Download Articulation point

COMP261 Lecture 8
Articulation Points 1 of 2 (Ideas)
Outline
• What/Why articulation point
• Find articulation points (bad algorithm)
• Find articulation points (a better idea)
Connectedness
• Is this graph connected or not?
D
BB
C
A
FF
U
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E
N
I
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AA
H
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M
K
B
X
F
L
G
Y
CC
J
DD
P
GG
Q
EE
O
Articulation points
• This graph is connected, but is it “fragile”?
Would deleting one node disconnect it?
D
C
A
BB
T
FF
U
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E
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AA
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DD
• Articulation point: node whose removal would disconnect part of the graph.
Applications
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•
•
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Wireless sensor networks
Road networks
Social networks
Military
Articulation Points: a bad algorithm
• Take each node out in turn, and test for connectedness
articulationPoints ← empty set
for each node in graph
node.visited ← true;
for all other nodes nd: nd.visited ← false
recDFS(first neighbour of node)
for each remaining neighbour of node
if not neighbour.visited then
add node to articulationPoints
recDFS(node):
if not node.visited then
node.visited ← true,
for each neighbour of node
if not neighbour.visited then
recDFS(neighbour )
Why is it bad?
• Cost of DFS:
– O(e)
= O(n2) for very dense graphs
• Cost of Alg:
– O(ne) = O(n3) for very dense graphs
• Why do we have to traverse the whole graph n times, once for
each node?
• Why not do a single traversal, identifying all articulation points
as we go?
Articulation Points
• What are we looking for?
Group 1
M
P
Nodes in a graph that separate
the graph into two groups, so that
all paths from nodes in one group
to nodes in the other group
go through the node.
R
S
U
Group 2
N
T
Q
W
Articulation Points: DFS
• Traverse the graph using DFS, keep track of the count number
of each node in the search tree
B
A 1
In the graph but not
in the search tree
A
B 2
C 3
C
D
G
D
F
E 5
E
F
6
4
G 7
Articulation Points: DFS
• In the search tree, each node separates all
the nodes into two subsets
– The subtree nodes
– The non-subtree nodes
• Check if each subtree node is connected to
the non-subtree nodes if the node breaks
down (whether there are alternative paths)
A 1
B 2
– Do we need to check {D,E,F} and {G}?
– No
C 3
• Alternative path from {D,E,F} to {A,B}
• Alternative path from {G} to {A,B}
• ⇒ Alternative path from {D,E,F} to {G}
F
D
E 5
6
4
G 7
Articulation Points: DFS
• A node is an articulation point if there exists a subtree
node that is connected to non-subtree nodes only
through the node, i.e. there is not alternative path
• An alternative path uses the edges that are in the
graph but not in the search tree (back edges)
• How to check that the back edge connects to nonsubtree node?
– The other end-node has a smaller count number
– All the nodes with smaller count numbers are non-subtree
nodes
Articulation Points: DFS
• C (count = 3) is an articulation point, because there is
not back edge from G to A (count = 1) or B (count = 2)
• B (count = 2) is not an articulation point, because all
the subtree nodes {C, D, E, F, G} can connect via a
back edge (D,A) to A (count = 1)
A 1
B 2
• D? E?
C 3
D
• Is it all?
• We haven’t checked the root
E 5
F
6
4
G 7
Articulation Points: DFS
• DFS with another root D
• How to check for D?
D 1
– No non-subtree node
B
A
A
B
C
D
G
F
E
C
G
3
4
5
• Whether there are more than one sub-trees
2
E 6
F
7
Articulation Points
• Key ideas of algorithm:
– Record count number of nodes as you search
– From each recursive search of a subtree, return the
minimum count number that the subtree nodes can "reach
back" to.
– Compare the "reach back" of each subtree to count number
of this node, if smaller then there is an alternative path