Snap-Stabilizing Detection of Cutsets

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

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

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 Snap-Stabilizing Detection of Cutsets

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

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

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 Snap-Stabilizing Detection of Cutsets

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

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

10. Using a DF Token Circulation for the cutset detection Cpt=1 because R is a CCRoot R H=0,Cpt=2 H=0,Cpt=1 H=1,Cpt=1 H=1,Cpt=2 H=1,Cpt=2,B=0 R decides that CS is a cutset because Cpt = 2 Cpt++ because H=B H=2,Cpt=2,B=2 H=2,Cpt=1 H=3,Cpt=1,B=2 H=3,Cpt=1 H=3,Cpt=1 H=4,Cpt=1,B=3 H=4,Cpt=1 H=5,Cpt=1,B=3 Snap-Stabilizing Detection of Cutsets

11. Using a DF Token Circulation for the cutset detection Cpt=0 because R is not a CCRoot R H=0,Cpt=1 H=0,Cpt=0 H=1,Cpt=0 H=1,Cpt=1 H=1,Cpt=1 R decides that CS is not a cutset because Cpt = 1 Cpt=1 because H=B H=2,Cpt=1,B=2 H=2,Cpt=0 H=3,Cpt=0,B=2 H=3,Cpt=0 H=3,Cpt=0,B=2 H=4,Cpt=0,B=3 H=4,Cpt=0 H=4,Cpt=0,B=3 H=5,Cpt=0,B=3 Snap-Stabilizing Detection of Cutsets

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

13. Self-Stabilisation, Dijkstra (1974) A self-stabilizing system, regardless of the initial state of the processors, is guaranteed to converge to the intended behavior in finite time. 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) Snap-Stabilizing Detection of Cutsets

14. Snap-Stabilisation, Bui et al (1999) A snap-stabilizing system, regardless of the initial state of the processors, always behaves according to its specifications. 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) Snap-Stabilizing Detection of Cutsets

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. Snap-Stabilizing Detection of Cutsets

16. Thank you! Snap-Stabilizing Detection of Cutsets