Problem Description
You are given an array nums which is a permutation of [0, 1, 2, ..., n - 1]. The score of any permutation of [0, 1, 2, ..., n - 1] named perm is defined as:
score(perm) = |perm[0] - nums[perm[1]]| + |perm[1] - nums[perm[2]]| + ... + |perm[n - 1] - nums[perm[0]]|
Return the permutation perm which has the minimum possible score. If multiple permutations exist with this score, return the one that is lexicographically smallest among them.
Key Insights
- The problem involves finding a permutation that minimizes a specific cost function based on absolute differences.
- The score function forms a cyclic sum, which suggests that certain ordering of the elements might lead to a lower score.
- The constraints (n ≤ 14) allow for exhaustive searching techniques such as backtracking or bit manipulation to explore all permutations.
- Lexicographically smallest permutations can be achieved by generating permutations in a sorted order.
Space and Time Complexity
Time Complexity: O(n!) - We must consider all permutations of the array. Space Complexity: O(n) - Space required for storing the permutation and other variables.
Solution
To solve this problem, we can generate all possible permutations of the array [0, 1, 2, ..., n - 1], compute the score for each permutation, and track the minimum score found along with the corresponding permutation. The key is to ensure that we check permutations in lexicographical order to easily find the smallest one when scores are tied.
We can use Python's itertools.permutations to generate the permutations efficiently. For each permutation, we compute the score based on the given formula and update our best score and permutation accordingly.