Problem:
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The Number of Inversions

For a given sequence $A = \{a_0, a_1, ... a_{n-1}\}$, the number of pairs $(i, j)$ where $a_i > a_j$ and $i < j$, is called the number of inversions. The number of inversions is equal to the number of swaps of Bubble Sort defined in the following program:

bubbleSort(A)
cnt = 0 // the number of inversions
for i = 0 to A.length-1
for j = A.length-1 downto i+1
if A[j] < A[j-1]
swap(A[j], A[j-1])
cnt++

return cnt

For the given sequence $A$, print the number of inversions of $A$. Note that you should not use the above program, which brings Time Limit Exceeded.

Input

In the first line, an integer $n$, the number of elements in $A$, is given. In the second line, the elements $a_i$ ($i = 0, 1, .. n-1$) are given separated by space characters.

output

Print the number of inversions in a line.

Constraints

$ 1 \leq n \leq 200,000$

$ 0 \leq a_i \leq 10^9$

$a_i$ are all different

Sample Input 1

5
3 5 2 1 4

Sample Output 1

6

Sample Input 2

3
3 1 2

Sample Output 2

2

Solution in C (judge verdict: Accepted)

#include<stdio.h>
#include<math.h>

#define MAX 200000
#define SENTINEL 2000000000

int L[MAX / 2 + 2], R[MAX / 2 + 2];

long merge(int A[], int n, int left, int mid, int right){
	int i, j, k;
	long cnt = 0;
	int n1 = mid - left;
	int n2 = right - mid;
	for(i = 0; i < n1; i++){
		L[i] = A[left + i];
	}
	for(i = 0; i < n2; i++){
		R[i] = A[mid + i];
	}
	L[n1] = SENTINEL;
	R[n2] = SENTINEL;
	i = 0;
	j = 0;
	for(k = left; k < right; k++){
		if( L[i] <= R[j]){
			A[k] = L[i++];
		}else{
			A[k] = R[j++];
			cnt += n1 - i;  // = mid + j - k - 1
		}
	}
	return cnt;
}

long mergeSort(int A[], int n, int left, int right){
	int mid;
	long v1, v2, v3;
	if(left + 1 < right){
		mid = (left + right) / 2;
		v1 = mergeSort(A, n, left, mid);
		v2 = mergeSort(A, n, mid, right);
		v3 = merge(A, n, left, mid, right);
		return v1 + v2 + v3;
	}else{
		return 0;
	}
}


int main(){
	int n;
	int val[MAX];
	
	scanf("%d",&n);
	for(int i = 0; i < n; i++){
		scanf("%d",&val[i]);
	}
	
	printf("%ld\n",mergeSort(val, n, 0, n));
	
	return 0;
}