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

  • Last Updated : 28 Dec, 2021

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

Examples :  

Input: arr[] = {1, 20, 2, 10}
Output: 72
We can get 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 get 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 Rj be value of i*arr[i] with j rotations. The idea is to calculate next rotation value from previous rotation, i.e., calculate Rj from Rj-1. We can calculate initial value of result as R0, then keep calculating next rotation values. 

How to efficiently calculate Rj from Rj-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

Rj - Rj-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 Rj from  Rj-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 :

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)

Output : 

Max sum is 330

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

Please refer complete article on Find maximum value of Sum( i*arr[i]) with only rotations on given array allowed for more details!


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