Prefix Sum Array – Implementation and Applications in Competitive Programming

Introduction and Implementation :

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

Examples of prefix sum array :

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 prefix sum array, we run through index 1 to last and keep on adding present element with previous value in prefix sum array.

Below is the implementation :

C/C++

// 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

// Implementing prefix sum arrayclass in Java
public 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


Output:

10 14 30 50

Applications :

An Example Problem :

Consider an array of size n with all initial values as 0. Perform given ‘m’ add operations from index ‘a’ to ‘b’ and evaluate highest element in 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.

Efficient approach using Prefix Sum Array :

1 : Run a loop for 'm' times, inputting 'a' and 'b'.
2 : Add 100 at index 'a' and subtract 100 from index 'b+1'.
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 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


Recent Articles on Prefix Sum Technique

This article is contributed by Rohit Thapliyal. If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.

GATE CS Corner    Company Wise Coding Practice

Recommended Posts:







Writing code in comment? Please use ide.geeksforgeeks.org, generate link and share the link here.