Snap-Stabilizing Detection of Cutsets

1
Snap-Stabilizing Detection of Cutsets
HIPC2005, December 18-21 2005, Goa (India)

• Alain Cournier, Stéphane Devismes, and Vincent
  Villain

2
What is a Cutset?

• Let G(V,E) be an undirected connected graph.
• Let CS be a subset of V.
• Let G be the subgraph induced by V\CS.
• CS is a cutset of G if and only if G is
  unconnected.

3
What is a Cutset?
1
2
3
4
CS is a cutset of G
G(V,E)
CS2,6,8
G (V \ CS, E n CS²)
6
5
7
8
9
4
Problem Given a network G and a subset of
processors CS.Is CS a Cutset of G?
This decition must be performed in a distributed
manner
5
Properties
G(V,E)
CS2,6,8
G (V \ CS, E n CS²)
6
DFS Spanning Tree
H0
H1
H1
CCRoots
H2
H3
H3
H4
H4
H5
7
Approach

• Theorem CS is a cutset of G if and only if there
  exists at least two CCRoots.

• Scheme of the algorithm
• To detect the CCRoots
• To count the CCRoots
• To decide if CS is a cutset

8
Detection of the CCRoots
H0 gt CCRoot
H0
H1
H1,B0
H1
HB gt CCRoot
H2
H2,B2
H3
H3,B2
H3
H4
H4,B3
H4
H5
H5,B3
9
Detection of the CCRoots
H0
H1
H1,B0
H1
H2
H2,B0
H3
H3,B0
H3
H4
H4,B0
H4
H5
H5,B3
10
Using a DF Token Circulation for the cutset
detection
Cpt1 because R is a CCRoot
R
H0,Cpt2
H0,Cpt1
H1,Cpt1
H1,Cpt2
H1,Cpt2,B0
R decides that CS is a cutset because Cpt 2
Cpt because HB
H2,Cpt1
H2,Cpt2,B2
H3,Cpt1,B2
H3,Cpt1
H3,Cpt1
H4,Cpt1,B3
H4,Cpt1
H5,Cpt1,B3
11
Using a DF Token Circulation for the cutset
detection
Cpt0 because R is not a CCRoot
R
H0,Cpt1
H0,Cpt0
H1,Cpt0
H1,Cpt1
H1,Cpt1
R decides that CS is not a cutset because Cpt 1
Cpt1 because HB
H2,Cpt0
H2,Cpt1,B2
H3,Cpt0
H3,Cpt0,B2
H3,Cpt0,B2
H4,Cpt0,B3
H4,Cpt0
H4,Cpt0,B3
H5,Cpt0,B3
12
What about Stabilization?
13
Self-Stabilization
If we use a Self-Stabilizing DFTC, Then the
cutset detection is Self-Stabilizing

• Huang and Chen (Distributed Computing, 1993)
• Johnen et al (WDAG, 1997)
• Datta et al (Distributed Computing, 2000)

14
Snap-Stabilization
If we use a Snap-Stabilizing DFTC, Then the
cutset detection is Snap-Stabilizing

• Cournier, Devismes, Petit, and Villain (OPODIS,
  2004)
• Cournier, Devismes, and Villain (SSS, 2005)

15
Conclusion

• The decision needs one traversal of the network
  only
• The time complexity of the solution corresponds
  to the time complexity of the DFTC
• Small memory overcost (two integers)
• The stabilization property depends on the DFTC
• Our solution can used in dynamic networks
• Our method can be adapted to solve some problem
  closed to the cutset detection cutpoint and
  bridge finding.

16
Thank you!