Sort an Array which contains 1 to n values using Mathematical Formula

Last Updated : 10 Oct, 2024

Given an array arr[] of size n, having distinct integers from 1 to n, the task is to sort the array.

Input: arr[] = { 2, 1, 5, 4, 3}
Output: {1, 2, 3, 4, 5}

Input: arr[] = {1, 2, 3, 4, 5, 6}
Output: {1, 2, 3, 4, 5, 6}

We have already discussed different approaches to solve the problem in the post Sort an array which contains 1 to n values. In this post, we will discuss a mathematical approach to sort the array.

Using Mathematical Formula – O(n) Time and O(1) Space

The idea is to traverse the input array and for each element arr[i], place it at its correct index, that is (arr[i] – 1). The problem with this approach is that if we update arr[arr[i] – 1] with arr[i] we will end up overwriting the element at index (arr[i] – 1). To solve this problem, we can find the maximum value in the array, that is (n + 1) and use it to store the original as well as the updated value at each index.

Let’s say for any index i, we have the current element as arr[i]. We know that the correct index for arr[i], say correctIdx is (arr[i] – 1). Now, instead of overwriting arr[correctIdx], we add (arr[i] * (n + 1)) to arr[correctIdx]. This is because we can get the original value by arr[i] % (n + 1) and updated value by arr[i] / (n + 1).

After traversing the array and modifying each index, traverse again and update arr[i] to arr[i] / (n + 1) to get the sorted values.

C++

// C++ Program to sort distinct integer array from  // 1 to n using Mathematical formula   #include <iostream> #include <vector> #include <algorithm> using namespace std;  // function to sort the array void sortArr(vector<int> &arr) {     int n = arr.size();          for(int i = 0; i < n; i++) {        int originalVal = arr[i] % (n + 1);        int correctIdx = originalVal - 1;        arr[correctIdx] += originalVal * (n + 1);     }        for(int i = 0; i < n; i++)          arr[i] = arr[i] / (n + 1);    }  int main() {     vector<int> arr = { 2, 1, 5, 4, 3 };     sortArr(arr);       for(int i = 0; i < arr.size(); i++)         cout << arr[i] << " ";        return 0; }

// C Program to sort distinct integer array from  // 1 to n using Mathematical formula   #include <stdio.h>  // function to sort the array void sortArr(int *arr, int n) {       for (int i = 0; i < n; i++) {         int originalVal = arr[i] % (n + 1);         int correctIdx = originalVal - 1;         arr[correctIdx] += originalVal * (n + 1);     }        for (int i = 0; i < n; i++) {         arr[i] = arr[i] / (n + 1);     } }  int main() {     int arr[] = { 2, 1, 5, 4, 3 };     int n = sizeof(arr) / sizeof(arr[0]);     sortArr(arr, n);       for (int i = 0; i < n; i++) {         printf("%d ", arr[i]);     }        return 0; }

Java

// Java Program to sort distinct integer array from  // 1 to n using Mathematical formula   import java.util.Arrays;  class GfG {      // function to sort the array     static void sortArr(int[] arr) {         int n = arr.length;                for (int i = 0; i < n; i++) {             int originalVal = arr[i] % (n + 1);             int correctIdx = originalVal - 1;             arr[correctIdx] += originalVal * (n + 1);         }          for (int i = 0; i < n; i++)             arr[i] = arr[i] / (n + 1);     }      public static void main(String[] args) {         int[] arr = { 2, 1, 5, 4, 3 };         sortArr(arr);           for (int i = 0; i < arr.length; i++)             System.out.print(arr[i] + " ");     } }

Python

# Python Program to sort distinct integer array from  # 1 to n using Mathematical formula   def sortArr(arr):     n = len(arr)          for i in range(n):         originalVal = arr[i] % (n + 1)         correctIdx = originalVal - 1         arr[correctIdx] += originalVal * (n + 1)          for i in range(n):         arr[i] = arr[i] // (n + 1)  if __name__ == "__main__":     arr = [2, 1, 5, 4, 3]     sortArr(arr)       for i in range(len(arr)):         print(arr[i], end=" ")

// C# Program to sort distinct integer array from  // 1 to n using Mathematical formula   using System;  class GfG {        // function to sort the array     static void sortArr(int[] arr) {         int n = arr.Length;         for (int i = 0; i < n; i++) {             int originalVal = arr[i] % (n + 1);             int correctIdx = originalVal - 1;             arr[correctIdx] += originalVal * (n + 1);         }          for (int i = 0; i < n; i++)             arr[i] = arr[i] / (n + 1);     }      static void Main() {         int[] arr = { 2, 1, 5, 4, 3 };         sortArr(arr);         for (int i = 0; i < arr.Length; i++) {             Console.Write(arr[i] + " ");         }     } }

JavaScript

// JavaScript Program to sort distinct integer array from  // 1 to n using Mathematical formula   function sortArr(arr) {     let n = arr.length;          for (let i = 0; i < n; i++) {         let originalVal = arr[i] % (n + 1);         let correctIdx = originalVal - 1;         arr[correctIdx] += originalVal * (n + 1);     }        for (let i = 0; i < n; i++)         arr[i] = Math.floor(arr[i] / (n + 1)); }  let arr = [2, 1, 5, 4, 3]; sortArr(arr);   console.log(arr.join(" "));

Output

1 2 3 4 5

Convert an Array to reduced form using Vector of pairs

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Sort an Array which contains 1 to n values using Mathematical Formula

Using Mathematical Formula – O(n) Time and O(1) Space

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