Minimum Array Sum

Minimum Array Sum

What is this problem about?

The Minimum Array Sum problem provides an array and two types of operations you can perform:

1. Divide an element by 2 (rounded up).
2. Subtract a value k from an element (if it is $\ge k$). You are given a limit on how many times you can use each operation (op1 and op2). Crucially, you can apply both operations to the same element, but the order matters. The objective is to minimize the total sum of the array.

Why is this asked in interviews?

This is a high-level Minimum Array Sum interview question asked at Amazon and Oracle. It is a multi-dimensional Dynamic Programming problem. It tests a candidate's ability to manage state (remaining operations) and handle the "overlapping subproblems" created by choosing different operations for each element.

Algorithmic pattern used

The Dynamic Programming interview pattern with state dp[i][o1][o2] is the standard solution.

• i: current index in the array.
• o1: number of Operation 1 uses remaining.
• o2: number of Operation 2 uses remaining. At each step, you have 4 choices: apply none, apply op1, apply op2, or apply both (in either order). You take the minimum of these choices and use memoization to avoid redundant work.

Example explanation

nums = [10], k = 5, op1 = 1, op2 = 1.

1. No ops: sum = 10.
2. Op 1 only: ceil(10/2) = 5.
3. Op 2 only: 10 - 5 = 5.
4. Op 1 then Op 2: ceil(10/2) - 5 = 5 - 5 = 0.
5. Op 2 then Op 1: ceil((10-5)/2) = 3. The minimum sum is 0. The DP will explore these choices for every element in the array.

Common mistakes candidates make

• Greedy approach: Trying to always apply the operation that gives the biggest immediate reduction. This doesn't work because using an operation on one element might prevent a much better use on a later element.
• Incorrect "Both" logic: Failing to check if applying both operations in different orders results in different values (e.g., ceil((x-k)/2) vs ceil(x/2)-k).
• State space explosion: Not using memoization or using an iterative DP with too much memory.

Interview preparation tip

When you see multiple types of operations with limited "budgets" (like op1 and op2), it's almost always a Knapsack-style DP. Practice defining the state carefully and identifying all possible transitions for a single element.