Dynamic Programming on matrix
Last updated: July 9, 2025
Quick Overview
Given a 2D matrix filled with integers, implement a dynamic programming solution to find the maximum sum of a path from the top-left corner to the bottom-right corner, where you can only move either down or right at any point in time. The function should return the maximum sum achievable along such a path.
Optiver
July 9, 202553
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Given a 2D matrix filled with integers, implement a dynamic programming solution to find the maximum sum of a path from the top-left corner to the bottom-right corner, where you can only move either down or right at any point in time. The function should return the maximum sum achievable along such a path.
Coding interviews at Optiver focus on problem-solving approach as much as the final solution. The interviewer wants to see you break down the problem, consider edge cases, and optimize iteratively. Communication throughout the process is key.
What the Interviewer Expects
- Quickly identify the optimal approach and its theoretical basis
- Handle complex algorithm design with multiple interacting components
- Write concise, elegant code under time pressure
- Prove correctness of your approach and discuss alternative solutions
- Optimize beyond the obvious: discuss constant factor improvements
- Address follow-up variations and explain how the solution generalizes
Key Topics to Cover
How to Approach This
- Clarify input constraints and edge cases before writing code.
- Walk through your approach verbally and confirm with the interviewer before coding.
- Start with a brute force solution, then optimize. Mention time and space complexity.
- Test your solution with examples, including edge cases like empty input or duplicates.
- Consider common patterns: sliding window, two pointers, hash map, BFS/DFS, dynamic programming.
Possible Follow-up Questions
- How would you modify your solution to handle streaming input?
- Can you optimize the space complexity of your solution?
- What is the worst-case input for your solution?
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Problem Analysis
This problem can be effectively tackled using Dynamic Programming (DP), specifically focusing on the 2D grid (or matrix) structure. The reason DP applies here is that the problem can be broken...
Approach
- Initialization: Create a DP table (2D list) of the same dimensions as the input matrix initialized to zero.
- Base Case: Set
dp[0][0]to the value ofmatrix[0][0]since the path start...
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