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BlackRock·Software Engineer·Technical Phone Screen·Intermediate

Intermediate
Apr 2026

Summary

BlackRock software engineer interview focused on a classic 2D prefix sum problem. Pretty algorithmic, not much else to it, but the follow-up about arbitrary range queries is where things got interesting.

Questions Asked (2)

Q1

Given an m×n integer matrix, design a solution that preprocesses it so any query for the sum of elements from (0,0) to (x,y) runs in O(1). Walk through the preprocessing time and space complexity.

Algorithms & Data StructuresTechnical Trade-offs
Author's notes

I knew prefix sums going in, so the core answer came out fine.

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AI HintsAI Generated

Suggested Approach

Use a 2D prefix sum array where each cell (i,j) stores the sum of the submatrix from (0,0) to (i,j). Preprocess in O(m*n) time and space, then answer each query in O(1) using inclusion-exclusion.

Pro tip: Mention that this technique is widely used in real-time analytics and risk calculations at BlackRock, where fast range queries on large matrices are critical. Also, discuss potential memory optimizations if the matrix is sparse or if queries are limited.

1. Define the prefix sum matrix

Create a 2D array `prefix` of size (m+1) x (n+1) to handle boundaries easily. `prefix[i][j]` will store the sum of all elements in the submatrix from (0,0) to (i-1,j-1).

2. Preprocess the matrix

Fill the prefix array using the recurrence: `prefix[i][j] = matrix[i-1][j-1] + prefix[i-1][j] + prefix[i][j-1] - prefix[i-1][j-1]`. This takes O(m*n) time.

3. Answer queries in O(1)

For a query (x,y), return `prefix[x+1][y+1]` if using 0-indexed coordinates. If the query is for a submatrix not starting at (0,0), use inclusion-exclusion: `sum = prefix[x2+1][y2+1] - prefix[x1][y2+1] - prefix[x2+1][y1] + prefix[x1][y1]`.

4. Analyze complexity

Preprocessing takes O(m*n) time and O(m*n) space. Each query takes O(1) time. Discuss trade-offs: if queries are frequent, this is optimal; if memory is constrained, consider alternatives like storing row-wise prefix sums (O(m*n) space but O(n) query time).

Key Points to Mention

  • 2D prefix sum (integral image) technique
  • Inclusion-exclusion principle for submatrix sums
  • Time complexity: O(m*n) preprocessing, O(1) per query
  • Space complexity: O(m*n) for the prefix array
  • Handling boundaries with an extra row and column
  • Trade-offs: memory vs. query speed, and alternative approaches for sparse matrices

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

Q2

How would you extend the prefix sum approach to support sum queries over an arbitrary submatrix from (r1,c1) to (r2,c2), not just from the origin?

Algorithms & Data StructuresTechnical Trade-offs
Author's notes

This is the inclusion-exclusion part and I blanked for a second.

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AI HintsAI Generated

Suggested Approach

Explain that a 2D prefix sum array can be built such that each cell stores the sum of all elements from (0,0) to (i,j). Then, to query any submatrix, use inclusion-exclusion: sum = P[r2][c2] - P[r1-1][c2] - P[r2][c1-1] + P[r1-1][c1-1], handling boundaries with a padded array or conditionals. Emphasize that this gives O(1) query time after O(m*n) preprocessing.

Pro tip: Mention that using a 1-indexed prefix array with a dummy row and column simplifies boundary handling and avoids off-by-one errors, which is crucial in production code.

1. Define the 2D prefix sum

Explain that P[i][j] represents the sum of all elements in the rectangle from (0,0) to (i,j). This allows any submatrix sum to be derived from four prefix values.

2. Derive the inclusion-exclusion formula

Show that the sum from (r1,c1) to (r2,c2) equals P[r2][c2] - P[r1-1][c2] - P[r2][c1-1] + P[r1-1][c1-1]. Explain why each term is added or subtracted.

3. Handle boundaries

Discuss how to handle cases where r1=0 or c1=0 by either using conditional checks or padding the prefix array with an extra row and column of zeros.

4. Analyze complexity

State that preprocessing takes O(m*n) time and O(m*n) space, while each query is O(1). Contrast with naive O((r2-r1+1)*(c2-c1+1)) per query.

5. Discuss trade-offs and extensions

Mention that this approach is ideal for static matrices; for dynamic updates, consider a 2D Fenwick tree. Also note memory usage and potential for compression if sparse.

Key Points to Mention

  • 2D prefix sum array construction: P[i][j] = A[i][j] + P[i-1][j] + P[i][j-1] - P[i-1][j-1]
  • Inclusion-exclusion principle for arbitrary submatrix sum
  • Boundary handling with 1-indexed array or conditionals
  • Time complexity: O(1) per query after O(m*n) preprocessing
  • Space complexity: O(m*n) for the prefix array
  • Trade-offs: static vs dynamic (e.g., 2D Fenwick tree for updates)

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