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Given an array nums of length n, we define a rotation function F on nums as follows:

F(k) = 0 * nums[k] + 1 * nums[k+1] + 2 * nums[k+2] + ... + (n-1) * nums[k+n-1]

where k is an index in the array. k+n-1 is calculated modulo n to ensure that it wraps around the array.

Return the maximum value of F(k) for k in the range 0 to n-1.

Example:

Input: nums = [4, 3, 2, 6]
Output: 26
Explanation:
F(0) = 0*4 + 1*3 + 2*2 + 3*6 = 26
F(1) = 0*6 + 1*4 + 2*3 + 3*2 = 25
F(2) = 0*2 + 1*6 + 2*4 + 3*3 = 24
F(3) = 0*3 + 1*2 + 2*6 + 3*4 = 23

Clarifying Questions

  1. What should we return if the input nums is empty?
    • Typically, for an empty array, the function should return 0 or it could be undefined.
  2. Are the elements of the array nums always integers?
    • Yes, nums is an array of integers.
  3. Is there any range constraint on the elements of nums?
    • Usually, there are constraints, but assume typical integer constraints (-10^7 to 10^7).

Strategy

  1. Initial Calculation:
    • Compute the sum of the array S.
    • Compute the initial value F(0).
  2. Recurrence Relation:
    • Use the relationship between F(k) and F(k-1) to compute the values in O(1) time.
    • The relationship can be derived from the definition of F(k): 
      F(k) = F(k-1) + S - n * nums[n-k]
    • This allows us to update the value of F efficiently for each step.
  3. Max Value Search:
    • Iterate over the computed F values to find the maximum.

Code

Here is a Python implementation of the optimized solution:

def maxRotateFunction(nums):
    if not nums:
        return 0
    
    n = len(nums)
    total_sum = sum(nums)
    
    # Initial F(0) computation
    F_0 = sum(i * nums[i] for i in range(n))
    
    max_value = F_0
    current_value = F_0
    
    # Compute F(k) for k from 1 to n-1
    for k in range(1, n):
        current_value = current_value + total_sum - n * nums[-k]
        max_value = max(max_value, current_value)
    
    return max_value

# Example usage:
nums = [4, 3, 2, 6]
print(maxRotateFunction(nums))  # Output: 26

Time Complexity

The total time complexity of the solution is O(n). The memory complexity is O(1), aside from the input storage, as we only use a fixed amount of additional space.

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