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Prefix Sum Array – Implementation and Applications in Competitive Programming

  • Difficulty Level : Medium
  • Last Updated : 25 May, 2021

Given an array arr[] of size n, its prefix sum array is another array prefixSum[] of the same size, such that the value of prefixSum[i] is arr[0] + arr[1] + arr[2] … arr[i].

Examples : 

Input  : arr[] = {10, 20, 10, 5, 15}
Output : prefixSum[] = {10, 30, 40, 45, 60}

Explanation : While traversing the array, update 
the element by adding it with its previous element.
prefixSum[0] = 10, 
prefixSum[1] = prefixSum[0] + arr[1] = 30, 
prefixSum[2] = prefixSum[1] + arr[2] = 40 and so on.

To fill the prefix sum array, we run through index 1 to last and keep on adding the present element with the previous value in the prefix sum array.
Below is the implementation : 

C++




// C++ program for Implementing
// prefix sum array
#include <bits/stdc++.h>
using namespace std;
 
// Fills prefix sum array
void fillPrefixSum(int arr[], int n, int prefixSum[])
{
    prefixSum[0] = arr[0];
 
    // Adding present element
    // with previous element
    for (int i = 1; i < n; i++)
        prefixSum[i] = prefixSum[i - 1] + arr[i];
}
 
// Driver Code
int main()
{
    int arr[] = { 10, 4, 16, 20 };
    int n = sizeof(arr) / sizeof(arr[0]);
    int prefixSum[n];
 
    fillPrefixSum(arr, n, prefixSum);
    for (int i = 0; i < n; i++)
        cout << prefixSum[i] << " ";
}

Java




// Java Program for Implementing
// prefix sum arrayclass
class Prefix {
    // Fills prefix sum array
    static void fillPrefixSum(int arr[], int n,
                              int prefixSum[])
    {
        prefixSum[0] = arr[0];
 
        // Adding present element
        // with previous element
        for (int i = 1; i < n; ++i)
            prefixSum[i] = prefixSum[i - 1] + arr[i];
    }
 
    // Driver code
    public static void main(String[] args)
    {
        int arr[] = { 10, 4, 16, 20 };
        int n = arr.length;
        int prefixSum[] = new int[n];
 
        fillPrefixSum(arr, n, prefixSum);
 
        for (int i = 0; i < n; i++)
            System.out.print(prefixSum[i] + " ");
        System.out.println("");
    }
}
 
// This Code is Contributed by Saket Kumar

Python3




# Python Program for Implementing
# prefix sum array
 
# Fills prefix sum array
def fillPrefixSum(arr, n, prefixSum):
 
    prefixSum[0] = arr[0]
 
    # Adding present element
    # with previous element
    for i in range(1, n):
        prefixSum[i] = prefixSum[i - 1] + arr[i]
 
# Driver code
arr =[10, 4, 16, 20 ]
n = len(arr)
 
prefixSum = [0 for i in range(n + 1)]
 
fillPrefixSum(arr, n, prefixSum)
 
for i in range(n):
    print(prefixSum[i], " ", end ="")
 
# This code is contributed
# by Anant Agarwal.

C#




// C# Program for Implementing
// prefix sum arrayclass
using System;
 
class GFG {
    // Fills prefix sum array
    static void fillPrefixSum(int[] arr, int n,
                              int[] prefixSum)
    {
        prefixSum[0] = arr[0];
 
        // Adding present element
        // with previous element
        for (int i = 1; i < n; ++i)
            prefixSum[i] = prefixSum[i - 1] + arr[i];
    }
 
    // Driver code
    public static void Main()
    {
        int[] arr = { 10, 4, 16, 20 };
        int n = arr.Length;
        int[] prefixSum = new int[n];
 
        fillPrefixSum(arr, n, prefixSum);
 
        for (int i = 0; i < n; i++)
            Console.Write(prefixSum[i] + " ");
        Console.Write("");
    }
}
 
// This Code is Contributed by nitin mittal

PHP




<?php
// PHP program for
// Implementing prefix
// sum array
 
// Fills prefix sum array
function fillPrefixSum($arr,
                       $n)
{
    $prefixSum = array();
    $prefixSum[0] = $arr[0];
 
    // Adding present element
    // with previous element
    for ($i = 1; $i < $n; $i++)
        $prefixSum[$i] = $prefixSum[$i - 1] +
                                    $arr[$i];
         
    for ($i = 0; $i < $n; $i++)
        echo $prefixSum[$i] . " ";
}
 
// Driver Code
$arr = array(10, 4, 16, 20);
$n = count($arr);
 
fillPrefixSum($arr, $n);
 
// This code is contributed
// by Sam007
?>

Javascript




<script>
 
    // JavaScript Program for Implementing
    // prefix sum arrayclass
     
    // Fills prefix sum array
    function fillPrefixSum(arr, n, prefixSum)
    {
        prefixSum[0] = arr[0];
  
        // Adding present element
        // with previous element
        for (let i = 1; i < n; ++i)
            prefixSum[i] = prefixSum[i - 1] + arr[i];
    }
     
    let arr = [ 10, 4, 16, 20 ];
    let n = arr.length;
    let prefixSum = new Array(n);
 
    fillPrefixSum(arr, n, prefixSum);
 
    for (let i = 0; i < n; i++)
        document.write(prefixSum[i] + " ");
    document.write("");
     
</script>

Output: 

10 14 30 50

Applications : 



An Example Problem : 
Consider an array of size n with all initial values as 0. Perform the given ‘m’ add operations from index ‘a’ to ‘b’ and evaluate the highest element in the array. An add operation adds 100 to all elements from a to b (both inclusive). 

Example:  

Input : n = 5 // We consider array {0, 0, 0, 0, 0}
        m = 3.
        a = 2, b = 4.
        a = 1, b = 3.
        a = 1, b = 2.
Output : 300

Explanation : 

After I operation -
A : 0 100 100 100 0

After II operation -
A : 100 200 200 100 0

After III operation -
A : 200 300 200 100 0

Highest element : 300

A simple approach is running a loop ‘m’ times. Inputting a and b and running a loop from a to b, adding all elements by 100. 
The efficient approach using Prefix Sum Array

1 : Run a loop for 'm' times, inputting 'a' and 'b'.
2 : Add 100 at index 'a-1' and subtract 100 from index 'b'.
3 : After completion of 'm' operations, compute the prefix sum array.
4 : Scan the largest element and we're done.

What we did was adding 100 at ‘a’ because this will add 100 to all elements while taking the prefix sum array. Subtracting 100 from ‘b+1’ will reverse the changes made by adding 100 to elements from ‘b’ onward.
For better understanding : 

After I operation -
A : 0 100 0 0 -100 
Prefix Sum Array : 0 100 100 100 0

After II operation -
A : 100 100 0 -100 -100
Prefix Sum Array : 100 200 200 100 0

After III operation -
A : 200 100 -100 -100 -100
Prefix Sum Array : 200 300 200 100 0

Final Prefix Sum Array : 200 300 200 100 0 

The required highest element : 300

C++14




#include <bits/stdc++.h>
using namespace std;
 
int find(int m, vector<pair<int,int>> q)
{
    int mx = 0;
    vector<int> pre(5,0);
     
    for (int i = 0; i < m; i++)
    {   
        // take input a and b
        int a = q[i].first, b = q[i].second;
       
        // add 100 at first index and
        // subtract 100 from last index
       
        // pre[1] becomes 100
        pre[a-1] += 100;
          
        // pre[4] becomes -100 and this
        pre[b] -=100;   
        // continues m times as we input diff. values of a and b
    }
    for (int i = 1; i < 5; i++)
    {
        // add all values in a cumulative way
        pre[i] += pre[i - 1];
         
        // keep track of max value
        mx = max(mx, pre[i]);
    }
      
                                
    return mx;
}
 
// Driver Code
int main()
{
     
    int m = 3;
    vector<pair<int,int>> q = {{2,4},{1,3},{1,2}};
   
   
    // Function call
    cout<< find(m,q);
    return 0;
}

Python3




# Python implementation of the approach
def find( m, q):
    mx = 0
    pre = [0 for i in range(5) ]
     
    for i in range(m):
        # take input a and b
        a,b = q[i][0], q[i][1]
       
        # add 100 at first index and
        # subtract 100 from last index
       
        # pre[1] becomes 100
        pre[a-1] += 100
          
        # pre[4] becomes -100 and this
        pre[b] -=100
         
        # continues m times as we input diff. values of a and b
    for i in range(1,5):
       
        # add all values in a cumulative way
        pre[i] += pre[i - 1]
         
        # keep track of max value
        mx = max(mx, pre[i])                            
    return mx
   
# Driver Code
m = 3
q = [[2,4],[1,3],[1,2]]
 
# Function call
print(find(m,q))
 
# This code is contributed by rohitsingh07052

Javascript




<script>
 
function find( m,q)
{
    let mx = 0;
    let pre = [0,0,0,0,0];
     
    for (let i = 0; i < m; i++)
    {   
        // take input a and b
        let a = q[i][0], b = q[i][1];
       
        // add 100 at first index and
        // subtract 100 from last index
       
        // pre[1] becomes 100
        pre[a-1] += 100;
          
        // pre[4] becomes -100 and this
        pre[b] -=100;   
        // continues m times as we input diff. values of a and b
    }
    for (let i = 1; i < 5; i++)
    {
        // add all values in a cumulative way
        pre[i] += pre[i - 1];
         
        // keep track of max value
        mx = Math.max(mx, pre[i]);
    }
      
                                
    return mx;
}
 
// Driver Code
let m = 3;
let q = [[2,4],[1,3],[1,2]];
   
   
    // Function call
    document.write(find(m,q));
 
</script>
Output
300

Recent Articles on Prefix Sum Technique
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