1. Snap-Stabilizing Detection of Cutsets Alain Cournier, Stéphane Devismes, and Vincent Villain HIPC’2005, December 18-21 2005, Goa (India)

2. Snap-Stabilizing Detection of Cutsets2 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. Snap-Stabilizing Detection of Cutsets3 What is a Cutset? 6 8 7 5 4 2 G=(V,E) CS={2,6,8} G’ = (V \ CS, E ∩ CS²) CS is a cutset of G 9 1 3

4. Snap-Stabilizing Detection of Cutsets4 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. Snap-Stabilizing Detection of Cutsets5 Properties G=(V,E) CS={2,6,8} G’ = (V \ CS, E ∩ CS²)

6. Snap-Stabilizing Detection of Cutsets6 DFS Spanning Tree H=1 H=0 H=1 H=2 H=4 H=3 H=5 CCRoots

7. Snap-Stabilizing Detection of Cutsets7 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. Snap-Stabilizing Detection of Cutsets8 Detection of the CCRoots H=1 H=0 H=1 H=2 H=4 H=3 H=5 H=5,B=3 H=1,B=0 H=4,B=3 H=2,B=2 H=3 H=3,B=2 H=0 => CCRoot H=B => CCRoot

9. Snap-Stabilizing Detection of Cutsets9 Detection of the CCRoots H=1 H=0 H=1 H=2 H=4 H=3 H=5 H=5,B=3 H=1,B=0 H=4,B=0 H=2,B=0 H=3 H=3,B=0

10. Snap-Stabilizing Detection of Cutsets10 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=3H=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

11. Snap-Stabilizing Detection of Cutsets11 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=3H=4,Cpt=0 H=5,Cpt=0,B=3 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

12. Snap-Stabilizing Detection of Cutsets12 What about Stabilization?

13. Snap-Stabilizing Detection of Cutsets13 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)

14. Snap-Stabilizing Detection of Cutsets14 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)

15. Snap-Stabilizing Detection of Cutsets15 Conclusion v The decision needs one traversal of the network only v The time complexity of the solution corresponds to the time complexity of the DFTC v Small memory overcost (two integers) v The stabilization property depends on the DFTC v Our solution can used in dynamic networks v Our method can be adapted to solve some problem closed to the cutset detection: cutpoint and bridge finding.

16. Snap-Stabilizing Detection of Cutsets16 Thank you!