Meta·Software Engineer·Onsite - Coding / Algorithms·Senior

SeniorPrefer not to say

Jun 2026

Summary

Meta SWE coding round with a 2D prefix sum problem. The question was well-scoped but they pushed hard on complexity analysis and edge cases, which is where things got interesting.

Questions Asked (1)

Given an immutable binary matrix, design a data structure that preprocesses it so that any subsequent query asking for the count of 1s in a given sub-rectangle can be answered as fast as possible. Explain your preprocessing and query complexity, implement the interface, and provide at least 5 test cases including edge cases.

Algorithms & Data StructuresTechnical Trade-offs

Author's notes

The core idea is a 2D prefix sum table built once at construction time, O(R*C), and then each query runs in O(1) using inclusion-exclusion on the four corners.

Create a free account to read the full note

AI HintsAI Generated

Suggested Approach

Use a 2D prefix sum (integral image) to preprocess the matrix in O(m*n) time, enabling each sub-rectangle sum query to be answered in O(1) time. Implement a class with a constructor that builds the prefix sum and a method that computes the sum using inclusion-exclusion. Provide test cases covering normal, edge, and boundary scenarios.

Pro tip: Mention that the matrix is immutable, so the prefix sum is built once and reused; also note that the same technique extends to other 2D range queries like sums or averages.

1. Clarify requirements and constraints

Confirm the matrix dimensions, immutability, and that queries are frequent. Discuss expected query patterns and any memory constraints.

2. Design preprocessing with 2D prefix sums

Explain how to build a (m+1) x (n+1) prefix sum array where each cell stores the sum of the sub-matrix from (0,0) to (i-1,j-1). Use the recurrence: P[i][j] = P[i-1][j] + P[i][j-1] - P[i-1][j-1] + matrix[i-1][j-1].

3. Implement O(1) query using inclusion-exclusion

For a query (r1, c1, r2, c2), compute sum = P[r2+1][c2+1] - P[r1][c2+1] - P[r2+1][c1] + P[r1][c1]. Explain why this works and its constant time complexity.

4. Analyze complexity and trade-offs

State preprocessing time O(m*n) and space O(m*n), query time O(1). Compare with naive O(m*n) per query and discuss when this is optimal.

5. Provide test cases and edge cases

Include tests for: empty matrix, single cell, full matrix query, single row/column, and queries at boundaries. Verify correctness with expected outputs.

Key Points to Mention

2D prefix sum (integral image) concept and construction
Inclusion-exclusion principle for rectangle sum
Time and space complexity: O(m*n) preprocessing, O(1) query, O(m*n) space
Immutability ensures prefix sum remains valid
Handling edge cases like empty matrix, 1x1 matrix, and queries covering entire matrix
Potential follow-up: if matrix were mutable, need a different structure like Fenwick tree 2D

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