How Dynamic Programming Solves the Maximum Subarray Problem in Linear Time

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A software developer explains the core logic behind dynamic programming (DP) through the classic maximum subarray problem, a common coding interview challenge. The brute-force approach using two nested loops runs in O(n²) time, becoming impractically slow on large inputs. The DP solution reframes the problem as a per-element decision: either start a new subarray at the current index or extend the best subarray from the previous index, expressed as dp[i] = max(nums[i], dp[i-1] + nums[i]). This recurrence relies on optimal substructure and overlapping subproblems, reducing time complexity to O(n) and space complexity to O(1). The article also highlights a common pitfall — initialising the best sum to zero, which produces incorrect results when all array elements are negative.

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