Minimum Partition Score II - Solution & Explanation

Problem Overview: You are given an array and must split it into multiple contiguous partitions so that the total partition score is minimized. The score of each segment depends on values inside that subarray, so the challenge is choosing optimal cut points.

Approach 1: Brute Force Partition Enumeration (O(n^3) time, O(n) space)

The most direct method tries every possible partition configuration. For each index i, treat it as the end of the current segment and iterate over all possible starting points j. Compute the segment score for nums[j..i] and combine it with the best score before j. If computing segment scores requires scanning the subarray, each check costs O(n), leading to O(n^3) total complexity. This approach is useful for understanding the recurrence but becomes impractical for large inputs.

Approach 2: Dynamic Programming with Prefix Precomputation (O(n^2) time, O(n) space)

Define dp[i] as the minimum partition score for the prefix ending at index i. For each i, iterate backward through possible previous partition points j. The transition becomes dp[i] = min(dp[j] + score(j+1, i)). Precompute segment statistics using prefix sums or other auxiliary arrays so the score of any subarray can be evaluated in constant time. This reduces the runtime to O(n^2). Most accepted solutions rely on this DP structure because it is straightforward and handles general scoring functions.

Approach 3: Optimized DP with Segment Tree / Monotonic Structure (O(n log n) time, O(n) space)

When the partition score has monotonic or decomposable properties, the transition can be optimized. Maintain candidate transitions using a segment tree or monotonic structure that stores the best previous DP states. Instead of scanning all j for every i, query the data structure for the optimal value over a valid range. Each update and query costs O(log n), reducing the total complexity to O(n log n). This pattern commonly appears in advanced dynamic programming problems where transitions depend on range queries.

Recommended for interviews: The O(n^2) dynamic programming approach is usually expected. It clearly demonstrates the recurrence and correct state definition. Starting with the brute-force idea shows understanding of the partition structure, while moving to optimized DP demonstrates algorithmic maturity. If constraints are large, discussing a segment tree or monotonic optimization signals strong problem-solving depth.