Minimum Partition Score - Solution & Explanation

Problem Overview: You are given an array and must split it into multiple contiguous partitions so the total partition score is minimized. Each partition contributes a score based on the values inside the segment, and the goal is to choose split points that produce the smallest total score.

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

The most direct way is to try every possible partitioning. For every pair of indices i and j, treat nums[i..j] as a segment and compute its score by scanning the subarray. Then recursively evaluate the remaining suffix. This approach repeatedly recomputes segment statistics such as minimum and maximum values, which leads to cubic time. It helps understand the partition structure but becomes infeasible once the array size grows.

Approach 2: Dynamic Programming with Precomputed Segment Scores (O(n^2) time, O(n) space)

Define dp[i] as the minimum score required to partition the prefix ending at index i. For each i, iterate over all previous split points j and evaluate the cost of the segment nums[j+1..i]. The transition becomes dp[i] = min(dp[j] + cost(j+1, i)). Segment statistics such as minimum, maximum, or sum can be tracked incrementally using techniques from array processing and prefix sum preprocessing. This reduces redundant work but still checks every possible split, resulting in quadratic time.

Approach 3: Monotonic Queue Optimized DP (O(n) time, O(n) space)

The optimal approach observes that segment scores depend on monotonic properties such as the minimum or maximum value within the current window. Instead of recomputing them for every candidate split, maintain two monotonic queues that track increasing and decreasing elements as the right boundary moves. While extending the segment, update the structures so the current min/max can be retrieved in constant time. Combine this with a rolling DP transition so outdated partition candidates are removed when they stop producing optimal scores. The monotonic behavior allows each element to be pushed and popped at most once, giving linear complexity.

This technique combines ideas from dynamic programming and monotonic queue optimization. Instead of scanning all previous split points, the queue maintains only candidates that can still produce the best score for upcoming indices.

Recommended for interviews: Start by describing the DP formulation because it clearly expresses how partitions contribute to the final score. Interviewers expect you to recognize that checking every split leads to O(n^2) work. The strong signal is identifying the monotonic property in the segment score and applying a monotonic queue to compress transitions to O(n). That shift from quadratic DP to linear optimization demonstrates strong problem‑solving ability.