Change one element in the given array to make it an Arithmetic Progression

Given an array which is an original arithmetic progression with one element changed. The task is to make it an arithmetic progression again. If there are many such possible sequences, return any one of them. The length of the array will always be greater than 2.
Examples:

Input : arr = [1, 3, 4, 7]
Output : arr = [1, 3, 5, 7]
The common difference for the arithmetic progression is 2, so the resulting
arithmetic progression is [1, 3, 5, 7].

Input : arr = [1, 3, 7]
Output : arr = [-1, 3, 7]
The common difference can be either 2 or 3 or 4.
Hence the resulting array can be either [1, 3, 5], [-1, 3, 7] or [1, 4, 7].
As any one can be chosen, [-1, 3, 7] is returned as the output.



Approach:

The key components of an Arithmetic Progression are initial term and the common difference. So our task is to find these two values to know the actual Arithmetic Progression.

  • If the length of the array is 3, take the common difference as the difference between any two elements.
  • Otherwise, try to find the common difference using the first four elements.
    • Check if the first three elements are in arithmetic progression using the formula a[1] – a[0] = a[2] – a[1]. If they are in arithmetic progression, the common difference d = a[1] – a[0] and the initial term would be a[0]
    • Check if the 2nd, 3rd and 4th elements are in arithmetic progression using the formula a[2] – a[1] = a[3] – a[2]. If they are in arithmetic progression, the common difference d = a[2] – a[1], which means that the first element has been changed, so initial term is a[0] = a[1] – d
    • In both of the above cases, we have checked if the first element or fourth element has been changed. If both the cases are false, then it means that first or fourth element was not changed. So the common difference can be taken as d = (a[3] – a[0])/3 and the initial term = a[0]
  • Print all the elements using the initial term and common difference.

Below is the implementation of the above approach:

C/C++

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// C++ program to change one element of an array such
// that the resulting array is in arithmetic progression.
#include <bits/stdc++.h>
using namespace std;
  
// Finds the initial term and common difference and
// prints the resulting sequence.
void makeAP(int arr[], int n)
{
    int initial_term, common_difference;
  
    if (n == 3) {
        common_difference = arr[2] - arr[1];
        initial_term = arr[1] - common_difference;
    }
  
    else if ((arr[1] - arr[0]) == arr[2] - arr[1]) {
  
        // Check if the first three elements are in
        // arithmetic progression
        initial_term = arr[0];
        common_difference = arr[1] - arr[0];
    }
    else if ((arr[2] - arr[1]) == (arr[3] - arr[2])) {
  
        // Check if the first element is not
        // in arithmetic progression
        common_difference = arr[2] - arr[1];
        initial_term = arr[1] - common_difference;
    }
    else {
  
        // The first and fourth element are
        // in arithmetic progression
        common_difference = (arr[3] - arr[0]) / 3;
        initial_term = arr[0];
    }
  
    // Print the arithmetic progression
    for (int i = 0; i < n; i++)
        cout << initial_term + (i * common_difference) << " ";
    cout << endl;
}
  
// Driver Program
int main()
{
    int arr[] = { 1, 3, 7 };
    int n = sizeof(arr) / sizeof(arr[0]);
  
    makeAP(arr, n);
  
    return 0;
}

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Java

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// Java program to change one element of an array such
// that the resulting array is in arithmetic progression.
import java.util.Arrays;
  
class AP {
    static void makeAP(int arr[], int n)
    {
        int initial_term, common_difference;
  
        // Finds the initial term and common difference and
        // prints the resulting array.
        if (n == 3) {
            common_difference = arr[2] - arr[1];
            initial_term = arr[1] - common_difference;
        }
        else if ((arr[1] - arr[0]) == arr[2] - arr[1]) {
  
            // Check if the first three elements are in
            // arithmetic progression
            initial_term = arr[0];
            common_difference = arr[1] - arr[0];
        }
        else if ((arr[2] - arr[1]) == (arr[3] - arr[2])) {
            // Check if the first element is not
            // in arithmetic progression
            common_difference = arr[2] - arr[1];
            initial_term = arr[1] - common_difference;
        }
        else {
            // The first and fourth element are
            // in arithmetic progression
            common_difference = (arr[3] - arr[0]) / 3;
            initial_term = arr[0];
        }
  
        // Print the arithmetic progression
        for (int i = 0; i < n; i++)
            System.out.print(initial_term +
                            (i * common_difference) + " ");
        System.out.println();
    }
  
    // Driver code
    public static void main(String[] args)
    {
        int arr[] = { 1, 3, 7 };
        int n = arr.length;
        makeAP(arr, n);
    }
}

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Python3

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# Python program to change one element of an array such
# that the resulting array is in arithmetic progression.
def makeAP(arr, n): 
    initial_term, common_difference = 0, 0
    if (n == 3):
        common_difference = arr[2] - arr[1]
        initial_term = arr[1] - common_difference
    elif((arr[1] - arr[0]) == arr[2] - arr[1]):
  
        # Check if the first three elements are in 
        # arithmetic progression
        initial_term = arr[0]
        common_difference = arr[1] - arr[0]
          
    elif((arr[2] - arr[1]) == (arr[3] - arr[2])):
  
        # Check if the first element is not 
        # in arithmetic progression
        common_difference = arr[2] - arr[1]
        initial_term = arr[1] - common_difference
          
    else:
        # The first and fourth element are 
        # in arithmetic progression
        common_difference = (arr[3] - arr[0]) / 3
        initial_term = arr[0]
  
    # Print the arithmetic progression
    for i in range(n):
        print(int(initial_term+
                 (i * common_difference)), end = " ")
    print()
    
# Driver code 
arr = [1, 3, 7
n = len(arr) 
makeAP(arr, n)

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C#

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// C# program to change one element of an array such
// that the resulting array is in arithmetic progression.
using System;
  
public class AP 
{
    static void makeAP(int []arr, int n)
    {
        int initial_term, common_difference;
  
        // Finds the initial term and common difference and
        // prints the resulting array.
        if (n == 3) 
        {
            common_difference = arr[2] - arr[1];
            initial_term = arr[1] - common_difference;
        }
        else if ((arr[1] - arr[0]) == arr[2] - arr[1]) 
        {
  
            // Check if the first three elements are in
            // arithmetic progression
            initial_term = arr[0];
            common_difference = arr[1] - arr[0];
        }
        else if ((arr[2] - arr[1]) == (arr[3] - arr[2]))
        {
            // Check if the first element is not
            // in arithmetic progression
            common_difference = arr[2] - arr[1];
            initial_term = arr[1] - common_difference;
        }
        else
        {
            // The first and fourth element are
            // in arithmetic progression
            common_difference = (arr[3] - arr[0]) / 3;
            initial_term = arr[0];
        }
  
        // Print the arithmetic progression
        for (int i = 0; i < n; i++)
            Console.Write(initial_term +
                            (i * common_difference) + " ");
        Console.WriteLine();
    }
  
    // Driver code
    public static void Main(String[] args)
    {
        int []arr = { 1, 3, 7 };
        int n = arr.Length;
        makeAP(arr, n);
    }
}
  
// This code contributed by Rajput-Ji

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Output:

-1 3 7

Time Complexity: O(N), where N is the number of elements in the array.



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Improved By : Rajput-Ji