#3149

Hard

auto_awesomeState transition dynamic programming

LeetCode Problem Workspace

Find the Minimum Cost Array Permutation

Determine the lexicographically smallest permutation of nums that minimizes a cyclic score using state transition DP techniques.

array dynamic programming bit manipulation bitmask

Problem Statement

You are given an array nums containing a permutation of integers from 0 to n - 1. The cost of a permutation perm is computed cyclically as the sum of absolute differences between consecutive elements mapped through nums: |perm[0] - nums[perm[1]]| + |perm[1] - nums[perm[2]]| + ... + |perm[n - 1] - nums[perm[0]]|.

Return the permutation perm that achieves the minimum possible cost. If multiple permutations yield the same minimal score, return the lexicographically smallest permutation. Ensure your solution handles n up to 14 using efficient state transition dynamic programming techniques combined with bitmasking to track used indices.

Examples

Example 1

Input: nums = [1,0,2]

Output: [0,1,2]

The lexicographically smallest permutation with minimum cost is [0,1,2] . The cost of this permutation is |0 - 0| + |1 - 2| + |2 - 1| = 2 .

Example 2

Input: nums = [0,2,1]

Output: [0,2,1]

The lexicographically smallest permutation with minimum cost is [0,2,1] . The cost of this permutation is |0 - 1| + |2 - 2| + |1 - 0| = 2 .

Constraints

2 <= n == nums.length <= 14
nums is a permutation of [0, 1, 2, ..., n - 1].

Solution Approach

Fix the First Element for Lexical Minimality

Since the score is cyclic, we can safely set perm[0] = 0 without losing generality. This ensures that the final permutation will be the lexicographically smallest among those with minimal cost, reducing the search space.

Use Bitmask Dynamic Programming

Implement DP where dp[mask][i] stores the minimum cost to reach element i using the subset of indices in mask. Transition by adding each unused index j, updating dp[mask | (1 << j)][j] with dp[mask][i] + |i - nums[j]|.

Reconstruct the Optimal Permutation

After filling the DP table, trace back from the full mask to build the permutation with minimal score. At each step, select the smallest index that preserves the DP cost to ensure lexicographical minimality.

Complexity Analysis

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

Time complexity is O(n * 2^n) due to iterating over all subsets and transitions for each state. Space complexity is O(n * 2^n) to store the DP table, which is feasible for n <= 14.

What Interviewers Usually Probe

Looking for recognition that the problem reduces to cyclic state transition DP with bitmasking.
Expecting the candidate to optimize for lexicographically minimal permutation by fixing perm[0].
Observing whether the candidate correctly implements DP transitions and reconstruction from bitmask states.

Common Pitfalls or Variants

Common pitfalls

Failing to fix perm[0] leading to incorrect lexicographical ordering.
Incorrectly updating DP states without considering the cost from previous element.
Trying to generate all permutations directly, causing timeouts for n = 14.

Follow-up variants

Return only the minimum cost instead of the permutation, focusing purely on DP value computation.
Consider a variation with non-cyclic score, requiring handling first and last elements differently.
Extend to larger n using heuristic pruning instead of exact DP to explore trade-offs in performance.

FAQ

What is the primary pattern used in Find the Minimum Cost Array Permutation?

The main pattern is state transition dynamic programming with bitmasking to handle all subsets of indices efficiently.

Why do we fix perm[0] to 0 in this problem?

Fixing perm[0] to 0 reduces the search space and guarantees the lexicographically smallest permutation without changing the minimum cost.

What is the time complexity for the optimal solution?

The time complexity is O(n * 2^n), iterating over all subsets and transitions, feasible for n up to 14.

Can this approach handle larger arrays efficiently?

Direct DP with bitmasking becomes infeasible for n > 14; alternative heuristics or pruning would be needed for larger n.

What common mistakes should I avoid in this problem?

Avoid generating all permutations, failing to fix perm[0], or incorrectly updating DP states, all of which can lead to wrong results or inefficiency.

terminal

Solution

Solution 1: Memoization Search

We notice that for any permutation $\textit{perm}$, if we cyclically shift it to the left any number of times, the score of the permutation remains the same. Since the problem requires returning the lexicographically smallest permutation, we can determine that the first element of the permutation must be $0$.

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class Solution:
    def findPermutation(self, nums: List[int]) -> List[int]:
        @cache
        def dfs(mask: int, pre: int) -> int:
            if mask == (1 << n) - 1:
                return abs(pre - nums[0])
            res = inf
            for cur in range(1, n):
                if mask >> cur & 1 ^ 1:
                    res = min(res, abs(pre - nums[cur]) + dfs(mask | 1 << cur, cur))
            return res

        def g(mask: int, pre: int):
            ans.append(pre)
            if mask == (1 << n) - 1:
                return
            res = dfs(mask, pre)
            for cur in range(1, n):
                if mask >> cur & 1 ^ 1:
                    if abs(pre - nums[cur]) + dfs(mask | 1 << cur, cur) == res:
                        g(mask | 1 << cur, cur)
                        break

        n = len(nums)
        ans = []
        g(1, 0)
        return ans

Continue Practicing

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