary array Total: 6

  8. Merge and Count • Merge and count step. • Given two sorted halves, count number of inversions where ai and aj are in different halves. • Combine two sorted halves into sorted whole. i = 4 3 7 10 14 18 19 2 11 16 17 23 25 two sorted halves 6 2 3 7 10 auxiliary array Total: 6

  9. Merge and Count • Merge and count step. • Given two sorted halves, count number of inversions where ai and aj are in different halves. • Combine two sorted halves into sorted whole. i = 3 3 7 10 14 18 19 2 11 16 17 23 25 two sorted halves 6 2 3 7 10 auxiliary array Total: 6

  10. Merge and Count • Merge and count step. • Given two sorted halves, count number of inversions where ai and aj are in different halves. • Combine two sorted halves into sorted whole. i = 3 3 7 10 14 18 19 2 11 16 17 23 25 two sorted halves 6 3 2 3 7 10 11 auxiliary array Total: 6 + 3

  11. Merge and Count • Merge and count step. • Given two sorted halves, count number of inversions where ai and aj are in different halves. • Combine two sorted halves into sorted whole. i = 3 3 7 10 14 18 19 2 11 16 17 23 25 two sorted halves 6 3 2 3 7 10 11 auxiliary array Total: 6 + 3

  12. Merge and Count • Merge and count step. • Given two sorted halves, count number of inversions where ai and aj are in different halves. • Combine two sorted halves into sorted whole. i = 3 3 7 10 14 18 19 2 11 16 17 23 25 two sorted halves 6 3 2 3 7 10 11 14 auxiliary array Total: 6 + 3

  13. Merge and Count • Merge and count step. • Given two sorted halves, count number of inversions where ai and aj are in different halves. • Combine two sorted halves into sorted whole. i = 2 3 7 10 14 18 19 2 11 16 17 23 25 two sorted halves 6 3 2 3 7 10 11 14 auxiliary array Total: 6 + 3

  14. Merge and Count • Merge and count step. • Given two sorted halves, count number of inversions where ai and aj are in different halves. • Combine two sorted halves into sorted whole. i = 2 3 7 10 14 18 19 2 11 16 17 23 25 two sorted halves 6 3 2 2 3 7 10 11 14 16 auxiliary array Total: 6 + 3 + 2

  15. Merge and Count • Merge and count step. • Given two sorted halves, count number of inversions where ai and aj are in different halves. • Combine two sorted halves into sorted whole. i = 2 3 7 10 14 18 19 2 11 16 17 23 25 two sorted halves 6 3 2 2 3 7 10 11 14 16 auxiliary array Total: 6 + 3 + 2

  16. Merge and Count • Merge and count step. • Given two sorted halves, count number of inversions where ai and aj are in different halves. • Combine two sorted halves into sorted whole. i = 2 3 7 10 14 18 19 2 11 16 17 23 25 two sorted halves 6 3 2 2 2 3 7 10 11 14 16 17 auxiliary array Total: 6 + 3 + 2 + 2

  17. Merge and Count • Merge and count step. • Given two sorted halves, count number of inversions where ai and aj are in different halves. • Combine two sorted halves into sorted whole. i = 2 3 7 10 14 18 19 2 11 16 17 23 25 two sorted halves 6 3 2 2 2 3 7 10 11 14 16 17 auxiliary array Total: 6 + 3 + 2 + 2

  18. Merge and Count • Merge and count step. • Given two sorted halves, count number of inversions where ai and aj are in different halves. • Combine two sorted halves into sorted whole. i = 2 3 7 10 14 18 19 2 11 16 17 23 25 two sorted halves 6 3 2 2 2 3 7 10 11 14 16 17 18 auxiliary array Total: 6 + 3 + 2 + 2

  19. Merge and Count • Merge and count step. • Given two sorted halves, count number of inversions where ai and aj are in different halves. • Combine two sorted halves into sorted whole. i = 1 3 7 10 14 18 19 2 11 16 17 23 25 two sorted halves 6 3 2 2 2 3 7 10 11 14 16 17 18 auxiliary array Total: 6 + 3 + 2 + 2

  20. Merge and Count • Merge and count step. • Given two sorted halves, count number of inversions where ai and aj are in different halves. • Combine two sorted halves into sorted whole. i = 1 3 7 10 14 18 19 2 11 16 17 23 25 two sorted halves 6 3 2 2 2 3 7 10 11 14 16 17 18 19 auxiliary array Total: 6 + 3 + 2 + 2

  21. Merge and Count • Merge and count step. • Given two sorted halves, count number of inversions where ai and aj are in different halves. • Combine two sorted halves into sorted whole. first half exhausted i = 0 3 7 10 14 18 19 2 11 16 17 23 25 two sorted halves 6 3 2 2 2 3 7 10 11 14 16 17 18 19 auxiliary array Total: 6 + 3 + 2 + 2

  22. Merge and Count • Merge and count step. • Given two sorted halves, count number of inversions where ai and aj are in different halves. • Combine two sorted halves into sorted whole. i = 0 3 7 10 14 18 19 2 11 16 17 23 25 two sorted halves 6 3 2 2 0 2 3 7 10 11 14 16 17 18 19 23 auxiliary array Total: 6 + 3 + 2 + 2 + 0

  23. Merge and Count • Merge and count step. • Given two sorted halves, count number of inversions where ai and aj are in different halves. • Combine two sorted halves into sorted whole. i = 0 3 7 10 14 18 19 2 11 16 17 23 25 two sorted halves 6 3 2 2 0 2 3 7 10 11 14 16 17 18 19 23 auxiliary array Total: 6 + 3 + 2 + 2 + 0

  24. Merge and Count • Merge and count step. • Given two sorted halves, count number of inversions where ai and aj are in different halves. • Combine two sorted halves into sorted whole. i = 0 3 7 10 14 18 19 2 11 16 17 23 25 two sorted halves 6 3 2 2 0 0 2 3 7 10 11 14 16 17 18 19 23 25 auxiliary array Total: 6 + 3 + 2 + 2 + 0 + 0

  25. Merge and Count • Merge and count step. • Given two sorted halves, count number of inversions where ai and aj are in different halves. • Combine two sorted halves into sorted whole. i = 0 3 7 10 14 18 19 2 11 16 17 23 25 two sorted halves 6 3 2 2 0 0 2 3 7 10 11 14 16 17 18 19 23 25 auxiliary array Total: 6 + 3 + 2 + 2 + 0 + 0 = 13

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