1 / 16

Snap-Stabilizing Detection of Cutsets

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

lazaro
Télécharger la présentation

Snap-Stabilizing Detection of Cutsets

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related