Remove Stones to Minimize the Total

Remove Stones to Minimize the Total

What is this problem about?

The Remove Stones to Minimize the Total interview question gives you a list of stone piles and an integer k. In each of k operations, you choose the pile with the most stones and remove exactly half of them (rounding down). Your goal is to minimize the total number of stones remaining after exactly k operations. This is a greedy problem that pairs naturally with a max-heap.

Why is this asked in interviews?

This problem is asked at Meta, Nvidia, Amazon, and Expedia because it is a textbook application of the greedy algorithm with a priority queue (max-heap). It evaluates whether candidates can identify that always targeting the largest pile yields the globally optimal result — a classic greedy correctness argument. It also tests practical heap usage, which is relevant to scheduling, load balancing, and simulation systems.

Algorithmic pattern used

The pattern is greedy with a max-heap (priority queue). Build a max-heap from all pile sizes. Repeat k times: extract the maximum value m, compute the stones removed as m // 2, push back m - m // 2 (the remainder). After k operations, sum all remaining values in the heap. Using Python's heapq with negated values simulates a max-heap.

Example explanation

Piles: [5, 4, 9], k = 2.

• Max = 9. Remove 9//2 = 4. Push back 5. Heap: [5, 5, 4].
• Max = 5 (either one). Remove 5//2 = 2. Push back 3. Heap: [5, 4, 3].

Total remaining: 5 + 4 + 3 = 12.

Without the greedy (removing from a small pile first): remove from 4 → push 2, then remove from 9 → push 5. Total: 5 + 2 + 5 = 12... depends on the example. In general, always targeting the max minimizes the total.

Common mistakes candidates make

• Using a min-heap instead of a max-heap, which removes stones from the smallest pile (incorrect greedy).
• Not pushing back the remainder after removal — forgetting that the reduced pile stays in the game.
• Computing floor(m/2) as the remainder instead of ceil(m/2) — the remainder is m - floor(m/2).
• Not handling k larger than necessary — once all piles are size 1, further operations have no effect.

Interview preparation tip

For the Remove Stones to Minimize the Total coding problem, the heap (priority queue) and greedy interview pattern is central. In Python, negate values to use heapq as a max-heap. Practice the greedy argument: "removing half of the largest pile always reduces the total more than removing half of any smaller pile." Interviewers at Expedia and Nvidia may ask for the time complexity — it's O(k log n), which you should state confidently.