Find maximum value of Sum( i*arr[i]) with only rotations on given array allowed

Given an array, only rotation operation is allowed on array. We can rotate the array as many times as we want. Return the maximum possbile of summation of i*arr[i].

Examples :

Input: arr[] = {1, 20, 2, 10}
Output: 72
We can 72 by rotating array twice.
{2, 10, 1, 20}
20*3 + 1*2 + 10*1 + 2*0 = 72

Input: arr[] = {10, 1, 2, 3, 4, 5, 6, 7, 8, 9};
Output: 330
We can 330 by rotating array 9 times.
{1, 2, 3, 4, 5, 6, 7, 8, 9, 10};
0*1 + 1*2 + 2*3 ... 9*10 = 330

We strongly recommend you to minimize your browser and try this yourself first.

A Simple Solution is to find all rotations one by one, check sum of every rotation and return the maximum sum. Time complexity of this solution is O(n²).

We can solve this problem in O(n) time using an Efficient Solution.
Let R_j be value of i*arr[i] with j rotations. The idea is to calculate next rotation value from previous rotation, i.e., calculate R_j from R_j-1. We can calculate initial value of result as R₀, then keep calculating next rotation values.

How to efficiently calculate R_j from R_j-1?
This can be done in O(1) time. Below are details.

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 = arrSum - n * arr[n-j]

Where arrSum is sum of all array elements, i.e., 

arrSum = ∑ arr[i]
        0<=i<=n-1

Below is complete algorithm:

1) Compute sum of all array elements. Let this sum be 'arrSum'.

2) Compute R₀ by doing i*arr[i] for given array. 
   Let this value be currVal.

3) Initialize result: maxVal = currVal // maxVal is result.

// This loop computes R_j from  R_j-1 
4) Do following for j = 1 to n-1
......a) currVal = currVal + arrSum-n*arr[n-j];
......b) If (currVal > maxVal)
            maxVal = currVal   

5) Return maxVal

Below is the implementation of above idea :

C++

// C++ program to find max value of i*arr[i] 
#include <iostream> 
using namespace std; 
  
// Returns max possible value of i*arr[i] 
int maxSum(int arr[], int n) 
{ 
    // Find array sum and i*arr[i] with no rotation 
    int arrSum = 0;  // Stores sum of arr[i] 
    int currVal = 0;  // Stores sum of i*arr[i] 
    for (int i=0; i<n; i++) 
    { 
        arrSum = arrSum + arr[i]; 
        currVal = currVal+(i*arr[i]); 
    } 
  
    // Initialize result as 0 rotation sum 
    int maxVal = currVal; 
  
    // Try all rotations one by one and find 
    // the maximum rotation sum. 
    for (int j=1; j<n; j++) 
    { 
        currVal = currVal + arrSum-n*arr[n-j]; 
        if (currVal > maxVal) 
            maxVal = currVal; 
    } 
  
    // Return result 
    return maxVal; 
} 
  
// Driver program 
int main(void) 
{ 
    int arr[] = {10, 1, 2, 3, 4, 5, 6, 7, 8, 9}; 
    int n = sizeof(arr)/sizeof(arr[0]); 
    cout << "\nMax sum is " << maxSum(arr, n); 
    return 0; 
} 

Java

// Java program to find max value of i*arr[i] 
  
import java.util.Arrays; 
  
class Test 
{ 
    static int arr[] = new int[]{10, 1, 2, 3, 4, 5, 6, 7, 8, 9}; 
      
    // Returns max possible value of i*arr[i] 
    static int maxSum() 
    { 
        // Find array sum and i*arr[i] with no rotation 
        int arrSum = 0;  // Stores sum of arr[i] 
        int currVal = 0;  // Stores sum of i*arr[i] 
        for (int i=0; i<arr.length; i++) 
        { 
            arrSum = arrSum + arr[i]; 
            currVal = currVal+(i*arr[i]); 
        } 
       
        // Initialize result as 0 rotation sum 
        int maxVal = currVal; 
       
        // Try all rotations one by one and find 
        // the maximum rotation sum. 
        for (int j=1; j<arr.length; j++) 
        { 
            currVal = currVal + arrSum-arr.length*arr[arr.length-j]; 
            if (currVal > maxVal) 
                maxVal = currVal; 
        } 
       
        // Return result 
        return maxVal; 
    } 
      
    // Driver method to test the above function 
    public static void main(String[] args)  
    { 
        System.out.println("Max sum is " + maxSum()); 
    } 
} 

Python

'''Python program to find maximum value of Sum(i*arr[i])'''
  
# returns max possible value of Sum(i*arr[i]) 
def maxSum(arr): 
  
    # stores sum of arr[i] 
    arrSum = 0    
  
    # stores sum of i*arr[i] 
    currVal = 0
      
    n = len(arr) 
  
    for i in range(0, n): 
        arrSum = arrSum + arr[i] 
        currVal = currVal + (i*arr[i]) 
  
    # initialize result 
    maxVal = currVal 
  
    # try all rotations one by one and find the maximum  
    # rotation sum 
    for j in range(1, n): 
        currVal = currVal + arrSum-n*arr[n-j] 
        if currVal > maxVal: 
            maxVal = currVal 
  
    # return result 
    return maxVal 
  
# test maxsum(arr) function 
arr = [10, 1, 2, 3, 4, 5, 6, 7, 8, 9] 
print "Max sum is: ", maxSum(arr) 

C#

// C# program to find max value of i*arr[i] 
using System; 
  
class Test 
{ 
    static int []arr = new int[]{10, 1, 2, 3, 4, 
                                  5, 6, 7, 8, 9}; 
      
    // Returns max possible value of i*arr[i] 
    static int maxSum() 
    { 
        // Find array sum and i*arr[i] 
        // with no rotation 
        int arrSum = 0; // Stores sum of arr[i] 
        int currVal = 0; // Stores sum of i*arr[i] 
          
        for (int i = 0; i < arr.Length; i++) 
        { 
            arrSum = arrSum + arr[i]; 
            currVal = currVal + (i * arr[i]); 
        } 
      
        // Initialize result as 0 rotation sum 
        int maxVal = currVal; 
      
        // Try all rotations one by one and find 
        // the maximum rotation sum. 
        for (int j = 1; j < arr.Length; j++) 
        { 
            currVal = currVal + arrSum - arr.Length * 
                                arr[arr.Length - j]; 
            if (currVal > maxVal) 
                maxVal = currVal; 
        } 
      
        // Return result 
        return maxVal; 
    } 
      
    // Driver Code 
    public static void Main()  
    { 
        Console.WriteLine("Max sum is " + maxSum()); 
    } 
} 
  
// This article is contributed by vt_m.  

PHP

<?php 
// PHP program to find max  
// value of i*arr[i]  
  
// Returns max possible  
// value of i*arr[i] 
function maxSum($arr, $n) 
{ 
    // Find array sum and 
    // i*arr[i] with no rotation 
      
    // Stores sum of arr[i] 
    $arrSum = 0;  
      
    // Stores sum of i*arr[i] 
    $currVal = 0;  
    for ($i = 0; $i < $n; $i++) 
    { 
        $arrSum = $arrSum + $arr[$i]; 
        $currVal = $currVal +  
                  ($i * $arr[$i]); 
    } 
  
    // Initialize result as 
    // 0 rotation sum 
    $maxVal = $currVal; 
  
    // Try all rotations one  
    // by one and find the  
    // maximum rotation sum. 
    for ($j = 1; $j < $n; $j++) 
    { 
        $currVal = $currVal + $arrSum -  
                   $n * $arr[$n - $j]; 
        if ($currVal > $maxVal) 
            $maxVal = $currVal; 
    } 
  
    // Return result 
    return $maxVal; 
} 
  
// Driver Code 
$arr = array (10, 1, 2, 3, 4, 
              5, 6, 7, 8, 9); 
$n = sizeof($arr); 
echo "Max sum is " , 
     maxSum($arr, $n); 
  
// This code is contributed by m_kit 
?> 

Output :

Max sum is 330

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

This article is contributed by Nitesh Singh. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above

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