Strongly Connected Components


















What's a Strongly Connected Component in a directed graph?

	A subgraph with a path from x to y
	and also a path from y to x
	for every pair of nodes x,y in the subgraph.

	Treat each strongly connected component as a meta-node.
	The resulting meta-graph is a Directed Acyclic Graph (DAG).

List the strongly connected components in the directed graph.
Draw the meta-graph (where each component is a meta-node).
(graph3.pdf)

	V = { a, b, c, d, e, f }
	E = { (a,b), (a,d), (b,d), (b,e), (c,b),
	      (c,f), (e,a), (e,d), (f,b), (f,e) }

	a---b---c
	| X | \ |
	d---e---f


















Finding the Components

Suppose you use Depth-First Search to find the nodes in a component.
Which component would you need to find first?

	the one that doesn't point to any other components

How do you identify a node within this component?

	first node in a topological-sort of the reverse-edge graph

What are the steps to find the strong components?

	1. Create the graph with the edges reversed.
	2. Run DFS-Forest on the reverse-edge graph.
	3. Order the nodes by decreasing post-order number.
	4. For each node n in the order from the last step,
		Run dfs(n) on the original-edge graph.
	5. Nodes reachable by dfs(n) form a component.
	   (remove last component before running next dfs)


















Finding the Components Example

Show the steps of finding the components in the directed graph.

1. Draw the reverse-edge graph.
2. Draw the DFS-Forest for the reverse-edge graph.
3. Give the postorder numbers for the nodes in the reverse-edge graph.
4. Draw the DFS-Tree for each component in the order it is found.
5. Draw the meta-graph.
(graph3.pdf)

	V = { a, b, c, d, e, f }
	E = { (a,b), (a,d), (b,d), (b,e), (c,b),
	      (c,f), (e,a), (e,d), (f,b), (f,e) }

	a---b---c
	| X | \ |
	d---e---f


















Classwork
You may work with a partner.

Show the steps of finding the components in the directed graph.

1. Draw the reverse-edge graph.
2. Draw the DFS-Forest for the reverse-edge graph.
3. Give the postorder numbers for the nodes in the reverse-edge graph.
4. Draw the DFS-Tree for each component in the order it is found.
5. Draw the meta-graph.
(graph2.pdf)

	V = { a, b, c, d, e }
	E = { (b,a), (b,c), (b,d), (b,e), (c,b), (d,c), (d,e), (e,a) }

	a---b---c
	 \ / \ /
	  e---d