Expert · Module 28 · Lesson 2 of 4
Network Flow, Min-Cut, and Dinic's Algorithm (Bipartite Matching)
Model capacity constraints, minimum cuts, and bipartite matching with Residual Graphs, understand why backward edges undo suboptimal greedy choices, and implement Dinic's BFS Level Graph + DFS Pointer-Advancing Blocking Flow.
By the end you can
- Explain why residual backward edges (capacity 0, increased by +f when forward edge sends f) are essential for Ford-Fulkerson correctness
- Apply the Max-Flow Min-Cut Theorem to extract the minimum s-t edge cut from reachable residual vertices
- Implement Dinic's Algorithm with BFS level graphs and the `ptr[u]` current-arc optimization in O(V^2 E) (and O(E sqrt(V)) on unit networks)
Builds on Breadth-first search, Depth-first search, Tarjan's Strongly Connected Components (SCC), Bridges, and Articulation Points.
01 · A question first
A greedy flow path pushes 1 unit diagonally across a bridge and blocks two 1-unit paths that could have carried 2 units total. How does adding a 'backward edge' with capacity +1 let the algorithm change its mind without ever backtracking?
Make a guess before you read on.
02 · The idea
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What's inside
- Intuitive Explanation & Real-World Analogy
- Core Concepts & Visual Walkthrough
- Pseudocode
- Multi-Language Implementations (C++, Java, Python)
- Time & Auxiliary Space Complexity Derivation
- Edge Cases & Common Beginner Mistakes
- Alternative Approaches (Brute Force vs. Optimal)
- Pattern Recognition Guide & Three-Stage Practice Plan
03 · Practice
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