Amazon·Software Engineer·Technical Phone Screen·Intermediate

Intermediate

Aug 2026

Summary

Amazon SWE interview with a tricky array/binary string optimization problem. The question stumped even the LLMs apparently, which was a weird comfort.

Questions Asked (1)

Given an integer array called population and a binary string called unit, both of size n, where unit[i] = 1 means population[i] is protected: each 1 in the binary string can shift left at most once (from index i to i-1). Find the maximum total protected population after optimally repositioning the 1s.

Algorithms & Data Structures

Author's notes

My approach was to find contiguous clumps of 1s and figure out whether the leftmost 1 in a clump should swap with the 0 to its left.

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AI HintsAI Generated

Suggested Approach

Recognize that each '1' can either stay or move left by one, so the problem reduces to selecting a subset of 1s to shift left such that no two 1s end up at the same index. Use dynamic programming with states representing the previous position's occupancy, or model as maximum weight matching on a path graph. Then compute the maximum sum of populations at the final protected indices.

Pro tip: Clarify with the interviewer whether multiple 1s can shift to the same index (they cannot, since each index can hold at most one protected unit). Also, consider edge cases like all 1s, no 1s, and 1s at boundaries.

1. Understand the movement constraints

Each '1' can either stay at its original index or move left by one. No two '1's can occupy the same index. The goal is to maximize the sum of populations at the final positions of the 1s.

2. Model as a decision problem

For each '1', decide whether to shift left or stay, ensuring no conflicts. This can be seen as selecting a set of indices to protect, where each selected index must be either an original '1' or the left neighbor of an original '1', and each original '1' can be used at most once.

3. Develop a dynamic programming solution

Process indices from left to right. Maintain DP states: dp[i][0] = max protected population up to index i when index i is not protected, dp[i][1] = max when index i is protected. Transition based on whether the current index can be protected by a 1 from i or i+1.

4. Implement and test

Write code for the DP, handle base cases, and test with examples including edge cases (all 1s, no 1s, 1s at start/end). Verify time complexity O(n) and space O(1) if optimized.

5. Discuss alternative approaches

Mention that this can also be solved via greedy or maximum matching on a path, but DP is straightforward. Be prepared to explain why greedy might fail (e.g., local optimal choices may not lead to global optimum).

Key Points to Mention

Each '1' can move left at most once, so the final position of a 1 is either its original index or one less.
No two 1s can occupy the same index, so conflicts must be resolved.
Dynamic programming with states tracking whether the current index is protected yields an O(n) solution.
The problem is equivalent to selecting a maximum weight independent set on a path graph with additional constraints.
Edge cases: all 1s, no 1s, 1s at index 0 (cannot move left), and 1s at the last index.
Time and space complexity: O(n) time, O(1) space if optimized with two variables.

AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.