- Definition:
- A directed graph G is
strongly connected if:
- For any two vertices
u and v:
- There is a directed path u → v, and
- There is a directed path v → u
- For any two vertices
u and v:
- A directed graph G is
strongly connected if:
- Example:
- The naive algorithm tests if
we can reach all nodes
starting at a node x
We do this for every node x
- Naive algorithm:
for ( each node "n" ∈ G ) { Find all nodes x that can be reached by a traveral starting in "n"; if ( # nodes reached != N ) return( G is not strongly connected ); } return( G is strongly connected );
- Running time of the naive algorithm:
- Each graph traversal is
O(N2)
- The naive algorithm performs
N graph traversals
- Running time of naive algorithm = O(N3)
- Each graph traversal is
O(N2)
- Reverse Path graph:
- The reverse path graph of
a directed graph is a graph
where the direction
of the edges are
reverse
- Example:
- The reverse path graph of
a directed graph is a graph
where the direction
of the edges are
reverse
- Traversing the forward path graph and
the reverse path graph:
Observation:
- A node y that is reached
by a traversal in the
forward path graph starting
at node x
<==>
there is a path
x → y
(This is pretty obvious)
- A node y that is reached
by a traversal in the
reverse path graph starting
at node x
<==>
there is a path
y → x
You can see this fact when you reverse the directions of the edges in the reverse path graph:
Note:
- Node 2 can be reached from
node 1 in the
reverse path graph
- If we reverse the direction of the edge, you find the path from 2 → 1
- A node y that is reached
by a traversal in the
forward path graph starting
at node x
<==>
there is a path
x → y
- How to detect a strongly connected directed graph:
- Recall that:
- A directed graph G is
strongly connected if:
- For any two vertices
u and v:
- There is a directed path u → v, and
- There is a directed path v → u
- For any two vertices
u and v:
- A directed graph G is
strongly connected if:
- A graph G
is strongly connected
if and only if:
- Every node can be reached
from
a node u in the
forward path graph,
and
- Every node can be reached from a node u in the reverse path graph
Proof:
- If G is
strongly connected then:
- There is a directed path
u → v, for
any node v
Therefore:
- Every node can be reached from a node u in the forward path graph
- There is a directed path
v → u
Therefore:
- Every node can be reached from a node u in the reverse path graph
- There is a directed path
u → v, for
any node v
- The reverse of the statement (claim) is very trivial
- Every node can be reached
from
a node u in the
forward path graph,
and
- Recall that:
- Algorithm to detect strongly connected di-graphs:
Select an arbitrary node x; A = set of node reached by a DFS/BFS in the forward path graph starting at x; if ( A ≠ all nodes in graph ) { return ( not strongly connected ); } B = set of node reached by a DFS/BFS in the reverse path graph starting at x; if ( B ≠ all nodes in graph ) { return ( not strongly connected ); } return ( strongly connected );