Maximum value of Sum(i*arr[i]) with array rotations allowed

Given an array arr[], the task is to determine the maximum possible value of the expression i*arr[i] after rotating the array any number of times (including zero).
Note: In each rotation, every element of the array shifts one position to the right, and the last element moves to the front.

Examples :  

Input: arr[] = [4, 3, 2, 6, 1, 5]
Output: 60
Explanation: After rotating the array 3 times, we get [1, 5, 4, 3, 2, 6]. Now, the sum of i*arr[i] = 0*1 + 1*5 + 2*4 + 3*3 + 4*2 + 5*6 = 0 + 5 + 8 + 9 + 8 + 30 = 60

Input: arr[] = [8, 3, 1, 2]
Output: 29
Explanation: After rotating the array 3 times, we get [3, 1, 2, 8]. Now, the sum of i*arr[i] = 0*3 + 1*1 + 2*2 + 3*8 = 0 + 1 + 4 + 24 = 29.

Input: arr[] = [10, 1, 2, 7, 9, 3]
Output: 105

[Naive Approach] Take Maximum of All Rotations - O(n^2) Time and O(1) Space 

The idea is to check all possible rotations of the array. As on each rotation, the value of the expression changes as the index positions shift. We rotate the array one step at a time and compute the new sum. By tracking the maximum of these sum values, we ensure we capture the best possible configuration.

// C++ Code to find maximum value of Sum of 
// i*arr[i] with rotations using Naive Approach
#include <iostream>
#include <vector>
#include <climits>
using namespace std;

// Function to calculate i*arr[i]
// for given array
int computeSum(vector<int> &arr) {
    
    int n = arr.size();
    int total = 0;

    // Calculate the sum of i*arr[i]
    for (int i = 0; i < n; i++) {
        total += i * arr[i];
    }

    return total;
}

// Function to find maximum value of i*arr[i]
// after any number of rotations
int maxRotateSum(vector<int> &arr) {
    
    int n = arr.size();
        
    int maxVal = INT_MIN;

    // Try all rotations
    for (int r = 0; r < n; r++) {

        // Calculate i*arr[i] for 
        // current rotation
        int currVal = computeSum(arr);

        // Update max value
        maxVal = max(maxVal, currVal);

        // Rotate array by 1 to right
        int last = arr[n - 1];
        for (int i = n - 1; i > 0; i--) {
            arr[i] = arr[i - 1];
        }
        arr[0] = last;
    }

    return maxVal;
}

// Driver code
int main() {
    
    vector<int> arr = {4, 3, 2, 6, 1, 5};
    cout << maxRotateSum(arr);

    return 0;
}

// Java Code to find maximum value of Sum of 
// i*arr[i] with rotations using Naive Approach
import java.util.*;

class GfG {

    // Function to calculate i*arr[i]
    // for given array
    static int computeSum(int[] arr) {

        int n = arr.length;
        int total = 0;

        // Calculate the sum of i*arr[i]
        for (int i = 0; i < n; i++) {
            total += i * arr[i];
        }

        return total;
    }

    // Function to find maximum value of i*arr[i]
    // after any number of rotations
    static int maxRotateSum(int[] arr) {

        int n = arr.length;

        int maxVal = Integer.MIN_VALUE;

        // Try all rotations
        for (int r = 0; r < n; r++) {

            // Calculate i*arr[i] for 
            // current rotation
            int currVal = computeSum(arr);

            // Update max value
            maxVal = Math.max(maxVal, currVal);

            // Rotate array by 1 to right
            int last = arr[n - 1];
            for (int i = n - 1; i > 0; i--) {
                arr[i] = arr[i - 1];
            }
            arr[0] = last;
        }

        return maxVal;
    }

    public static void main(String[] args) {

        int[] arr = {4, 3, 2, 6, 1, 5};
        System.out.println(maxRotateSum(arr));
    }
}

# Python Code to find maximum value of Sum of 
# i*arr[i] with rotations using Naive Approach

# Function to calculate i*arr[i]
# for given array
def computeSum(arr):

    n = len(arr)
    total = 0

    # Calculate the sum of i*arr[i]
    for i in range(n):
        total += i * arr[i]

    return total

# Function to find maximum value of i*arr[i]
# after any number of rotations
def maxRotateSum(arr):

    n = len(arr)

    maxVal = float('-inf')

    # Try all rotations
    for r in range(n):

        # Calculate i*arr[i] for 
        # current rotation
        currVal = computeSum(arr)

        # Update max value
        maxVal = max(maxVal, currVal)

        # Rotate array by 1 to right
        last = arr[n - 1]
        for i in range(n - 1, 0, -1):
            arr[i] = arr[i - 1]
        arr[0] = last

    return maxVal

if __name__ == "__main__":

    arr = [4, 3, 2, 6, 1, 5]
    print(maxRotateSum(arr))

// C# Code to find maximum value of Sum of 
// i*arr[i] with rotations using Naive Approach
using System;

class GfG {

    // Function to calculate i*arr[i]
    // for given array
    static int computeSum(int[] arr) {

        int n = arr.Length;
        int total = 0;

        // Calculate the sum of i*arr[i]
        for (int i = 0; i < n; i++) {
            total += i * arr[i];
        }

        return total;
    }

    // Function to find maximum value of i*arr[i]
    // after any number of rotations
    static int maxRotateSum(int[] arr) {

        int n = arr.Length;

        int maxVal = int.MinValue;

        // Try all rotations
        for (int r = 0; r < n; r++) {

            // Calculate i*arr[i] for 
            // current rotation
            int currVal = computeSum(arr);

            // Update max value
            maxVal = Math.Max(maxVal, currVal);

            // Rotate array by 1 to right
            int last = arr[n - 1];
            for (int i = n - 1; i > 0; i--) {
                arr[i] = arr[i - 1];
            }
            arr[0] = last;
        }

        return maxVal;
    }

    static void Main(string[] args) {

        int[] arr = {4, 3, 2, 6, 1, 5};
        Console.WriteLine(maxRotateSum(arr));
    }
}

// JavaScript Code to find maximum value of Sum of 
// i*arr[i] with rotations using Naive Approach

// Function to calculate i*arr[i]
// for given array
function computeSum(arr) {

    let n = arr.length;
    let total = 0;

    // Calculate the sum of i*arr[i]
    for (let i = 0; i < n; i++) {
        total += i * arr[i];
    }

    return total;
}

// Function to find maximum value of i*arr[i]
// after any number of rotations
function maxRotateSum(arr) {

    let n = arr.length;

    let maxVal = -Infinity;

    // Try all rotations
    for (let r = 0; r < n; r++) {

        // Calculate i*arr[i] for 
        // current rotation
        let currVal = computeSum(arr);

        // Update max value
        maxVal = Math.max(maxVal, currVal);

        // Rotate array by 1 to right
        let last = arr[n - 1];
        for (let i = n - 1; i > 0; i--) {
            arr[i] = arr[i - 1];
        }
        arr[0] = last;
    }

    return maxVal;
}

// Driver Code
let arr = [4, 3, 2, 6, 1, 5];
console.log(maxRotateSum(arr));

60

[Expected Approach] Using Mathematical Formula - O(n) Time and O(1) Space

The idea is to compute the sum of i*arr[i] for each possible rotation without recalculating it from scratch each time. Instead, we calculate the next rotation value from the previous rotation, i.e., calculate R_j from R_j-1. So, we can calculate the initial value of the result as R₀, then keep calculating the next rotation values.

How to Efficiently Calculate R_j from R_j-1? 

Let us calculate initial value of i*arr[i] with no rotation

R₀ = 0*arr[0] + 1*arr[1] +...+ (n-1)*arr[n-1]

After 1 rotation arr[n-1], becomes first element of array, 

• arr[0] becomes second element, arr[1] becomes third element and so on.
• R₁ = 0*arr[n-1] + 1*arr[0] +...+ (n-1)*arr[n-2]
• R₁ - R₀ = arr[0] + arr[1] + ... + arr[n-2] - (n-1)*arr[n-1]

After 2 rotations arr[n-2], becomes first element of array, 

• arr[n-1] becomes second element, arr[0] becomes third element and so on.
• R₂ = 0*arr[n-2] + 1*arr[n-1] +...+ (n-1)*arr[n-3]
• R₂ - R₁ = arr[0] + arr[1] + ... + arr[n-3] - (n-1)*arr[n-2] + arr[n-1]

If we take a closer look at above values, we can observe below pattern:

R_j - R_j-1 = totalSum - n * arr[n-j]

Where totalSum is sum of all array elements

Illustration

Given arr[]={10, 1, 2, 3, 4, 5, 6, 7, 8, 9}, |
arrSum = 55, currVal = summation of (i*arr[i]) = 285
In each iteration the currVal is currVal = currVal + arrSum-n*arr[n-j] ,

1st rotation: currVal = 285 + 55 - (10 * 9) = 250

2nd rotation: currVal = 250 + 55 - (10 * 8) = 225

3rd rotation: currVal = 225 + 55 - (10 * 7) = 210
.......

Last rotation: currVal = 285 + 55 - (10 * 1) = 330

Previous currVal was 285, now it becomes 330.

It's the maximum value we can find hence return 330.

Steps to implement the above idea:

• Start by computing the total sum of all elements and the initial value of i * arr[i] as currVal i.e. R₀.
• Initialize maxVal with currVal to track the maximum.
• Loop from j = 1 to n-1 to simulate all possible rotations using a formula.
• In each iteration, update currVal using the formula: currVal = currVal + totalSum - n * arr[n - j].
• After each update, compare and store the maximum value in maxVal.
• Finally, return maxVal which holds the result for the best rotation.

// C++ Code to find maximum value of Sum of 
// i*arr[i] with rotations using Optimized Approach
#include <iostream>
#include <vector>
#include <climits>
using namespace std;

// Function to find maximum value of i*arr[i]
// after any number of rotations
int maxRotateSum(vector<int> &arr) {

    int n = arr.size();

    int totalSum = 0;
    int currVal = 0;
    
    // Compute initial value of i*arr[i]
    // and total sum
    for (int i = 0; i < n; i++) {
        totalSum += arr[i];
        currVal += i * arr[i];
    }

    // Initialize result with intial sum value
    int maxVal = currVal;

    // Compute sum values for each configuration 
    // and update max
    for (int j = 1; j < n; j++) {
        
        // Current sum value for current configuration
        currVal = currVal + totalSum - n * arr[n - j];

        maxVal = max(maxVal, currVal);
    }

    return maxVal;
}

// Driver code
int main() {
    
    vector<int> arr = {4, 3, 2, 6, 1, 5};
    cout << maxRotateSum(arr);

    return 0;
}

// Java Code to find maximum value of Sum of 
// i*arr[i] with rotations using Optimized Approach
class GfG {

    // Function to find maximum value of i*arr[i]
    // after any number of rotations
    static int maxRotateSum(int[] arr) {

        int n = arr.length;

        int totalSum = 0;
        int currVal = 0;
        
        // Compute initial value of i*arr[i]
        // and total sum
        for (int i = 0; i < n; i++) {
            totalSum += arr[i];
            currVal += i * arr[i];
        }

        // Initialize result with intial sum value
        int maxVal = currVal;

        // Compute sum values for each configuration 
        // and update max
        for (int j = 1; j < n; j++) {
            
            // Current sum value for current configuration
            currVal = currVal + totalSum - n * arr[n - j];

            maxVal = Math.max(maxVal, currVal);
        }

        return maxVal;
    }

    public static void main(String[] args) {

        int[] arr = {4, 3, 2, 6, 1, 5};
        System.out.println(maxRotateSum(arr));
    }
}

# Python Code to find maximum value of Sum of 
# i*arr[i] with rotations using Optimized Approach

def maxRotateSum(arr):

    n = len(arr)

    totalSum = 0
    currVal = 0
    
    # Compute initial value of i*arr[i]
    # and total sum
    for i in range(n):
        totalSum += arr[i]
        currVal += i * arr[i]

    # Initialize result with intial sum value
    maxVal = currVal

    # Compute sum values for each configuration 
    # and update max
    for j in range(1, n):
        
        # Current sum value for current configuration
        currVal = currVal + totalSum - n * arr[n - j]

        maxVal = max(maxVal, currVal)

    return maxVal

if __name__ == "__main__":

    arr = [4, 3, 2, 6, 1, 5]
    print(maxRotateSum(arr))

// C# Code to find maximum value of Sum of 
// i*arr[i] with rotations using Optimized Approach
using System;

class GfG {

    // Function to find maximum value of i*arr[i]
    // after any number of rotations
    public static int maxRotateSum(int[] arr) {

        int n = arr.Length;

        int totalSum = 0;
        int currVal = 0;
        
        // Compute initial value of i*arr[i]
        // and total sum
        for (int i = 0; i < n; i++) {
            totalSum += arr[i];
            currVal += i * arr[i];
        }

        // Initialize result with intial sum value
        int maxVal = currVal;

        // Compute sum values for each configuration 
        // and update max
        for (int j = 1; j < n; j++) {
            
            // Current sum value for current configuration
            currVal = currVal + totalSum - n * arr[n - j];

            maxVal = Math.Max(maxVal, currVal);
        }

        return maxVal;
    }

    public static void Main(string[] args) {

        int[] arr = {4, 3, 2, 6, 1, 5};
        Console.WriteLine(maxRotateSum(arr));
    }
}

// JavaScript Code to find maximum value of Sum of 
// i*arr[i] with rotations using Optimized Approach

function maxRotateSum(arr) {

    let n = arr.length;

    let totalSum = 0;
    let currVal = 0;
    
    // Compute initial value of i*arr[i]
    // and total sum
    for (let i = 0; i < n; i++) {
        totalSum += arr[i];
        currVal += i * arr[i];
    }

    // Initialize result with intial sum value
    let maxVal = currVal;

    // Compute sum values for each configuration 
    // and update max
    for (let j = 1; j < n; j++) {
        
        // Current sum value for current configuration
        currVal = currVal + totalSum - n * arr[n - j];

        maxVal = Math.max(maxVal, currVal);
    }

    return maxVal;
}

// Driver Code
let arr = [4, 3, 2, 6, 1, 5];
console.log(maxRotateSum(arr));

60