snap-stabilizing detection of cutsets alain cournier, stéphane devismes, and vincent villain...

Snap-Stabilizing Detection of Cutsets

Alain Cournier, Stéphane Devismes, and Vincent Villain

HIPC’2005, December 18-21 2005, Goa (India)

Snap-Stabilizing Detection of Cutsets 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.


What is a Cutset?

6

87

5

4

2

G=(V,E)

CS={2,6,8}

G’ = (V \ CS, E ∩ CS²)

CS is a cutset of G

9

1

3


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


Properties

G=(V,E)

CS={2,6,8}

G’ = (V \ CS, E ∩ CS²)


DFS Spanning Tree

H=1

H=0

H=1

H=2

H=4 H=4

H=3H=3

H=5

CCRoots


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


Detection of the CCRoots

H=1

H=0

H=1

H=2

H=4 H=4

H=3

H=5H=5,B=3

H=1,B=0

H=4,B=3

H=2,B=2

H=3H=3,B=2

H=0 => CCRoot

H=B => CCRoot


Detection of the CCRoots

H=1

H=0

H=1

H=2

H=4 H=4

H=3

H=5H=5,B=3

H=1,B=0

H=4,B=0

H=2,B=0

H=3H=3,B=0


Using a DF Token Circulation for the cutset detection

R

H=0,Cpt=1

H=1,Cpt=1

H=2,Cpt=1

H=3,Cpt=1

H=4,Cpt=1,B=3 H=4,Cpt=1

H=5,Cpt=1,B=3

H=3,Cpt=1,B=2 H=3,Cpt=1

H=2,Cpt=2,B=2

H=1,Cpt=2

H=0,Cpt=2

H=1,Cpt=2,B=0

R decides that

CS is a cutset

because Cpt =

2

Cpt=1 because R is a CCRoot

Cpt++ because H=B


Using a DF Token Circulation for the cutset detection

R

H=0,Cpt=0

H=1,Cpt=0

H=2,Cpt=0

H=3,Cpt=0

H=4,Cpt=0,B=3 H=4,Cpt=0

H=5,Cpt=0,B=3

H=3,Cpt=0,B=2 H=3,Cpt=0,B=2

H=2,Cpt=1,B=2

H=1,Cpt=1

H=0,Cpt=1

H=1,Cpt=1

R decides that

CS is not a

cutset because

Cpt = 1

Cpt=0 because R is not a CCRoot

Cpt=1 because H=B

H=4,Cpt=0,B=3


What about Stabilization?


Self-Stabilization

A self-stabilizing system, regardless of the initial state of the processors, is guaranteed to converge to the intended behavior in finite time.

Self-Stabilisation, Dijkstra (1974)

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)


Snap-Stabilization

A snap-stabilizing system, regardless of the initial state of the processors, always behaves according to its specifications.

Snap-Stabilisation, Bui et al (1999)

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)


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.


Thank you!

snap-stabilizing detection of cutsets alain cournier, stéphane devismes, and vincent villain...

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