Affirm·Software Engineer·Technical Phone Screen·Intermediate

Intermediate

Jun 2026

Summary

Affirm software engineer interview with a classic game theory DP problem. The follow-up asking for an actual move sequence on top of the score was the part that tripped me up a bit.

Questions Asked (1)

Given a row of N integers, two players take turns picking either the leftmost or rightmost card, both playing optimally. Implement a function that returns the maximum score difference (first player minus second player). Then explain your time and space complexity, and modify the solution to also return one optimal sequence of picks as a list of indices.

Algorithms & Data StructuresTechnical Trade-offs

Author's notes

The core DP wasn't too bad once I recognized it as a minimax problem with memoization.

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AI HintsAI Generated

Suggested Approach

Start by clarifying the problem and constraints, then present a dynamic programming solution that computes the maximum score difference using a 2D DP table. After explaining time and space complexity, extend the DP to track choices for reconstructing one optimal sequence of picks.

Pro tip: Emphasize that the DP state represents the score difference (current player minus opponent) from the subarray, which elegantly handles optimal play without tracking both players' scores separately. Also, mention that you can optimize space to O(N) if only the difference is needed, but reconstruction requires storing choices.

1. Clarify and Define

Restate the problem: two players pick from ends optimally, return max score difference. Define DP state: dp[i][j] = max difference (current player - other) from subarray i..j.

2. Formulate Recurrence

Derive recurrence: dp[i][j] = max(arr[i] - dp[i+1][j], arr[j] - dp[i][j-1]). Base case: dp[i][i] = arr[i].

3. Implement and Analyze Complexity

Implement bottom-up DP with O(N^2) time and space. Explain that each state is computed once, and space can be optimized to O(N) if only the difference is needed.

4. Extend for Reconstruction

Modify DP to store the choice (left or right) for each state. After computing dp[0][N-1], backtrack from (0, N-1) to reconstruct one optimal sequence of indices.

5. Test and Discuss Trade-offs

Walk through a small example to verify. Discuss trade-offs: storing choices increases space to O(N^2) but enables reconstruction; alternative approaches like memoization with recursion.

Key Points to Mention

Optimal substructure and overlapping subproblems justify DP.
State definition: dp[i][j] as score difference for current player.
Recurrence relation: dp[i][j] = max(arr[i] - dp[i+1][j], arr[j] - dp[i][j-1]).
Time complexity O(N^2), space complexity O(N^2) for reconstruction (or O(N) for difference only).
Reconstruction: store choice per state and backtrack from (0, N-1).
Edge cases: N=0, N=1, and handling even/odd N.

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