Maximal Rectangle
Given a binary matrix, find the largest rectangle containing only 1s and return its area. This is a Hard DP problem that reduces to 'Largest Rectangle in Histogram'.
Why Interviewers Ask This
Oracle evaluates candidates on their ability to decompose complex spatial problems into manageable sub-problems. This question specifically tests mastery of dynamic programming, stack-based optimization, and the capacity to recognize pattern reduction from a 2D grid problem to a 1D histogram challenge under strict time constraints.
How to Answer This Question
1. Clarify the input constraints and edge cases immediately, such as an empty matrix or all zeros, demonstrating Oracle's focus on robustness. 2. Propose the core insight: reducing the 2D problem by treating each row as the base of a histogram where bar heights represent consecutive ones above. 3. Explain the algorithm for finding the largest rectangle in a histogram using a monotonic stack to achieve O(N) efficiency per row. 4. Outline the full solution flow: iterate rows, update height array, compute max area, and track the global maximum. 5. Analyze complexity rigorously, confirming O(M*N) time and O(N) space, then offer to write clean code with proper variable naming conventions.
Key Points to Cover
- Recognizing the reduction from 2D matrix to 1D histogram problems
- Implementing the monotonic stack for O(N) histogram area calculation
- Maintaining a cumulative height array that updates row by row
- Demonstrating O(M*N) time complexity analysis
- Handling edge cases like empty matrices or single rows
Sample Answer
To solve the Maximal Rectangle problem efficiently, I would first break down the two-dimensional binary matrix into a series of one-dimensional histogram problems. My approach relies on the observation that if we fix a specific row as the base, the number of consecutive ones extending upwards forms a histogram. For each cell in the current row, if the value is '1', we increment the height from the previous row; otherwise, we reset it to zero. Once we construct this height array for a given row, the problem reduces to finding the largest rectangle in that histogram. To solve the histogram part optimally, I would use a monotonic stack. This allows us to find the left and right boundaries for each bar in linear time, avoiding the naive O(N^2) approach. By iterating through every row, updating our height array, and calculating the max area for that specific histogram, we ensure we explore all possible rectangles. The global maximum across all rows will be our answer. This strategy achieves O(M*N) time complexity, which is optimal since we must visit every element at least once, and uses O(N) auxiliary space for the stack and height array. This method demonstrates strong dynamic programming skills and the ability to transform a difficult 2D constraint into a solvable 1D structure, a skill highly valued in Oracle's infrastructure engineering roles.
Common Mistakes to Avoid
- Attempting a brute-force O(N^4) solution checking every possible rectangle without optimization
- Failing to explain how the height array is updated when encountering a zero in the matrix
- Neglecting to analyze space complexity, leading to unnecessary memory usage
- Overlooking boundary conditions where the stack becomes empty during the histogram traversal
Practice This Question with AI
Answer this question orally or via text and get instant AI-powered feedback on your response quality, structure, and delivery.