Given an array arr[] of size N, find the prefix sum of the array. A 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.
Approach: To solve the problem follow the given steps:
- Declare a new array prefixSum[] of the same size as the input array
- Run a for loop to traverse the input array
- For each index add the value of the current element and the previous value of the prefix sum array
Below is the implementation of the above approach:
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]; // Function call fillPrefixSum(arr, n, prefixSum); for (int i = 0; i < n; i++) cout << prefixSum[i] << " "; } // This code is contributed by Aditya Kumar (adityakumar129) |
C
// C program for Implementing prefix sum array #include <stdio.h> // 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]; // Function call fillPrefixSum(arr, n, prefixSum); for (int i = 0; i < n; i++) printf("%d ", prefixSum[i]); } // This code is contributed by Aditya Kumar (adityakumar129) |
Java
// Java Program for Implementing prefix sum arrayclass import java.io.*; 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]; // Function call fillPrefixSum(arr, n, prefixSum); for (int i = 0; i < n; i++) System.out.print(prefixSum[i] + " "); System.out.println(""); } } // This code is contributed by Aditya Kumar (adityakumar129) |
Python3
# Python3 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 if __name__ == '__main__': arr = [10, 4, 16, 20] n = len(arr) # Function call 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]; // Function call 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); // Function call 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
Time Complexity: O(N)
Auxiliary Space: O(N)
Sum of an array between indexes L and R using Prefix Sum:
Given an array arr[] of size N. Given Q queries and in each query given L and R, Print the sum of array elements from index L to R.
Follow the given steps to solve the problem:
- Create the prefix sum array of the given input array
- Now for every query (1-based indexing)
- If L is greater than 1, then print prefixSum[R] – prefixSum[L-1]
- else print prefixSum[R]
Below is the implementation of the above approach.
C++14
// C++ program for the above approach #include <bits/stdc++.h> using namespace std; const int N = 1e5 + 10; int a[N]; int pf[N]; // Driver code int main() { int n = 6; int a[] = { 3, 6, 2, 8, 9, 2 }; //Calculating prefix sum pf[0] = a[0]; for (int i = 1; i < n; i++) { pf[i] = pf[i - 1] + a[i]; } int q = 4; //Creating a 2D vector to store queries and display output vector<vector<int> > query = { { 2, 3 }, { 4, 6 }, { 1, 5 }, { 3, 6 } }; //Printing sum from Queries for (int i = 0; i < q; i++) { int l = query[i][0], r = query[i][1]; if (r > n || l < 1) { cout << "Please input in range 1 to " << n << endl; continue; } if (l == 1) cout << pf[r - 1] << endl; else cout << pf[r - 1] - pf[l - 2] << endl; } return 0; } |
Java
// Java program for the above approach import java.util.*; class GFG { // Driver code public static void main(String[] args) { int n = 6; int[] a = { 3, 6, 2, 8, 9, 2 }; int[] pf = new int[n + 2]; pf[0] = 0; for (int i = 0; i < n; i++) { pf[i + 1] = pf[i] + a[i]; } int[][] q = { { 2, 3 }, { 4, 6 }, { 1, 5 }, { 3, 6 } }; for (int i = 0; i < q.length; i++) { int l = q[i][0]; int r = q[i][1]; // Calculating sum from r to l. System.out.print(pf[r] - pf[l - 1] + "\n"); } } } // This code is contributed by umadevi9616 |
Python3
# Python3 program for the above approach # Driver code if __name__ == '__main__': n = 6 a = [3, 6, 2, 8, 9, 2] pf = [0 for i in range(n+2)] for i in range(n): pf[i + 1] = pf[i] + a[i] q = [[2, 3], [4, 6], [1, 5], [3, 6]] for i in range(4): l = q[i][0] r = q[i][1] # Calculating sum from r to l. print(pf[r] - pf[l - 1]) # This code is contributed by gauravrajput1 |
C#
// C# program for the above approach using System; public class GFG { // Driver code public static void Main(String[] args) { int n = 6; int[] a = { 3, 6, 2, 8, 9, 2 }; int[] pf = new int[n + 2]; pf[0] = 0; for (int i = 0; i < n; i++) { pf[i + 1] = pf[i] + a[i]; } int[, ] q = { { 2, 3 }, { 4, 6 }, { 1, 5 }, { 3, 6 } }; for (int i = 0; i < q.GetLength(0); i++) { int l = q[i, 0]; int r = q[i, 1]; // Calculating sum from r to l. Console.Write(pf[r] - pf[l - 1] + "\n"); } } } // This code is contributed by gauravrajput1 |
Javascript
<script> var n = 6; var a = [ 3, 6, 2, 8, 9, 2 ]; var pf = Array(n+2).fill(0); pf[0] = 0; for (i = 0; i < n; i++) { pf[i+1] = pf[i] + a[i]; } var q = [ [ 2, 3 ], [ 4, 6 ], [ 1, 5 ], [ 3, 6 ] ]; for (i = 0; i < q.length; i++) { var l = q[i][0]; var r = q[i][1]; // Calculating sum from r to l. document.write(pf[r] - pf[l - 1] + "<br/>"); } // This code is contributed by gauravrajput1 </script> |
Output
8 19 28 21
Time Complexity: O(N)
Auxiliary Space: O(N)
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 the elements from a to b (both inclusive).
Example:
Input: n = 5, 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
Naive Approach: To solve the problem follow the below idea:
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.
Time Complexity: O(N * M)
Auxiliary Space: O(1)
Efficient Approach: To solve the problem follow the below idea:
The efficient approach is to use Prefix Sum Array
Follow the given steps to solve the problem:
- Run a loop for ‘m‘ times, inputting ‘a‘ and ‘b‘.
- Add 100 at index ‘a-1‘ and subtract 100 from index ‘b‘.
- After completion of ‘m‘ operations, compute the prefix sum array.
- Scan the largest element and we’re done.
Explanation: We added 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.
Below is the illustration of the above approach:
After I operation –
A[] = {0, 100, 0, 0, -100}After II operation –
A[] = {100, 100, 0, -100, -100}After III operation –
A[] = {200, 100, -100, -100, -100}Final Prefix Sum Array : 200 300 200 100 0
The required highest element : 300
Below is the implementation of the above approach:
C++14
// C++ program for the above approach #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; } |
Java
// Java program for the above approach import java.util.*; class GFG { static class pair { int first, second; public pair(int first, int second) { this.first = first; this.second = second; } } static int find(int m, pair[] q) { int mx = 0; int[] pre = new int[5]; 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 = Math.max(mx, pre[i]); } return mx; } // Driver Code public static void main(String[] args) { int m = 3; pair[] q = { new pair(2, 4), new pair(1, 3), new pair(1, 2) }; // Function call System.out.print(find(m, q)); } } // This code is contributed by gauravrajput1 |
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 = 3q = [[2, 4], [1, 3], [1, 2]] # Function call print(find(m, q)) # This code is contributed by rohitsingh07052 |
C#
// C# program for the above approach using System; public class GFG { public class pair { public int first, second; public pair(int first, int second) { this.first = first; this.second = second; } } static int find(int m, pair[] q) { int mx = 0; int[] pre = new int[5]; 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 = Math.Max(mx, pre[i]); } return mx; } // Driver Code public static void Main(String[] args) { int m = 3; pair[] q = { new pair(2, 4), new pair(1, 3), new pair(1, 2) }; // Function call Console.Write(find(m, q)); } } // This code is contributed by gauravrajput1. |
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
Time Complexity: O(N + M), where N is the size of the array and M is the number of operations
Auxiliary Space: O(N)
Applications of Prefix Sum:
- Equilibrium index of an array: The equilibrium index of an array is an index such that the sum of elements at lower indexes is equal to the sum of elements at higher indexes.
- Find if there is a subarray with 0 sums: Given an array of positive and negative numbers, find if there is a subarray (of size at least one) with 0 sum.
- Maximum subarray size, such that all subarrays of that size have a sum less than k: Given an array of n positive integers and a positive integer k, the task is to find the maximum subarray size such that all subarrays of that size have the sum of elements less than k.
- Find the prime numbers which can be written as sum of most consecutive primes: Given an array of limits. For every limit, find the prime number which can be written as the sum of the most consecutive primes smaller than or equal to the limit.
- Longest Span with same Sum in two Binary arrays: Given two binary arrays, arr1[] and arr2[] of the same size n. Find the length of the longest common span (i, j) where j >= i such that arr1[i] + arr1[i+1] + …. + arr1[j] = arr2[i] + arr2[i+1] + …. + arr2[j].
- Maximum subarray sum modulo m: Given an array of n elements and an integer m. The task is to find the maximum value of the sum of its subarray modulo m i.e find the sum of each subarray mod m and print the maximum value of this modulo operation.
- Maximum subarray size, such that all subarrays of that size have sum less than k: Given an array of n positive integers and a positive integer k, the task is to find the maximum subarray size such that all subarrays of that size have the sum of elements less than k.
- Maximum occurred integer in n ranges : Given n ranges of the form L and R, the task is to find the maximum occurring integer in all the ranges. If more than one such integer exits, print the smallest one.
- Minimum cost for acquiring all coins with k extra coins allowed with every coin: You are given a list of N coins of different denominations. you can pay an amount equivalent to any 1 coin and can acquire that coin. In addition, once you have paid for a coin, we can choose at most K more coins and can acquire those for free. The task is to find the minimum amount required to acquire all the N coins for a given value of K.
- Random number generator in arbitrary probability distribution fashion: Given n numbers, each with some frequency of occurrence. Return a random number with a probability proportional to its frequency of occurrence.
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