# Count the number of pairs that have column sum greater than row sum in C++

C++Server Side ProgrammingProgramming

We are given a matrix of size NXN. The goal is to find the count of all valid pairs of indexes (i,j) such that the sum elements of column j is greater than the sum of elements of row i.

We will do this by traversing the matrix and calculate sums of elements of each row and column.

Store sum of elements of each in rowsum[N] and sum of elements of each column in colsum[N].

Now make pairs of rowsum[i] and colsum[j] and check if colsum[j]>rowsum[i]. If true increment count for such pairs.

Let’s understand with examples.

Input-: matrix= {
{ 1,2,0,1},
{ 3,3,0,2},
{ 1,3,0,2},
{ 3,0,0,2}
};

Output − Count of valid pairs − 9

Explanation

Rowsum= 1+2+0+1=5 Colsum= 1+3+1+3=8
Rowsum=3+3+0+2=8 Colsum= 2+3+3+0=8
Rowsum= 1+3+0+2=6 Colsum= 0+0+0+0=0
Rowsum=3+0+0+2=5 Colsum= 1+2+2+2=7
Pairs of (i,j) such that rowsum[i] < colsum[j].
(0,0), (0,1), (0,3), (2,0), (2,1), (2,3), (3,0) (3,1), (3,3)

Input

Arr[]= { {1,1,1}, {1,1,1}, {1,1,1} } N=3

Output − Count of valid pairs − 0

Explanation

Rowsum= 1+1+1=3 Colsum= 1+1+1=3
Rowsum= 1+1+1=3 Colsum= 1+1+1=3
Rowsum= 1+1+1=3 Colsum= 1+1+1=3
No pairs possible where rowsum[i]<colsum[j]

## Approach used in the below program is as follows

• We take an integer array Arr[] initialized with random numbers.

• Take a variable n which stores the length of Arr[].

• Function countPairs(int arr[], int n) takes an array, its length as input and returns the pairs which are valid and meet desired conditions.

• We take two arrays rowsum[n] and colsum[n].

• Traverse matrix and add arr[i][j] to rowsum[i] and colsum[j] to calculate sum of row i and column j.

• Now traverse arrays colsum[] and rowsum[] using two for loops.

• If any colsum[j]>rowsum[i]. Increment count.

• Return the count as result.

## Example

Live Demo

#include <bits/stdc++.h>
using namespace std;
int countPairs(int arr[], int n){
// Count of pairs
int count = 0;
int rowsum[n]={0};
int colsum[n]={0};
int i,j;
for (i = 0; i < n; i++){
for (j = 0; j < n; j++){
rowsum[i]+=arr[i][j];
colsum[j]+=arr[i][j];
}
}
for(i=0;i<n;i++){
for(j=0;j<n;j++)
if(colsum[j]>rowsum[i])
{ count++; }
}
return count;
}
int main(){
int Arr[] = { {1,3,5},{2,4,6},{3,5,7} };
int side=3;
cout <<endl<<"Count of number of pairs : "<< countPairs(Arr, side);
return 0;
}

## Output

If we run the above code it will generate the following output −

Count of number of pairs : 4