Construct an array from its pair-sum array

Given a pair-sum array arr[], construct the original array res[] where each element in arr represents the sum of a unique pair from res in a specific order, starting with res[0] + res[1], then res[0] + res[2], and so on through all combinations where the first index is less than the second.

Note: The size of the input array will always be n * (n - 1) / 2, which corresponds to the number of such unique pairs for an original array of size n. It's important to note that not every array arr[] corresponds to a valid original array res[]. It's also possible for multiple valid original arrays to exist that produce the same pair-sum array.

Examples

Input: arr[] = [4, 5, 3]
Output: [3, 1, 2]
Explanation: For [3, 1, 2], pairwise sums are (3 + 1), (3 + 2) and (1 + 2)

Input: arr[] = [3]
Output: [1, 2]
Explanation: There may be multiple valid original arrays that produce the same pair-sum array, such as [1, 2] and [2, 1]. Both are considered correct since they yield the same set of pairwise sums.

Approach:

The approach begins by calculating the size n of the original array using the length of the pair-sum array. It then uses the first few sums in arr to compute the first element of the original array by subtracting one sum from the others in a way that isolates it. Once the first element is known, the rest of the elements can be found by subtracting it from the corresponding pair sums.

Illustration:

Once the value of n (the size of the original array) and res[0] (the first element of the original array) is determined, the rest of the array can be reconstructed by using the structure of the pair-sum array. Since the initial elements of arr represent sums of res[0] with each of the remaining elements, we can simply subtract res[0] from each of these to get the other values. Specifically, for each i from 1 to n-1, res[i] can be found as arr[i-1] - res[0].

#include <iostream>
#include <vector>
#include <cmath>
using namespace std;

vector<int> constructArr(vector<int>& arr) {
    
    if(arr.size() == 1) 
        return {1, arr[0]-1} ;
    
    // Size of the result array
    int n = (1 + sqrt(1 + 8 * arr.size())) / 2;
    
    // Find the result array
    vector<int> res(n);
    res[0] = (arr[0] + arr[1] - arr[n - 1]) / 2;
    for (int i = 1; i < n; i++)
        res[i] = arr[i - 1] - res[0];
        
    return res;
}

// Driver program to test the function
int main() {
    vector<int> arr = {4, 5, 3};
    vector<int> res = constructArr(arr);
    for (int x : res)
        cout << x << " ";
    return 0;
}

import java.util.ArrayList;
import java.util.List;

class GfG {

    public static ArrayList<Integer> constructArr(int[] arr) {
        
        // only one pair-sum ⇒ original array has two numbers
        if (arr.length == 1) {
            return new ArrayList<>(List.of(1, arr[0] - 1));
        }

        // Compute n from m = n·(n−1)/2  ⇒  n = (1 + √(1 + 8m)) / 2
        int n = (int) ((1 + Math.sqrt(1 + 8 * arr.length)) / 2);

        int[] res = new int[n];
        
        // res[0] is obtained from the first three relevant sums
        res[0] = (arr[0] + arr[1] - arr[n - 1]) / 2;

        // Remaining elements: subtract res[0] from successive pair sums
        for (int i = 1; i < n; i++) {
            res[i] = arr[i - 1] - res[0];
        }

        ArrayList<Integer> ans = new ArrayList<>(n);
        for (int x : res) ans.add(x);
        return ans;
    }

    public static void main(String[] args) {
    
        int[] arr = {4, 5, 3};
        ArrayList<Integer> res = constructArr(arr);
    
        for (int x : res) System.out.print(x + " ");
    }
}

import math

def constructArr(arr):
    
    # When the pair-sum array has only one element,
    # the original array must have exactly two numbers.
    if len(arr) == 1:
        return [1, arr[0] - 1]

    # Size of the original array 
    n = int((1 + math.isqrt(1 + 8 * len(arr))) / 2)

    # Reconstruct the original array
    res = [0] * n
    res[0] = (arr[0] + arr[1] - arr[n - 1]) // 2
    for i in range(1, n):
        res[i] = arr[i - 1] - res[0]

    return res

# Driver code
if __name__ == "__main__":
    arr = [4, 5, 3]
    res = constructArr(arr)
    print(' '.join(map(str, res)))

using System;
using System.Collections.Generic;

class GfG{
    
    static List<int> constructArr(int[] arr){
        
        // only one pair-sum ⇒ original array has two numbers
        if (arr.Length == 1)
            return new List<int> { 1, arr[0] - 1 };

        // Compute n from m = n·(n−1)/2  ⇒  n = (1 + √(1 + 8m)) / 2
        int n = (int)((1 + Math.Sqrt(1 + 8 * arr.Length)) / 2);

        int[] res = new int[n];
        
        // res[0] is obtained from the first three relevant sums
        res[0] = (arr[0] + arr[1] - arr[n - 1]) / 2;

        // Remaining elements: subtract res[0] from successive pair sums
        for (int i = 1; i < n; i++)
            res[i] = arr[i - 1] - res[0];

        return new List<int>(res);
    }

    static void Main(){
        
        int[] arr = { 4, 5, 3 };
        List<int> res = constructArr(arr);
        foreach (int x in res) Console.Write(x + " ");
    }
}

function constructArr(arr) {
    
    // only one pair-sum 
    // the original array has exactly two numbers
    if (arr.length === 1) {
        return [1, arr[0] - 1];
    }

    // Size of the original array
    const n = Math.floor((1 + Math.sqrt(1 + 8 * arr.length)) / 2);

    // Reconstruct the original array
    const res = new Array(n);
    res[0] = (arr[0] + arr[1] - arr[n - 1]) / 2;
    for (let i = 1; i < n; i++) {
        res[i] = arr[i - 1] - res[0];
    }

    return res;
}

// Driver code
const arr = [4, 5, 3];
const res = constructArr(arr);
console.log(res.join(' ')); 

3 1 2 

Time Complexity: O(n), where n is the size of the original array, since we compute each element of res[] in a single pass.
Auxiliary Space: O(n), for storing the output array res[], no extra space is used beyond that.