Algorithm 2 Count Inversions

Input: An array $A[1..n]$ representing a sequence of $n$ elements.

Output: The number of inversions in $A$ .

1: // return the number of inversions in $A[p..r]$ and merge sort $A[p..r]$ .

2:procedure MergeCount( $A, p, r$ )

3:if $p \ge r$ then

4:return $0$

5:end if

6: $q = \lfloor(p + r)/2\rfloor$

7: $left =$ MergeCount( $A, p, q$ )

8: $right =$ MergeCount( $A, q + 1, r$ )

9: $cross =$ Merge( $A, p, q, r$ )

10:return $left + right + cross$

11:end procedure

12:

13: // merge two subarrays and count the inversion across two subarrays

14:procedure Merge( $A, p, q, r$ )

15: $n_1 = q - p + 1$

16: $n_2 = r - q$

17:Let $L[1..n_1 + 1]$ and $R[1..n_2 + 1]$ be two new arrays.

18:for $i = 1$ to $n_1$ do

19: $L[i] = A[p + i - 1]$

20:end for

21:for $j = 1$ to $n_2$ do

22: $R[j] = A[q + j]$

23:end for

24: $L[n_1 + 1] = \infty$

25: $R[n_2 + 1] = \infty$

26: $i = 1$

27: $j = 1$

28: $count = 0$ // the number of inversions

29:for $k = p$ to $r$ do

30:if $L[i] \le R[j]$ then

31: $A[k] = L[i]$

32: $i = i + 1$

33:else

34: $count = count + n_1 - i + 1$

35: $A[k] = R[j]$

36: $j = j + 1$

37:end if

38:end for

39:return $count$

40:end procedure

41:

42:MergeCount( $A, 1, n$ )