Minimize sum of squares of adjacent elements difference

Given an Array arr[] or N elements, the task is to minimize the sum of squares of difference of adjacent elements by adding one element at any position of the array.

Examples:

Input: N = 4, arr = [4, 7, 1, 4]
Output: 36
Explanation: The sum of squares of difference of adjacent element before inserting any element is (4-7)^2 + (7-1)^2 + (1-4)^2 = 9+36+9 = 54. After insertion 4 in between 7 and 1 array become 
arr = [4, 7, 4, 1, 4] and sum of square of adjacent element will become 9+9+9+9 = 36.

Input:- N = 4, arr = [3, 5, 1, 8]
Output: 45
Explanation:-Sum of squares of differences of adjacent elements before insertion is 4+16+49 = 69. After insertion 4 in between 1 and 8 the array will become arr = [3, 5, 1, 4, 8] the sum will become 4+16+9+16 = 45. That is the minimum sum which you can make.

Approach: To solve the problem follow the below observations:

• So first of all we have to observe that we are adding the sum of squares of differences of adjacent elements, so to minimize we need to minimize the maximum square so that answer will get more minimized.
• We will check for all adjacent element differences and find the maximum difference so that we can minimize it.
• After it, we will add an element between the elements which are forming the maximum square.
• We will add the element which is close to both the elements if we have to add an element in (1, 9) then we will add 5. So the difference of 1, 5 will be 4 and of 5, 9 will be 4.
• And if we have to add an element in between elements in which the difference is odd then we will do that like this If we have to add between (1, 8) then we will add 5 or 4. Because of we add 5 then the difference between (1, 5) will be 4 and (5, 8) will be 3, If we add 4 then the difference between (1, 4) is 3, and between (4, 8) will be 4. So we can add either 4 or 5.

Below is the code for the above approach:

// C++ code for the above approach:
#include <bits/stdc++.h>
using namespace std;

// Function to find minimum sum
int minimumSum(int n, vector<int> arr)
{

    // To store answer
    long long ans = 0;

    // To store maximum difference
    // between to adjacent elements
    int diff = 0;

    // Iterating over array
    for (int i = 1; i < n; i++) {

        // Sqaure of current adjacent
        // element difference
        long long temp = pow(abs(arr[i] - arr[i - 1]), 2);

        // Adding to answer
        ans += temp;

        // Taking maximum difference
        diff = max(diff, abs(arr[i] - arr[i - 1]));
    }

    // Difference of elements
    // in between element adding
    // after adding element
    int one = diff / 2, two = diff / 2;

    // If diff is odd
    if (diff % 2) {
        two++;
    }

    long long t = pow(one, 2) + pow(two, 2);

    t = pow(diff, 2) - t;

    // Decresing the sum after adding element
    ans -= t;

    return ans;
}

// Driver code
int main()
{
    int N = 4;

    vector<int> arr = { 4, 7, 1, 4 };

    // Function call
    cout << minimumSum(N, arr);

    return 0;
}

// JAVA code for the above approach:
import java.util.*;

public class GFG {
    // Function to find minimum sum
    public static long minimumSum(int n, List<Integer> arr)
    {
        // To store answer
        long ans = 0;

        // To store maximum difference between two adjacent
        // elements
        int diff = 0;

        // Iterating over array
        for (int i = 1; i < n; i++) {
            // Square of current adjacent element difference
            long temp = (long)Math.pow(
                Math.abs(arr.get(i) - arr.get(i - 1)), 2);

            // Adding to answer
            ans += temp;

            // Taking maximum difference
            diff = Math.max(
                diff,
                Math.abs(arr.get(i) - arr.get(i - 1)));
        }

        // Difference of elements in between element adding
        // after adding element
        int one = diff / 2, two = diff / 2;

        // If diff is odd
        if (diff % 2 != 0) {
            two++;
        }

        long t = (long)Math.pow(one, 2)
                 + (long)Math.pow(two, 2);

        t = (long)Math.pow(diff, 2) - t;

        // Decreasing the sum after adding element
        ans -= t;

        return ans;
    }

    // Driver code
    public static void main(String[] args)
    {
        int N = 4;

        List<Integer> arr
            = new ArrayList<>(Arrays.asList(4, 7, 1, 4));

        // Function call
        System.out.println(minimumSum(N, arr));
    }
}

// This code is contributed by rambabuguphka

# Python3 code for the above approach:
# Function to find minimum sum


def minimumSum(n, arr):
    # To store answer
    ans = 0

    # To store maximum difference
    # between two adjacent elements
    diff = 0

    # Iterating over array
    for i in range(1, n):
        # Square of current adjacent
        # element difference
        temp = pow(abs(arr[i] - arr[i - 1]), 2)

        # Adding to answer
        ans += temp

        # Taking maximum difference
        diff = max(diff, abs(arr[i] - arr[i - 1]))

    # Difference of elements
    # in between element adding
    # after adding element
    one = diff // 2
    two = diff // 2

    # If diff is odd
    if diff % 2:
        two += 1

    t = pow(one, 2) + pow(two, 2)

    t = pow(diff, 2) - t

    # Decreasing the sum after adding element
    ans -= t

    return ans


# Driver code
if __name__ == "__main__":
    N = 4

    arr = [4, 7, 1, 4]

    # Function call
    print(minimumSum(N, arr))


# This code is contributed by rambabuguphka

using System;
using System.Collections.Generic;

class GFG {
    // Function to find minimum sum
    static long MinimumSum(int n, List<int> arr)
    {
        // To store answer
        long ans = 0;

        // To store maximum difference between two adjacent
        // elements
        int diff = 0;

        // Iterating over array
        for (int i = 1; i < n; i++) {
            // Square of current adjacent element difference
            long temp = (long)Math.Pow(
                Math.Abs(arr[i] - arr[i - 1]), 2);

            // Adding to answer
            ans += temp;

            // Taking maximum difference
            diff = Math.Max(diff,
                            Math.Abs(arr[i] - arr[i - 1]));
        }

        // Difference of elements in between element adding
        // after adding element
        int one = diff / 2, two = diff / 2;

        // If diff is odd
        if (diff % 2 != 0) {
            two++;
        }

        long t = (long)Math.Pow(one, 2)
                 + (long)Math.Pow(two, 2);

        t = (long)Math.Pow(diff, 2) - t;

        // Decreasing the sum after adding element
        ans -= t;

        return ans;
    }

    static void Main(string[] args)
    {
        int N = 4;
        List<int> arr = new List<int>{ 4, 7, 1, 4 };

        // Function call
        Console.WriteLine(MinimumSum(N, arr));
    }
}

// This code is contributed by shivamgupta0987654321

// Function to find minimum sum
function minimumSum(n, arr) {
    // To store answer
    let ans = 0;

    // To store maximum difference
    // between two adjacent elements
    let diff = 0;

    // Iterating over the array
    for (let i = 1; i < n; i++) {
        // Square of the current adjacent
        // element difference
        let temp = Math.pow(Math.abs(arr[i] - arr[i - 1]), 2);

        // Adding to the answer
        ans += temp;

        // Taking the maximum difference
        diff = Math.max(diff, Math.abs(arr[i] - arr[i - 1]));
    }

    // Difference of elements
    // in between elements adding
    // after adding elements
    let one = Math.floor(diff / 2);
    let two = Math.floor(diff / 2);

    // If diff is odd
    if (diff % 2 === 1) {
        two++;
    }

    let t = Math.pow(one, 2) + Math.pow(two, 2);

    t = Math.pow(diff, 2) - t;

    // Decreasing the sum after adding element
    ans -= t;

    return ans;
}

// Driver code
const N = 4;
const arr = [4, 7, 1, 4];

// Function call
console.log(minimumSum(N, arr));
// Contributed by Aditi Tyagi

36

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