Minimum Array Sum - Solution & Explanation

Problem Overview: You are given an array of integers and a limited number of operations that modify element values. Each operation reduces a number in a specific way (for example halving it or subtracting a constant). The goal is to apply these operations optimally so the total array sum becomes as small as possible.

Approach 1: Greedy with Priority Queue (O((n + k) log n) time, O(n) space)

The greedy strategy always applies the operation that produces the largest immediate reduction in value. Push all elements into a max heap so the largest element is processed first. For each available operation, pop the current maximum, compute the reduction produced by the allowed operation, update the value, and push it back into the heap. This works well when the best choice at each step is the element producing the biggest decrease in the sum. The heap ensures efficient extraction and reinsertion while keeping the array effectively ordered by impact.

This approach is intuitive and fast in practice. You repeatedly evaluate which operation gives the maximum decrease and apply it to the largest candidate. Since heap operations take O(log n), the total cost depends on the number of operations performed.

Approach 2: Dynamic Programming (O(n * k1 * k2) time, O(n * k1 * k2) space)

When operations interact or their order matters, greedy choices may not always lead to the global minimum. Dynamic programming models every valid state explicitly. Define a DP state representing the minimum sum achievable after processing the first i elements while using a certain number of operations of each type. For each element, evaluate all valid combinations: no operation, operation A, operation B, or both if allowed.

Transitions update the state by applying the operation’s transformation to the current number and adding the resulting value to the accumulated sum. The DP table ensures every combination of operations is considered exactly once, guaranteeing the globally minimal sum. This approach is more expensive but reliable when greedy assumptions break.

The state transitions iterate through elements and update remaining operation counts. This makes the solution deterministic and safe for edge cases where applying a smaller local reduction enables a larger future gain.

Recommended for interviews: Start by explaining the greedy intuition since minimizing the largest contributors to the sum is a natural first idea. Then discuss why overlapping operation choices may require a dynamic programming formulation. Interviewers typically expect you to recognize the greedy optimization quickly, but implementing a correct DP over an array with operation counts demonstrates stronger problem‑solving depth.