NURO·Machine Learning Engineer·Technical Phone Screen·Senior

SeniorPrefer not to say

May 2026

Summary

Interviewed for an MLE role at Nuro and got a fairly classic sparse matrix problem. Nothing too wild but the complexity analysis at the end tripped me up more than I expected.

Questions Asked (1)

Given two sparse matrices A (m x k) and B (k x n) as 2D arrays, compute the matrix product A * B. Optimize for sparsity by using a compact representation and only processing non-zero entries. Then analyze the time complexity in terms of nnz(A) and nnz(B).

Algorithms & Data StructuresTechnical Trade-offs

Author's notes

I went straight to building a row-to-list-of-(col, val) map for A and a similar structure for B.

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AI HintsAI Generated

Suggested Approach

First, clarify the sparse matrix representation (e.g., list of (row, col, value) or CSR) and the expected output format. Then describe an algorithm that iterates over non-zero entries of A and B, using a hash map or dense accumulator to compute only necessary dot products. Finally, derive the time complexity in terms of nnz(A) and nnz(B), discussing trade-offs between different multiplication orders.

Pro tip: Mention that for very sparse matrices, iterating over non-zero entries of A and for each, over non-zero entries of B in the same row can be more efficient than the standard row-by-column approach, and that using a hash map for the output avoids dense intermediate storage.

1. Clarify representation and constraints

Ask whether the input matrices are given in a specific sparse format (e.g., list of triplets, CSR) and whether the output should be dense or sparse. Confirm the dimensions and that multiplication is valid (A's columns = B's rows).

2. Choose an efficient algorithm

Describe an algorithm that processes only non-zero entries. For example, for each non-zero A[i][k], iterate over non-zero B[k][j] and accumulate the product into a hash map keyed by (i,j). Alternatively, use CSR and iterate over rows.

3. Implement with appropriate data structures

Use a hash map or dictionary to store the output sparsely, avoiding a dense m x n matrix. If using CSR, leverage row pointers and column indices to efficiently access non-zeros.

4. Analyze time complexity

Derive the complexity: O(nnz(A) * average non-zeros per row of B) or more precisely O(sum over non-zero A[i][k] of nnz(B[k][:])). Discuss best and worst cases, and compare to dense O(m*k*n).

5. Discuss trade-offs and optimizations

Mention that the order of multiplication (A*B vs B*A) can affect efficiency, and that using a dense accumulator for rows with many non-zeros might be faster. Also note memory vs time trade-offs.

Key Points to Mention

Sparse matrix representations: coordinate list (COO), compressed sparse row (CSR), and their impact on iteration efficiency.
Algorithm: iterate over non-zero entries of A, and for each, iterate over non-zero entries of B in the corresponding row, accumulating into a hash map.
Time complexity: O(nnz(A) * nnz(B) / k) in the worst case, but more accurately O(sum_{i,k: A[i][k]!=0} nnz(B[k][:])).
Space complexity: O(nnz(A) + nnz(B) + nnz(C)) where C is the result, using sparse output storage.
Trade-offs: dense accumulation for rows with many non-zeros vs. sparse hash map; choosing multiplication order based on sparsity patterns.
Edge cases: zero matrices, empty rows/columns, and ensuring no unnecessary multiplications by zero.

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