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Space-Optimal Deterministic Rendezvous

Space-Optimal Deterministic Rendezvous. Stéphane Devismes VERIMAG UJF, Grenoble I. Joint work with Fabienne Carrier, Yvan Rivierre (VERIMAG, UJF, Grenoble I), and Franck Petit (LIP6, UPMC, Paris 6). System Settings. Graph G=(V,E) of n nodes and m bidirectional links

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Space-Optimal Deterministic Rendezvous

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  1. Space-Optimal Deterministic Rendezvous Stéphane Devismes VERIMAGUJF, Grenoble I Joint work withFabienne Carrier, Yvan Rivierre (VERIMAG, UJF, Grenoble I), and Franck Petit (LIP6, UPMC, Paris 6)

  2. System Settings • Graph G=(V,E) of n nodes and mbidirectional links • Set of kmobile agents • Nodes are anonymous • Agents are autonomous and oblivious

  3. System Settings • The agents move asynchronously • They cannot (explicitly) communicate together (even being located at the same node) • They have no knowledge of each other, in particular they do not know k • They have no knowledge aboutG, in particular they know nothing about n,m, the diameter or maximum degree of G, etc.

  4. Rendezvous • The agent are required to eventually meet and stop at the same node. • Initially, no agent is present in G • Agents can be inserted at any time • Deterministic solutions

  5. Related Works • Twosynchronous non oblivious agents[Alpern 76] [De Marco et al., 06] [Kowalski and Pelc 04] • k asynchronous agents provided that k and n are coprime and the edge labeling has sense of direction [Barrière et al., 07] • k oblivious agents able to take a snapshot of the whole system in a ring [Klasing et al., 08]

  6. Impossibility Result[De Marco et al., 06][Barrière et al., 07] • Anonymous, oblivious agents • No a priori conditions on n and k • No knowledge 1 2 1 2 2 1 1 2 1 2 2 1

  7. Impossibility Result[De Marco et al., 06][Barrière et al., 07] 1 2 2 1 2 1 1 2 1 2 2 1 Anonymous, oblivious agents No a priori conditions on n and k No knowledge

  8. Impossibility Result[De Marco et al., 06][Barrière et al., 07] 1 2 2 1 2 1 1 2 1 2 2 1 Anonymous, oblivious agents No a priori conditions on n and k No knowledge

  9. Impossibility Result[De Marco et al., 06][Barrière et al., 07] 1 2 2 1 2 1 1 2 1 2 2 1 Anonymous, oblivious agents No a priori conditions on n and k No knowledge

  10. Impossibility Result[De Marco et al., 06][Barrière et al., 07] 1 2 2 1 2 1 1 2 1 2 2 1 Anonymous, oblivious agents No a priori conditions on n and k No knowledge Semi-anonymous, oblivious agents, i.e., exactly one agent has the minimum label Nodes equipped with whiteboards

  11. Contribution • Time and space complexity lower bounds • Space-optimal and asymptotically time optimal algorithm • Necessary conditions to deterministically solve the rendezvous problem

  12. Lower Bounds • Any deterministic rendezvous algorithm must guarantee that at least one agent explore the whole graph. Ga Gb b1 a1 ua1 ub1 b2 a2 vb1 va1 bk' ak

  13. Lower Bounds • Any deterministic rendezvous algorithm must guarantee that at least one agent explores the whole graph. • Any deterministic rendezvous algorithmterminates in Ω(m) rounds.

  14. v' v 1 4 la 6 2 3 5 Lower Bounds • Any deterministic graph exploration made by an agent a terminates at the starting node of a. la 6 dv

  15. Lower Bounds • Any deterministic graph exploration made by an agent a terminates at the starting node of a. • Any deterministic rendezvous algorithm requires that each agent a writes its label la on the whiteboard. la la la lb lb lb ✓ ✓ ✓ ✓ ✓ ✓

  16. Lower Bounds • Any deterministic graph exploration made by an agent a terminates at the starting node of a. • Any deterministic rendezvous algorithm requires that each agent a writes its label la on the whiteboard. • Any deterministic rendezvous algorithm requires at least log(dv+1) + log(Lmax) + 1 bits.

  17. Algorithm • 3 variables on the whiteboard of each node v: • Currentv ∈ {0,…,dv-1} ∪ {⊥}, init. ⊥ • Homev ∈ {F,T}, init. F • Hostv: Set of labels • 3 primitives for each agent a: • Go(e): Sends athrough the edge e • From()∈ {0,…,dv-1} ∪ {⊥}: return the edge from which a comes, ⊥ otherwise (initial state) • Next() :Return the next edge label according to From(), e.g., (From()+1 mod dv) + 1

  18. Algorithm • Basic idea: • Each agent a tries to make the deterministic DFS traversal induced by the local labels of edges • Only the agent with the minimum label lmin eventually succeeds its traversal • The other agents eventually follow the traversal of lmin

  19. Algorithm Home=F Host= 3 2 1 Home=F Host= Home=F Host= 2 1 2 3 1 Home=F Host= 2 1 3 Home=F Host= 1 2 Home=F Host= 1

  20. Algorithm Home=T Host=X 3 2 1 Home=F Host= Home=F Host= 2 1 2 3 1 Home=F Host= 2 1 3 Home=F Host= 1 2 Home=F Host= 1

  21. Algorithm Home=T Host=X 3 2 1 Home=F Host= Home=F Host= 2 1 2 3 1 Home=F Host= 2 1 3 Home=F Host=X 1 2 Home=F Host= 1

  22. Algorithm Home=T Host=X 3 2 1 Home=F Host=X Home=T Host=L 2 1 2 3 1 Home=F Host= 2 1 3 Home=F Host=X 1 2 Home=F Host= 1

  23. Algorithm Home=T Host=X 3 2 1 Home=F Host=X Home=T Host=L 2 1 2 3 1 Home=F Host= 2 1 3 Home=F Host=X 1 2 Home=F Host=L 1

  24. Algorithm Home=T Host=X 3 2 1 Home=F Host=X Home=T Host=L 2 1 2 3 1 Home=F Host= 2 1 3 Home=F Host=L 1 2 Home=F Host=L 1

  25. Algorithm Home=F Host=L 3 2 1 Home=F Host=X Home=T Host=L 2 1 2 3 1 Home=F Host=X 2 1 3 Home=F Host=L 1 2 Home=F Host=L 1

  26. Algorithm Home=F Host=L 3 2 1 Home=F Host=X Home=T Host=L 2 1 2 3 1 Home=F Host=X 2 1 3 Home=F Host=L 1 2 Home=F Host=L 1

  27. Algorithm Home=F Host=L 3 2 1 Home=F Host=X Home=T Host=L 2 1 2 3 1 Home=F Host=X 2 1 3 Home=F Host=L 1 2 Home=F Host=L 1

  28. Algorithm Home=F Host=L 3 2 1 Home=F Host=L Home=T Host=L 2 1 2 3 1 Home=F Host=X 2 1 3 Home=F Host=L 1 2 Home=F Host=L 1

  29. Algorithm Home=F Host=L 3 2 1 Home=F Host=L Home=T Host=L 2 1 2 3 1 Home=F Host=L 2 1 3 Home=F Host=L 1 2 Home=F Host=L 1

  30. Algorithm Home=F Host=L 3 2 1 Home=F Host=L Home=T Host=L 2 1 2 3 1 Home=F Host=L 2 1 3 Home=F Host=L 1 2 Home=F Host=L 1

  31. Algorithm Home=F Host=L 3 2 1 Home=F Host=L Home=T Host=L 2 1 2 3 1 Home=F Host=L 2 1 3 Home=F Host=L 1 2 Home=F Host=L 1

  32. Algorithm Home=F Host=L 3 2 1 Home=F Host=L Home=T Host=L 2 1 2 3 1 Home=F Host=L 2 1 3 Home=F Host=L 1 2 Home=F Host=L 1

  33. Algorithm Home=F Host=L 2 1 0 Home=F Host=L Home=T Host=L 1 0 1 2 0 Home=F Host=L 1 0 2 Home=F Host=L 0 1 Home=F Host=L 0

  34. Algorithm • Performs a Rendezvous in θ(m) rounds. • 2log(dv+1) + log(Lmax) + 1 bits on each node. • Asymptotically optimal in time. • Optimal in space.

  35. Necessary Conditions • Labeled edges • Labels and whiteboards • Unique minimum label • (Local) Determinism • [Barriere et al., 07] • Lemma 3

  36. Conclusion • Time and space complexity lower bounds • Asymptotically space and time optimal algorithm • Future Work : directed graphs

  37. Thank you.

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