Dynamic Programming on graph
Last updated: September 1, 2025
Quick Overview
Given a directed graph represented as an adjacency matrix, implement a dynamic programming solution to find the longest path from a specified starting node to any other node. The function should return the length of the longest path and the corresponding sequence of nodes. The input will include the adjacency matrix and the starting node, while the output should be the length of the longest path and the list of nodes in that path.
Optiver
September 1, 202520
12
296 solved
Given a directed graph represented as an adjacency matrix, implement a dynamic programming solution to find the longest path from a specified starting node to any other node. The function should return the length of the longest path and the corresponding sequence of nodes. The input will include the adjacency matrix and the starting node, while the output should be the length of the longest path and the list of nodes in that path.
Optiver uses this problem in the Take-home Project to evaluate your algorithmic thinking. They expect you to discuss multiple approaches, analyze trade-offs between them, and implement the optimal solution with clean, readable code.
What the Interviewer Expects
- Recognize the underlying problem pattern (sliding window, two pointers, BFS/DFS, etc.)
- Discuss multiple approaches and trade-offs before coding
- Implement an optimal solution with clean, production-quality code
- Handle all edge cases including boundary conditions and invalid input
- Optimize both time and space complexity with clear justification
- Test your solution systematically with well-chosen examples
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 test this solution thoroughly?
- How would you parallelize this solution?
- What if the input doesn't fit in memory?
- What happens if the input contains duplicates?
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Practice DSA ProblemsSample Answer
Problem Analysis
The problem involves finding the longest path in a directed graph represented as an adjacency matrix. This scenario is well-suited for a Dynamic Programming (DP) approach, specifically since the longe...
Approach
- Topological Sorting: First, we perform a topological sort of the graph. This linear ordering of vertices ensures that for every directed edge u -> v, vertex u comes before v in the ordering. Th...
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