Advanced Graph Algorithms Questions

Higher level and combined graph algorithm topics frequently expected at senior or competitive programming levels. Topics include strongly connected components and algorithms such as Kosaraju and Tarjan, minimum spanning trees using Kruskal and Prim with Union Find optimizations, network flow fundamentals including Ford Fulkerson and Edmonds Karp and applications to bipartite matching, graph reductions and transformations, graph coloring and bipartite checks, advanced traversal techniques such as bidirectional search and multi source traversals, and strategies to combine algorithms for complex problems. Emphasize time and space complexity, algorithm correctness proofs, implementation pitfalls, and when to prefer one algorithm or data structure over another.

EasyTechnical

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Compare Kosaraju and Tarjan algorithms for computing strongly connected components (SCCs) in a directed graph. Describe the high-level steps of each, their time and space complexities, recursion/stack behavior, memory implications for graphs with millions of nodes, and practical reasons you might pick one over the other in production AI pipelines (e.g., knowledge graphs).

MediumSystem Design

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Design a distributed workflow to compute SCCs and connected components for graphs with 1 billion edges using tools like Spark/GraphX or Gelly. Describe partitioning strategy, communication minimization, memory limits, checkpointing, and how to handle skewed degree distributions.

EasyTechnical

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Write a Python function that takes an undirected graph represented as an adjacency list (dict: node -> list of neighbors) and returns all connected components. The function must run in O(n + m) time and handle graphs with up to 1e6 nodes (take recursion limits and memory into account). Include sample input/output and explain iterative vs recursive choices.

MediumTechnical

0 practiced

Given a directed graph problem requiring the maximum number of vertex-disjoint s-t paths, describe how to reduce it to a max-flow instance. Explain node-splitting, capacity assignments, complexity implications, and pitfalls for implementation in code.

MediumTechnical

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State and sketch a proof of the max-flow min-cut theorem. Then explain how this principle is applied to graph-cut segmentation in computer vision and one practical consideration when applying it in production.

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