Prefix Sum of Matrix (Or 2D Array) in C++

In this problem, we are given a 2D array of integer values mat[][]. Our task is to print the prefix sum matrix of mat.

Prefix sum matrix: every element of the matrix is the sum elements above and left of it. i.e

prefixSum[i][j] = mat[i][j] + mat[i-1][j]...mat[0][j] + mat[i][j-1] +... mat[i][0].

Let’s take an example to understand the problem

Input: arr =[
   [4   6   1]
   [5   7   2]
   [3   8   9]
]
Output:[
   [4   10   11]
   [9   22   25]
   [12   33   45]
]

To solve this problem, one simple solution is finding prefixSum by traversing all elements till i,j position and add them. But it is a bit complex for the system.

A more effective solution will be using the formula for finding the values of elements of prefixSum matrix.

The general formula for element at ij position is

prefixSum[i][j] = prefixSum[i-1][j] + prefixSum[i][j-1] - prefixSum[i-1][j-1] + a[i][j]

Some specail cases are

For i = j = 0, prefixSum[i][j] = a[i][j]
For i = 0 and j > 0, prefixSum[i][j] = prefixSum[i][j-1] + a[i][j]
For i > 0 and j = 0, prefixSum[i][j] = prefixSum[i-1][j] + a[i][j]

The code to show the implementation of our solution

Example

 Live Demo

#include <iostream>
using namespace std;
#define R 3
#define C 3
void printPrefixSum(int a[][C]) {
   int prefixSum[R][C];
   prefixSum[0][0] = a[0][0];
   for (int i = 1; i < C; i++)
   prefixSum[0][i] = prefixSum[0][i - 1] + a[0][i];
   for (int i = 0; i < R; i++)
   prefixSum[i][0] = prefixSum[i - 1][0] + a[i][0];
   for (int i = 1; i < R; i++) {
      for (int j = 1; j < C; j++)
      prefixSum[i][j]=prefixSum[i- 1][j]+prefixSum[i][j- 1]-prefixSum[i- 1][j- 1]+a[i][j];
   }
   for (int i = 0; i < R; i++) {
      for (int j = 0; j < C; j++)
      cout<<prefixSum[i][j]<<"\t";
      cout<<endl;
   }
}
int main() {
   int mat[R][C] = {
      { 1, 2, 3},
      { 4, 5, 6},
      { 7, 8, 9}
   };
   cout<<"The prefix Sum Matrix is :\n";
   printPrefixSum(mat);
   return 0;
}

Output

The prefix Sum Matrix is :
1   3   6
5   12   21
12   27   45