Maximum Sum of M Non-Overlapping Subarrays I - Solution & Explanation

Problem Overview: Given an array of integers, select m non-overlapping subarrays such that the total sum of their elements is maximized. Each chosen subarray must not intersect with another, so once you pick a range, the next must start strictly after it ends.

Approach 1: Brute Force Enumeration (Exponential Time, O(2^n) time, O(n) space)

The most straightforward idea is to try every possible set of subarrays and keep the configuration with the largest total sum. You recursively choose whether to start a subarray at each index and explore all valid combinations until m segments are selected. While conceptually simple, this approach quickly becomes impractical because the number of possible segment combinations grows exponentially with the array length. It mainly serves as a conceptual baseline for understanding why more structured optimization is required.

Approach 2: Dynamic Programming with Prefix Sum (O(n*m) time, O(n*m) space)

The key observation is that the decision at index i depends only on previous choices. Precompute prefix sums so any subarray sum can be retrieved in O(1). Define dp[i][j] as the maximum sum achievable using the first i elements with exactly j subarrays selected. For each index, either skip the element or end a subarray at that position and add its sum to a previous state. This structured transition removes redundant work and ensures every state is computed once. Prefix sums from prefix sum make range-sum calculations constant time, keeping the overall complexity manageable.

Approach 3: Optimized DP with Running Maximum (O(n*m) time, O(m) or O(n*m) space)

The DP transition can be optimized by maintaining the best possible previous value while scanning the array. Instead of recomputing every possible start index for each subarray, track the maximum candidate value that could extend into the current position. This idea resembles the optimization used in dynamic programming variants of Kadane's algorithm. By updating a running best value while iterating forward, each index contributes constant work, reducing nested scanning while preserving the O(n*m) complexity. Memory can also be reduced by keeping only the previous DP layer.

Recommended for interviews: The dynamic programming approach with prefix sums is what most interviewers expect. Starting with the brute-force idea shows you understand the constraints, but transitioning to the optimized DP demonstrates algorithmic maturity. Mentioning prefix sums and state transitions clearly communicates why the solution runs in O(n*m) time instead of exponential time.