Python Program to Find the Maximum sum of i*arr[i] among all rotations of a given array

Given an array arr[] of n integers, find the maximum that maximizes the sum of the value of i*arr[i] where i varies from 0 to n-1.

Examples:

Input: arr[] = {8, 3, 1, 2}
Output: 29
Explanation: Lets look at all the rotations,
{8, 3, 1, 2} = 8*0 + 3*1 + 1*2 + 2*3 = 11
{3, 1, 2, 8} = 3*0 + 1*1 + 2*2 + 8*3 = 29
{1, 2, 8, 3} = 1*0 + 2*1 + 8*2 + 3*3 = 27
{2, 8, 3, 1} = 2*0 + 8*1 + 3*2 + 1*3 = 17

Input: arr[] = {3, 2, 1}
Output: 7
Explanation: Lets look at all the rotations,
{3, 2, 1} = 3*0 + 2*1 + 1*2 = 4
{2, 1, 3} = 2*0 + 1*1 + 3*2 = 7
{1, 3, 2} = 1*0 + 3*1 + 2*2 = 7

Method 1: This method discusses the Naive Solution which takes O(n²) amount of time. 
The solution involves finding the sum of all the elements of the array in each rotation and then deciding the maximum summation value. 

• Approach:A simple solution is to try all possible rotations. Compute sum of i*arr[i] for every rotation and return maximum sum.
• Algorithm:
  1. Rotate the array for all values from 0 to n.
  2. Calculate the sum for each rotations.
  3. Check if the maximum sum is greater than the current sum then update the maximum sum.
• Implementation:

Python3

# A Naive Python3 program to find
# maximum sum rotation
import sys

# Returns maximum value of i * arr[i]
def maxSum(arr, n):

    # Initialize result
    res = -sys.maxsize

    # Consider rotation beginning with i
    # for all possible values of i.
    for i in range(0, n):


        # Initialize sum of current rotation
        curr_sum = 0

        # Compute sum of all values. We don't
        # actually rotation the array, but
        # compute sum by finding indexes when
        # arr[i] is first element
        for j in range(0, n):

            index = int((i + j)% n)
            curr_sum += j * arr[index]


        # Update result if required
        res = max(res, curr_sum)
    return res

# Driver code
arr = [8, 3, 1, 2]
n = len(arr)

print(maxSum(arr, n))

# This code is contributed by
# Smitha Dinesh Semwal

Output :

29

• Complexity Analysis:
  • Time Complexity : O(n²), as we are using nested loops.
  • Auxiliary Space : O(1), as we are not using any extra space.

Method 2: This method discusses the efficient solution which solves the problem in O(n) time. In the naive solution, the values were calculated for every rotation. So if that can be done in constant time then the complexity will decrease.

• Approach: The basic approach is to calculate the sum of new rotation from the previous rotations. This brings up a similarity where only the multipliers of first and last element change drastically and the multiplier of every other element increases or decreases by 1. So in this way, the sum of next rotation can be calculated from the sum of present rotation.
• Algorithm:
  The idea is to compute the value of a rotation using values of previous rotation. When an array is rotated by one, following changes happen in sum of i*arr[i]. 
  1. Multiplier of arr[i-1] changes from 0 to n-1, i.e., arr[i-1] * (n-1) is added to current value.
  2. Multipliers of other terms is decremented by 1. i.e., (cum_sum – arr[i-1]) is subtracted from current value where cum_sum is sum of all numbers.

next_val = curr_val - (cum_sum - arr[i-1]) + arr[i-1] * (n-1);

next_val = Value of ∑i*arr[i] after one rotation.
curr_val = Current value of ∑i*arr[i] 
cum_sum = Sum of all array elements, i.e., ∑arr[i].

Lets take example {1, 2, 3}. Current value is 1*0+2*1+3*2
= 8. Shifting it by one will make it {2, 3, 1} and next value
will be 8 - (6 - 1) + 1*2 = 5 which is same as 2*0 + 3*1 + 1*2

• Implementation:
  

Python3

# An efficient Python3 program to
# compute maximum sum of i * arr[i]

def maxSum(arr, n):

    # Compute sum of all array elements
    cum_sum = 0

    for i in range(0, n):
        cum_sum += arr[i]

    # Compute sum of i * arr[i] for
    # initial configuration.
    curr_val = 0

    for i in range(0, n):
        curr_val += i * arr[i]

    # Initialize result
    res = curr_val

    # Compute values for other iterations
    for i in range(1, n):

        # Compute next value using previous
        # value in O(1) time
        next_val = (curr_val - (cum_sum - arr[i-1]) +
                                    arr[i-1] * (n-1))

        # Update current value
        curr_val = next_val

        # Update result if required
        res = max(res, next_val)

    return res


# Driver code
arr = [8, 3, 1, 2]
n = len(arr)

print(maxSum(arr, n))

# This code is contributed by
# Smitha Dinesh Semwal

Output:

29

• Complexity analysis:
  • Time Complexity: O(n). 
    Since one loop is needed from 0 to n to check all rotations and the sum of the present rotation is calculated from the previous rotations in O(1) time).
  • Auxiliary Space: O(1). 
    As no extra space is required to so the space complexity will be O(1)
• Approach: Let’s assume the case of a sorted array. As we know for an array the maximum sum will be when the array is sorted in ascending order. In case of a sorted rotated array, we can rotate the array to make it in ascending order. So, in this case, the pivot element is needed to be found following which the maximum sum can be calculated.
• Algorithm:
  1. Find the pivot of the array: if arr[i] > arr[(i+1)%n] then it is the pivot element. (i+1)%n is used to check for the last and first element.
  2. After getting pivot the sum can be calculated by finding the difference with the pivot which will be the multiplier and multiply it with the current element while calculating the sum
• Implementations:
• Complexity analysis:
  • Time Complexity : O(n) 
    As only one loop was needed to traverse from 0 to n to find the pivot. To find the sum another loop was needed, so the complexity remains O(n).
  • Auxiliary Space : O(1). 
    We do not require extra space to so the Auxiliary space is O(1)