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  1. Self-stabilizing (f,g)-Alliances with Safe Convergence Fabienne Carrier Ajoy K. Datta Stéphane Devismes Lawrence L. Larmore YvanRivierre

  2. Co-Autors Ajoy K. Datta & Lawrence L. Larmore Fabienne Carrier & YvanRivierre SSS 2013, Osaka

  3. Roadmap • Safe convergence • The (f,g)-alliance problem • Contribution • Algorithm • Perspectives SSS 2013, Osaka

  4. Safe convergence SSS 2013, Osaka

  5. Pros and Cons of Self-Stabilization • Tolerate any finite number of transient faults • No initialization • Large-scale network • Self-organization in sensor network • Dynamicity • Topological change ≈ Transient fault • Tolerate onlytransient faults • Eventual safety • No stabilization detection SSS 2013, Osaka

  6. Pros and Cons of Self-Stabilization • Tolerate any finite number of transient faults • No initialization • Large-scale network • Self-organization in sensor network • Dynamicity • Topological change ≈ Transient fault • Tolerate onlytransient faults • Eventual safety • No stabilization detection SSS 2013, Osaka

  7. Related Work • Enhancing safety: • Fault-containment [Ghoshet al, PODC’96] • Superstabilization[Dolev & Herman, CJTCS’97] • Time-adaptive Self-stabilization [Kutten & Patt-Shamir, PODC’97] • Self-Stab + safe convergence [Kakugawa & Masuzawa, IPDPS’06] • Etc. SSS 2013, Osaka

  8. Back to self-stabilization SSS 2013, Osaka

  9. Back to self-stabilization No safety guarantee SSS 2013, Osaka

  10. Back to self-stabilization Ω(D) SSS 2013, Osaka

  11. Back to self-stabilization Are all illegitimate configurations identically bad ? SSS 2013, Osaka

  12. Back to self-stabilization Are all illegitimate configurations identically bad ? Of course, NO ! SSS 2013, Osaka

  13. Self-stabilization + Safe Convergence Really bad Not so bad good SSS 2013, Osaka

  14. Self-stabilization + Safe Convergence Quick convergence time SSS 2013, Osaka

  15. Self-stabilization + Safe Convergence • Optimal LC ⊆ feasable LC • Set of feasable LC: CLOSED • Set of optimal LC: CLOSED • Quick convergence to a feasable LC • (O(1) expected) • Convergence to an optimal LC SSS 2013, Osaka

  16. Self-stabilization + Safe Convergence: example [Kakugawa & Masuzawa, IPDPS’06] • Construction of a minimal dominating set • 1-round convergence to a dominating set • (not necessarily a minimal one) • Then,O(D)-rounds convergence to a MINIMAL dominating set • During this phase, all configurations contain a dominating set SSS 2013, Osaka

  17. The problem: (f,g)-Alliance[Douradoet al, SSS’11] • Alliance: subset of nodes • f, g: 2 functions mapping nodes to natural integers • For every process p: • p∉ Alliance ⇒ at leastf(p) neighbors ∈ Alliance • p∈ Alliance ⇒ at least g(p) neighbors ∈ Alliance SSS 2013, Osaka

  18. Example: (f,g)-Alliance Rednodes form a (1,0)-Alliance SSS 2013, Osaka

  19. Example: (f,g)-Alliance Rednodes DO NOT form a (1,0)-Alliance SSS 2013, Osaka

  20. (f,g)-Alliance: generalization of several problems • Dominating sets • K-domination sets • K-tuple domination sets • Global defensive alliance • Global offensive alliance E.g., Dominating set = (1,0)-alliance SSS 2013, Osaka

  21. Minimality & 1-Minimality • Let A be a set of nodes • A is a minimal (f,g)-Alliance iffevery proper subset of A is not an (f,g)-Alliance • A is a 1-minimal (f,g)-Alliance iff ∀p ∈ A, A-{p} is not an (f,g)-Alliance SSS 2013, Osaka

  22. Example: (0,1)-Alliance Red nodes form a (0,1)-Alliance, but NEITHER a minimal NOR a 1-minimal (0,1)-Alliance SSS 2013, Osaka

  23. Example: (0,1)-Alliance Red nodes form a 1-minimal (0,1)-Alliance but not a minimal one SSS 2013, Osaka

  24. Example: (0,1)-Alliance Red nodes (empty set) both form a minimal AND a 1-minimal (0,1)-Alliance SSS 2013, Osaka

  25. Property[Douradoet al, SSS’11] • Every minimal (f,g)-Alliance is a 1-minimal (f,g)-Alliance • If for every node p, f(p) ≥ g(p), then • A is a minimal (f,g)-Alliance iff A is a 1-minimal (f,g)-Alliance SSS 2013, Osaka

  26. Contribution • Self-Stabilizing Safe Converging Algorithm for computing: a minimal (f,g)-Alliance in identified networks • Safe Convergence • Stabilization in 4 rounds to a configuration, where an (f,g)-Alliance is defined • Stabilization in 4n+4 additional rounds to a configuration, where minimal (f,g)-Alliance is defined • Assumptions: • If for every node p, f(p) ≥ g(p) and δ(p) ≥ g(p) • Locally shared memory model, unfair daemon • Other complexities • Memory requirement: O(logn) bits per process • Step complexity: O(Δ3n) SSS 2013, Osaka

  27. Algorithm’s main ideas SSS 2013, Osaka

  28. ``Naïve Idea” • One Boolean • Red: ∈ A • Green: ∉ A • Two actions: • Join • Leave SSS 2013, Osaka

  29. ``Naïve Idea” To obtain safe convergence, it should be harder to leave than to join • One boolean • Red: ∈ A • Green: ∉ A • Two actions: • Join • Leave SSS 2013, Osaka

  30. Leave the alliance • p can leave if : • At least f(p) neighbors ∈ A after pleaves AND • Each neighbor q still have enough neighbors ∈ A after pleaves • i.e., g(q) or f(q) depending whether q belongs or not to A SSS 2013, Osaka

  31. At least f(p) neighbors ∈ A after pleaves • Leaving should be locally sequential • Example: (2,1)-Alliance p SSS 2013, Osaka

  32. At least f(p) neighbors ∈ A after pleaves • Leaving should be locally sequential • Example: (2,1)-Alliance p p SSS 2013, Osaka

  33. At least f(p) neighbors ∈ A after pleaves • Leaving should be locally sequential • Example: (2,1)-Alliance p SSS 2013, Osaka

  34. At least f(p) neighbors ∈ A after pleaves • Leaving should be locally sequential • Example: (2,1)-Alliance p p SSS 2013, Osaka

  35. Pointer: authorization to leave Nil p Nil SSS 2013, Osaka

  36. Pointer: authorization to leave Nil p Nil SSS 2013, Osaka

  37. Each neighbor still have enough neighbor ∈ A after pleaves • A neighbor q gives an authorization only if q still have enough neighbors ∈ A without p p (1,0)-Alliance q SSS 2013, Osaka

  38. Each neighbor still have enough neighbor ∈ A after pleaves • A neighbor q gives an authorization only if q still have enough neighbors ∈ A without p p (1,0)-Alliance If q has several choices ID breaks ties q SSS 2013, Osaka

  39. Deadlock problems Nil Nil (1,0)-Alliance SSS 2013, Osaka

  40. Deadlock problems Busy! Nil Nil (1,0)-Alliance SSS 2013, Osaka

  41. Deadlock problems Busy! Nil Nil (1,0)-Alliance SSS 2013, Osaka

  42. Deadlock problems Busy! Nil Nil Tie break! (1,0)-Alliance SSS 2013, Osaka

  43. Deadlock problems Busy! Nil Nil Tie break! Nil (1,0)-Alliance SSS 2013, Osaka

  44. Deadlock problems Busy! Nil Nil Tie break! Nil (1,0)-Alliance SSS 2013, Osaka

  45. How to evaluate Busy? • NP∩ A < f(p) Busy! p (2,0)-Alliance SSS 2013, Osaka

  46. How to evaluate Busy? • NP∩ A < f(p) • A neighbor qof pneeds that pstays in the alliance Busy! p q (2,0)-Alliance SSS 2013, Osaka

  47. How to evaluate Busy? • NP∩ A < f(p) • A neighbor qof pneeds that pstays in the alliance 0 2 2 1 Busy! p q 2 3 2 1 (2,0)-Alliance Nb SSS 2013, Osaka

  48. Last problem … • (1,0)-Alliance Nil Nil Nil SSS 2013, Osaka

  49. Last problem … • (1,0)-Alliance Nil Nil Nil Nil Nil Nil SSS 2013, Osaka

  50. Last problem … • (1,0)-Alliance Nil Nil Nil Nil Nil Nil Nil Nil Nil SSS 2013, Osaka