Minimum Cost to Split into Ones II - Solution & Explanation

Problem Overview: You are given a binary structure (string or array) and need to split it into segments so that every resulting segment contains exactly one 1. Each split or transformation contributes to a total cost. The task is to compute the minimum cost required to isolate every 1 into its own segment.

Approach 1: Brute Force Partitioning (O(n^2) time, O(n) space)

Start by enumerating every possible split position and evaluate whether the resulting segments each contain exactly one 1. Use prefix counts to quickly determine how many 1s appear inside a candidate segment. A dynamic programming array dp[i] stores the minimum cost needed to process the prefix ending at index i. For each index, iterate backward to test all valid segment boundaries. This works because every segment must contain exactly one 1, but the nested scanning results in O(n^2) time.

Approach 2: Optimized DP with Prefix Tracking (O(n) time, O(n) space)

A key observation: valid segments are determined entirely by the positions of 1s. Instead of scanning all boundaries, track the indices where 1 appears and process the gaps between them. Maintain a running DP value representing the minimum cost to isolate all previous 1s. Each time you encounter another 1, compute the cost contribution from the zeros or characters between consecutive 1s. Because each index is processed once, the algorithm runs in linear time. Prefix counts or cumulative metrics help compute costs for gaps instantly.

Approach 3: Greedy Gap Processing (O(n) time, O(1) space)

When the cost structure depends only on the distance between consecutive 1s, the problem reduces to processing gaps between them. Iterate through the array and track the previous 1. Each new 1 determines a segment boundary and the gap contributes directly to the final cost. This avoids maintaining a full DP table and works when the cost of isolating segments depends purely on local transitions.

Recommended for interviews: The optimized DP or gap-based linear scan is the expected solution. Brute force shows you understand the segmentation constraint, but interviewers want to see the observation that only the relative positions of 1s matter. That insight converts a quadratic partition problem into a linear pass using prefix counts or simple index tracking. These patterns appear frequently in problems involving binary arrays, segmentation, or prefix statistics, especially with dynamic programming, arrays, and greedy algorithms.