Merge and Count

1. Merge and Count current • 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. smallesti = 6 smallestj = 6 3 7 10 14 18 19 2 11 16 17 23 25 N/2 = 6 auxiliary array Inversions: Total: 0

2. Merge and Count current • 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. smallesti = 6 smallestj = 6 3 7 10 14 18 19 2 11 16 17 23 25 N/2 = 6 auxiliary array Inversions: Total: 0

3. Merge and Count current • 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. smallesti = 6 smallestj = 5 3 7 10 14 18 19 2 11 16 17 23 25 N/2 = 6 2 auxiliary array Inversions: 6 Total: 6

4. Merge and Count current • 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. smallesti = 6 smallestj = 5 3 7 10 14 18 19 2 11 16 17 23 25 N/2 = 6 2 auxiliary array Inversions: 6 Total: 6

5. Merge and Count current • 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. smallesti = 5 smallestj = 5 3 7 10 14 18 19 2 11 16 17 23 25 N/2 = 6 2 3 auxiliary array Inversions: 6 Total: 6

6. Merge and Count current • 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. smallesti = 5 smallestj = 5 3 7 10 14 18 19 2 11 16 17 23 25 N/2 = 6 2 3 auxiliary array Inversions: 6 Total: 6

7. Merge and Count current • 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. smallesti = 4 smallestj = 5 3 7 10 14 18 19 2 11 16 17 23 25 N/2 = 6 2 3 7 auxiliary array Inversions: 6 Total: 6

8. Merge and Count current • 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. smallesti = 4 smallestj = 5 3 7 10 14 18 19 2 11 16 17 23 25 N/2 = 6 2 3 7 auxiliary array Inversions: 6 Total: 6

9. Merge and Count current • 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. smallesti = 3 smallestj = 5 3 7 10 14 18 19 2 11 16 17 23 25 N/2 = 6 2 3 7 10 auxiliary array Inversions: 6 Total: 6

10. Merge and Count current • 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. smallesti = 3 smallestj = 5 3 7 10 14 18 19 2 11 16 17 23 25 N/2 = 6 2 3 7 10 auxiliary array Inversions: 6 Total: 6

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. smallesti = 3 smallestj = 4 3 7 10 14 18 19 2 11 16 17 23 25 N/2 = 6 2 3 7 10 11 auxiliary array Inversions: 6 + 3 Total: 9

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. smallesti = 3 smallestj = 4 3 7 10 14 18 19 2 11 16 17 23 25 N/2 = 6 2 3 7 10 11 auxiliary array Inversions: 6 + 3 Total: 9

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. smallesti = 2 smallestj = 4 3 7 10 14 18 19 2 11 16 17 23 25 N/2 = 6 2 3 7 10 11 14 auxiliary array Inversions: 6 + 3 Total: 9

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. smallesti = 2 smallestj = 4 3 7 10 14 18 19 2 11 16 17 23 25 N/2 = 6 2 3 7 10 11 14 auxiliary array Inversions: 6 + 3 Total: 9

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. smallesti = 2 smallestj = 3 3 7 10 14 18 19 2 11 16 17 23 25 N/2 = 6 2 3 7 10 11 14 16 auxiliary array Inversions: 6 + 3 + 2 Total: 11

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. smallesti = 2 smallestj = 3 3 7 10 14 18 19 2 11 16 17 23 25 N/2 = 6 2 3 7 10 11 14 16 auxiliary array Inversions: 6 + 3 + 2 Total: 11

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. smallesti = 2 smallestj = 2 3 7 10 14 18 19 2 11 16 17 23 25 N/2 = 6 2 3 7 10 11 14 16 17 auxiliary array Inversions: 6 + 3 + 2 + 2 Total: 13

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. smallesti = 2 smallestj = 2 3 7 10 14 18 19 2 11 16 17 23 25 N/2 = 6 2 3 7 10 11 14 16 17 auxiliary array Inversions: 6 + 3 + 2 + 2 Total: 13

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. smallesti = 1 smallestj = 2 3 7 10 14 18 19 2 11 16 17 23 25 N/2 = 6 2 3 7 10 11 14 16 17 18 auxiliary array Inversions: 6 + 3 + 2 + 2 Total: 13

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. smallesti = 1 smallestj = 2 3 7 10 14 18 19 2 11 16 17 23 25 N/2 = 6 2 3 7 10 11 14 16 17 18 auxiliary array Inversions: 6 + 3 + 2 + 2 Total: 13

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. smallesti = 0 smallestj = 2 first half exhausted 3 7 10 14 18 19 2 11 16 17 23 25 N/2 = 6 2 3 7 10 11 14 16 17 18 19 auxiliary array Inversions: 6 + 3 + 2 + 2 Total: 13

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. smallesti = 0 smallestj = 2 first half exhausted 3 7 10 14 18 19 2 11 16 17 23 25 N/2 = 6 2 3 7 10 11 14 16 17 18 19 auxiliary array Inversions: 6 + 3 + 2 + 2 Total: 13

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. smallesti = 0 smallestj = 2 first half exhausted 3 7 10 14 18 19 2 11 16 17 23 25 N/2 = 6 2 3 7 10 11 14 16 17 18 19 23 auxiliary array Inversions: 6 + 3 + 2 + 2 + 0 Total: 13

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. smallesti = 0 smallestj = 1 first half exhausted 3 7 10 14 18 19 2 11 16 17 23 25 N/2 = 6 2 3 7 10 11 14 16 17 18 19 23 auxiliary array Inversions: 6 + 3 + 2 + 2 + 0 Total: 13

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. smallesti = 0 smallestj = 0 second half exhausted first half exhausted 3 7 10 14 18 19 2 11 16 17 23 25 N/2 = 6 2 3 7 10 11 14 16 17 18 19 23 25 auxiliary array Inversions: 6 + 3 + 2 + 2 + 0 + 0 Total: 13