1. Strongly Connected Components

2. Strongly Connected Components (SCC)
Directed Graph
A set of nodes, such that every node is reachable from every other node in the SCC
The assumption is that the SCC is the maximal set of such nodes – i.e. the largest sets where all nodes are reachable from each other.
Typically we want to get a list of the SCCs
Notice: if we replace an SCC by a single node, where edges to any node in the SCC are in-edges, and edges from any node in the SCC are out-edges, then we end up with a DAG

3. Computing Strongly Connected Components
Perform DFS like before. SCCs are identified when a back edge creates a cycle (and therefore connects everything in that subtree).
Keep track of order visited, and “low” value, similar to bridges/articulation points
Update Low only for Exploring (not Visited/Finished) vertices.
Add each node to a stack
When you finish a node where Num == Low, it marks the root of a SCC
Pop nodes off the stack until you get that node off; those nodes are a SCC – can store them separately,

4. Low is lowest EXPLORING (not yet fully Visited) seen in that subgraph1 2
d is when visited
Low is lowest EXPLORING (not yet fully
Visited) seen in that subgraph 6 3 5 8 9 7 4 3
Notice that the Low value is NOT updated to the
previously node value, since the node it sees is
already finished/visited, not still exploring

5. Alternative Approach
To see if two components are in same SCC, check whether i->j and j->i are both possible
Can compute transitive closure of the graph (use Floyd-Warshall, which we discuss in Shortest Path)
Then, check for connectivity is trivial.
But, pulling into lists of all SCCs takes O(n^2) time