Graph Algorithms: Flows and Strongly Connected Components

This lecture covers essential graph algorithms, focusing on flow networks and strongly connected components. The instructor begins by discussing the Ford-Fulkerson method for finding maximum flows in a network, emphasizing the importance of selecting augmenting paths. The lecture then transitions to the concept of strongly connected components in directed graphs, explaining how to identify them using depth-first search (DFS) and the significance of topological sorting. The instructor illustrates the process of contracting strongly connected components and demonstrates how to analyze the runtime of these algorithms. Additionally, the lecture addresses practical applications, such as finding maximum matchings in bipartite graphs and edge-disjoint paths. The instructor provides examples and exercises to reinforce the concepts, ensuring students understand the algorithms' implications in real-world scenarios. The session concludes with a discussion on the relationship between maximum flow and minimum cut, highlighting the duality principle in flow networks.