Smallest Rotation with Highest Score

Problem Description

You are given an array nums. You can rotate it by a non-negative integer k so that the array becomes [nums[k], nums[k + 1], ... nums[nums.length - 1], nums[0], nums[1], ..., nums[k-1]]. Afterward, any entries that are less than or equal to their index are worth one point. Return the rotation index k that corresponds to the highest score we can achieve if we rotated nums by it. If there are multiple answers, return the smallest such index k.

Key Insights

• Each rotation of the array can yield a different score based on the condition of elements being less than or equal to their index.
• The score can be calculated for each possible rotation to determine the best rotation.
• Efficiently calculating scores can avoid re-evaluating the entire array for each rotation.
• A prefix sum or difference array approach can help in optimizing the score calculation.

Space and Time Complexity

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

Solution

To solve this problem, we will use an efficient approach to calculate the scores for all possible rotations without directly rotating the array. We can maintain a score for the initial configuration and then adjust this score as we consider each subsequent rotation.

1. Calculate the initial score for k = 0 by iterating through the array and counting how many elements satisfy the condition nums[i] <= i.
2. For each subsequent rotation k, update the score based on how many elements change their positions relative to their indices due to the rotation.
3. Keep track of the maximum score and the corresponding rotation index k. In case of ties in score, choose the smallest k.

This approach avoids the need for multiple full iterations over the array, thus optimizing the solution.

Code Solutions