1. CS344: Lecture 16 S. Muthu Muthukrishnan

2. Graph Navigation BFS: DFS: DFS numbering by start time or finish time. –tree, back, forward and cross edges. –How to do a DFS and identify what each edge type is? –Prove that a directed graph G is acyclic if and only if DFS of G has no back edges. A C E B D A C E B D

3. Details Iterative algorithm for BFS uses a queue, that for DFS uses a stack. How to use BFS or DFS to test whether a undirected graph is connected? Both BFS and DFS take time O(|V|+|E|) given graph in the adjacency list format.

4. DAGs DAG Topological ordering of a DAG G is a linear ordering of the vertices of G such that if (u,v) is an edge in G than v appears before u in the linear ordering. This can be thought of as numbering vertices in the topological order, and we will refer to it as topological numbering. Use DFS to do topological numbering.

5. Another application A directed graph G is strongly connected if there is a path from u to v and v to u for all pairs of vertices u,v. Cute trick to check if G is strongly connected: –Pick some v. –Do DFS(v). If some vertex w is not reachable, print NOT strongly connected. –Reverse edges of G to get G’. –Do DFS(v) in G’. If some vertex w is not reachable, print NOT strongly connected. Else, print YES. –Time: O(|V|+|E|)

6. Strongly connected graph? Example G: G’: a d c b e f g a d c b e f g

7. SC components How to partition a graph into strongly connected components (SCC)? SCC: Maximal subgraphs such that each vertex can reach all other vertices in the subgraph { a, c, g } { f, d, e, b } a d c b e f g

8. SC Components Algorithm Do DFS of G and put vertices in a stack at their finishing times. Reverse edges of G to get G’. Do DFS of G’ starting from vertices that get popped off the stack from the first step! Each DFS tree generated will be a SCC. Question: What is the graph of SCC components of a directed graph G?