#396

Medium

auto_awesomeState transition dynamic programming

LeetCode Problem Workspace

Rotate Function

Maximize the rotation function by rotating the array and calculating the weighted sum for all rotations.

Problem Statement

Given an array of integers, nums, you are to compute a rotation function F(k) for each rotation k. The function F(k) is defined as a weighted sum of the array elements, where each element's value is multiplied by its index in the rotated array. Your task is to find the maximum value of F(0), F(1), ..., F(n-1).

The function is designed to compute the weighted sum of elements in an array after rotating it k positions clockwise. You must determine the maximum value that can be obtained by rotating the array, considering the constraints where n can be as large as 100,000 and the values of the array elements range between -100 and 100.

Examples

Example 1

Input: nums = [4,3,2,6]

Output: 26

F(0) = (0 * 4) + (1 * 3) + (2 * 2) + (3 * 6) = 0 + 3 + 4 + 18 = 25 F(1) = (0 * 6) + (1 * 4) + (2 * 3) + (3 * 2) = 0 + 4 + 6 + 6 = 16 F(2) = (0 * 2) + (1 * 6) + (2 * 4) + (3 * 3) = 0 + 6 + 8 + 9 = 23 F(3) = (0 * 3) + (1 * 2) + (2 * 6) + (3 * 4) = 0 + 2 + 12 + 12 = 26 So the maximum value of F(0), F(1), F(2), F(3) is F(3) = 26.

Example 2

Input: nums = [100]

Output: 0

Example details omitted.

Constraints

n == nums.length
1 <= n <= 105
-100 <= nums[i] <= 100

Solution Approach

Initial Function Calculation

Start by calculating the first rotation F(0), which involves summing up the array with weights based on the indices. This gives you the initial result and will serve as the base case for further rotations.

Efficient State Transition

For each subsequent rotation, compute the new function value F(k) from the previous value F(k-1). This is done by adjusting the sum to account for the shift in indices, which can be computed using a dynamic programming approach to avoid recalculating the sum from scratch each time.

Track Maximum Value

Keep track of the maximum value encountered during each iteration of the function. By maintaining the maximum rotation value throughout the process, you ensure the final result is the optimal rotation function value.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

The time complexity depends on how efficiently the rotation function is computed. Using dynamic programming, the time complexity can be reduced to O(n), as each rotation's result is derived from the previous one. The space complexity is O(1) as only a few variables are required to store intermediate values, making it efficient for large input sizes.

What Interviewers Usually Probe

Look for the candidate's understanding of state transition dynamic programming and its application to rotating arrays.
Evaluate how well the candidate avoids recalculating the sum for each rotation and instead uses previous results.
Assess if the candidate can optimize the algorithm to meet the problem's constraints, especially with a time complexity of O(n).

Common Pitfalls or Variants

Common pitfalls

Failing to utilize dynamic programming, leading to redundant calculations for each rotation.
Not maintaining the maximum function value during the iterations, which can result in incorrect answers.
Overcomplicating the solution by recalculating the weighted sum from scratch for each rotation.

Follow-up variants

Rotate the array k positions to the left instead of the right.
Consider variations where the array elements are all the same, simplifying the rotation function calculation.
Change the weights used in the rotation function from simple indices to custom values based on a given formula.

FAQ

What is the primary pattern used in solving the Rotate Function problem?

The primary pattern used is state transition dynamic programming, where each rotation's result is computed based on the previous one.

How can I optimize the solution to handle large arrays?

Use dynamic programming to reduce the time complexity to O(n) by avoiding recalculating the sum for each rotation.

Why does the solution need to keep track of the maximum rotation function value?

Tracking the maximum ensures you find the optimal rotation value, which is the solution to the problem.

Can the Rotate Function problem be solved without dynamic programming?

While it is possible, a brute force solution would require recalculating the sum for each rotation, leading to a time complexity of O(n^2), which is inefficient for large inputs.

How does the Rotate Function solution apply to other array-based dynamic programming problems?

The problem illustrates how to optimize array operations through dynamic programming, a technique applicable to other problems involving rotating or shifting array elements.

terminal

Solution

Solution 1

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class Solution:
    def maxRotateFunction(self, nums: List[int]) -> int:
        f = sum(i * v for i, v in enumerate(nums))
        n, s = len(nums), sum(nums)
        ans = f
        for i in range(1, n):
            f = f + s - n * nums[n - i]
            ans = max(ans, f)
        return ans

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