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Space-Optimal Deterministic Rendezvous

This research presents a space-optimal and asymptotically time-optimal algorithm for the rendezvous problem involving k autonomous and oblivious mobile agents within an anonymous graph structure. The agents operate under various constraints including lack of knowledge about each other and the graph's topology, necessitating a robust exploration strategy. We establish lower bounds detailing that any deterministic rendezvous algorithm must encompass the exploration of the entire graph by at least one agent and demonstrate conditions under which agents can successfully rendezvous.

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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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