Original Array from given Prefix Sums

You are given a prefix sum array presum[] of an array arr[]. The task is to find the original array arr[] whose prefix sum is presum[].

Examples: 

Input:  presum[] = {5, 7, 10, 11, 18}
Output: [5, 2, 3, 1, 7]
Explanation: Original array {5, 2, 3, 1, 7} 
Prefix sum array = {5, 5+2, 5+2+3, 5+2+3+1, 5+2+3+1+7} = {5, 7, 10, 11, 18}
Each element of original array is replaced by the sum of the prefix of current index.

Input: presum[] = {45, 57, 63, 78, 89, 97}
Output: [45, 12, 6, 15, 11, 8] 

The problem can be solved using the following observation:

Given a prefix sum array presum[] and the original array arr[] of size n, the prefix sum array is calculated as:

• presum[0] = arr[0]
• presum[i] = arr[0] + arr[1] + ... + arr[i] for all i in the range [1, N-1]

This means: presum[i] = presum[i-1] + arr[i]. Hence, the original array elements can be calculated as:

• arr[0] = presum[0]
• For all i in the range [1, n-1], arr[i] = presum[i] - presum[i-1]

Steps to Solve:

1. Traverse the presum[] array starting from the beginning.
2. If the index i = 0, then arr[i] = presum[i].
3. Else, for all i > 0, arr[i] = presum[i] - presum[i-1].

#include <bits/stdc++.h>
using namespace std;

// Function to decode and return the original array as a vector
vector<int> decodeArray(vector<int>& presum) {
    int n = presum.size();
    vector<int> arr(n);

    // Calculating elements of original array
    arr[0] = presum[0];
    for (int i = 1; i < n; i++) {
        arr[i] = presum[i] - presum[i - 1];
    }

    return arr;
}

int main() {
    vector<int> presum = { 45, 57, 63, 78, 89, 97 };

    // Function call and storing the result
    vector<int> arr = decodeArray(presum);

    // Displaying elements of the original array
    for (int num : arr) {
        cout << num << " ";
    }

    return 0;
}

// Function to decode and return the original array as a list
import java.util.ArrayList;
import java.util.List;

public class GfG {
    public static List<Integer> decodeArray(int[] presum) {
        int n = presum.length;
        List<Integer> arr = new ArrayList<>(n);

        // Calculating elements of original array
        arr.add(presum[0]);
        for (int i = 1; i < n; i++) {
            arr.add(presum[i] - presum[i - 1]);
        }

        return arr;
    }

    public static void main(String[] args) {
        int[] presum = { 45, 57, 63, 78, 89, 97 };

        // Function call and storing the result
        List<Integer> arr = decodeArray(presum);

        // Displaying elements of the original array
        for (int num : arr) {
            System.out.print(num + " ");
        }
    }
}

# Function to decode and return the original array as a list
def decode_array(presum):
    
    # Calculating elements of original array
    arr = [presum[0]]
    for i in range(1, len(presum)):
        arr.append(presum[i] - presum[i - 1])
    return arr

if __name__ == '__main__':
    presum = [45, 57, 63, 78, 89, 97]

    # Function call and storing the result
    arr = decode_array(presum)

    # Displaying elements of the original array
    for num in arr:
        print(num, end=' ')

using System;

class GfG {
    // Function to decode and return the original 
    // array as an array
    static int[] DecodeArray(int[] presum) {
        int n = presum.Length;
        int[] arr = new int[n];

        // Calculating elements of original array
        arr[0] = presum[0];
        for (int i = 1; i < n; i++) {
            arr[i] = presum[i] - presum[i - 1];
        }

        return arr;
    }

    static void Main() {
        int[] presum = new int[] { 45, 57, 63, 78, 89, 97 };

        // Function call and storing the result
        int[] arr = DecodeArray(presum);

        // Displaying elements of the original array
        foreach (int num in arr) {
            Console.Write(num + " ");
        }
    }
}

// Function to decode and return the original array
function decodeArray(presum) {
    const n = presum.length;
    let arr = new Array(n);

    // Calculating elements of the original array
    arr[0] = presum[0];
    for (let i = 1; i < n; i++) {
        arr[i] = presum[i] - presum[i - 1];
    }

    return arr;
}

// Driver code to test the function
const presum = [45, 57, 63, 78, 89, 97];

// Function call and storing the result
const arr = decodeArray(presum);

// Displaying elements of the original array
console.log(arr.join(" "));

45 12 6 15 11 8 

Time Complexity : O(n)
Auxiliary Space : O(1)

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Article Tags :

• Algo Geek
• Arrays
• DSA
• Mathematical
• Algo-Geek 2021
• prefix-sum

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Practice Tags :

• Arrays
• Mathematical
• prefix-sum