View
219
Download
0
Category
Preview:
Citation preview
Connectivity 1
Connectivity and Biconnectivity
• connected components
• cutvertices
• biconnected components
Connectivity 2
Connected Components• Connected Graph: any two vertices are connected
by a path.
• Connected Component: maximal connected subgraph of a graph.
connected not connected
3
Equivalence Relations
• A relation on a set S is a set R of ordered pairs of elements of S defined by some property
• Example: S = {1,2,3,4} R = {(i,j) S X S such that i < j} = {(1,2),(1,3),(1,4),(2,3),
(2,4),{3,4)}
4
Equivalence RelationsAn equivalence relation is a relation with the following properties:
(x,x) R, x S (reflexive)(x,y) R (y,x) R (symmetric)(x,y), (y,z) R (x,z) R (transitive)
The relation C can be defined as follows on the set of vertices of a graph: (u,v) C u and v are in the same connected component
C is an equivalence relation.
5
DFS on a Disconnected Graph
• DFS(v) visits all the vertices and edges in the connected component of v.• To compute the connected components:
k = 0 // component counter
for each (vertex v)
if unvisited(v)
// add to component k the vertices
//reached by v
DFS(v, k++)
ab
d g
3
65
c
fe
6
DFS on a Disconnected Graph
ab
d g
3
f5
c
e
ab
d g
3
65
c
fe
A DFS from vertex a gives us...
Connectivity 7
Cutvertices
A Cutvertex (separation vertex) is one whose removal disconnects the graph.
In the above graph, cutvertices are: ORD, DEN
If the Chicago airport (ORD) is closed, then there is no way to get from Providence to cities on the west coast. Similarly for Denver.
MIA
SEA
SFO
ATL
PVD
LGA
STLLAXLAX
DFW
ORD
MSN
DEN
Connectivity 8
BiconnectivityBiconnected graph: has no cutvertices
• New flights:LGA-ATL and DFW-LAX make the graph biconnected.
MIA
SEA
SFO
ATL
PVD
LGA
STLLAXLAX
DFW
ORD
MSN
DEN
Connectivity 9
Properties of Biconnected Graphs
• There are two disjoint paths between any two vertices.
• There is a cycle through any two vertices.
By convention, two nodes connected by an edge form a biconnected graph, but this does not verify the above properties.
MIA
SEA
SFO
ATL
PVD
LGA
STLLAXLAX
DFW
MSN
DEN ORD
Connectivity 10
Biconnected Components
• A biconnected component (block) is a maximal biconnected subgraph
MIA
SEA
SFO
ATL
PVD
LGA
STLLAXLAX
DFW
ORD
MSN
DEN
•Biconnected components are edge-disjoint but share cutvertices.
Connectivity 11
Characterization of the Biconnected Components
We define an equivalence relation R on the edges of G: (e', e") R if there is a cycle containing both e' and e"
How do we know this is an equivalence relation? Proof (by pretty picture) of the transitive property:
e1 e2 e3
Connectivity 12
Characterization of the Biconnected Components
• What does our equivalence relation R give us?
• Each equivalence class corresponds to:
– a biconnected component of G
– a connected component of a graph H whose vertices are the edges of G and whose edges are the pairs in relation R.
Connectivity 13
DFS and Biconnected Components
• Graph H has O(m2) edges in the worst case.• Instead of computing the entire graph H, we use a smaller proxy
graph K.
c
b d
af
ei
h
g
g
h
if
ea
b
c d
•The connected components of K are the same as those of H!
Connectivity 14
DFS and Biconnected Components
• Start with an empty graph K whose vertices are the edges of G.• Given a DFS on G, consider the (m -n +1) cycles of G induced by
the back edges.• For each such cycle C = (e0, e1, ... , ep) add edges (e0, e1) ... (e0, ep) to
K.
c
b d
af
ei
h
g
g
h
if
ea
b
c d
Connectivity 15
A Linear Time Algorithm
We can further reduce the size of the proxy graph to O(m)– Process the back edges according to a preorder visit of their destination vertex in the DFS tree– Mark the discovery edges forming the cycles– Stop adding edges to the proxy graph after the first marked edge is encountered.– The resulting proxy graph is a forest!– This algorithm runs in O(n+m) time.
The size of K is O(mn) in the worst case.
Connectivity 16
Example• Back edges labeled according to the preorder visit of their
destination vertex in the DFS tree
c
b
d
a
f
e1
g
h
e2 e3
e4
e6
e5
a g
e1 a b g h
e1
c
e2Processing e1 Processing e2
Connectivity 17
Example (contd.)
c
b
d
a
f
g
h
e2 e3
e4
e6
e5
a b g h
e1
c
e2 e3
f
e4
d
e5
e6
DFS tree
final proxy graph (a tree since the graph is biconnected)
Connectivity 18
Why Preorder?The order in which the back edges are processed is essential for the correctness of the algorithmUsing a different order ...
c
b
d
a
g
e1
e2
e3
g d b a
e2
e3
e1... yields a graph that provides incorrect information
Connectivity 19
Try the Algorithm on this Graph!
MIA
SEA
SFO
ATL
PVD
LGA
STLLAXLAX
DFW
MSN
DEN
LAXSAN
SJU STT
ORD
Recommended