Anthropic·Software Engineer·Technical Phone Screen·Intermediate

IntermediatePrefer not to say

Jul 2026Remote

Summary

Interviewed for a software engineering role at Anthropic and got a matrix problem that seemed straightforward at first but kept growing. The follow-ups were where things got interesting.

Questions Asked (1)

Given an n x n integer matrix, write a function that returns the sum of its main diagonal. Then: if you get q point updates changing individual cells, design a data structure to return the current trace after each update in O(1) time. Also extend it to handle the anti-diagonal sum, and discuss complexity and edge cases like non-square inputs or very large n.

Algorithms & Data StructuresTechnical Trade-offs

Author's notes

The base function was fine, just loop the diagonal.

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AI HintsAI Generated

Suggested Approach

Start by explaining the straightforward O(n) trace computation, then introduce the O(1) update design by maintaining running sums for the main and anti-diagonals. For each update, adjust the sums based on whether the changed cell lies on either diagonal, and discuss edge cases and complexity.

Pro tip: Emphasize that the O(1) update is achieved by precomputing the initial sums and then applying constant-time adjustments per update, and proactively mention how to handle non-square matrices by defining diagonals only for square submatrices or returning an error.

1. Clarify requirements and assumptions

Confirm the definition of main diagonal (i == j) and anti-diagonal (i + j == n - 1), and ask whether the matrix is guaranteed square. Discuss how to handle non-square inputs (e.g., return error or use min(rows, cols)).

2. Design the data structure

Maintain the matrix itself, plus two running sums: main_sum and anti_sum. Initialize them by iterating over the diagonals in O(n) time. For updates, adjust the sums in O(1) by subtracting the old value and adding the new value if the cell belongs to the respective diagonal.

3. Implement update and query operations

For update(i, j, val): if i == j, main_sum += val - matrix[i][j]; if i + j == n - 1, anti_sum += val - matrix[i][j]; then set matrix[i][j] = val. For trace(), return main_sum; for anti_trace(), return anti_sum.

4. Analyze complexity and edge cases

Initialization O(n), each update O(1), each query O(1). Space O(n^2) for the matrix. Edge cases: n=0 or 1, non-square input, very large n (memory concerns), and updates to cells not on diagonals (no change to sums).

5. Discuss extensions and trade-offs

Mention that if updates are frequent and queries rare, recomputing on demand might be simpler; if memory is tight, consider storing only diagonal elements and a sparse representation for off-diagonal updates. Also note that for non-square matrices, the concept of main diagonal is ambiguous.

Key Points to Mention

Definition of main diagonal (i == j) and anti-diagonal (i + j == n - 1).
Initial O(n) computation of diagonal sums.
O(1) update by adjusting running sums only when the updated cell lies on a diagonal.
Handling of non-square matrices: either reject or define diagonals for the largest square submatrix.
Edge cases: n=0, n=1, updates to off-diagonal cells, and very large n causing memory issues.
Trade-offs: space vs. time, and when recomputation might be preferable.

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