Space-Optimal%20Deterministic%20Rendezvous

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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)
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System Settings

• Graph G(V,E) of n nodes and m bidirectional
  links
• Set of k mobile agents
• Nodes are anonymous
• Agents are autonomous and oblivious

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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 about G, in particular
  they know nothing about n, m, the diameter or
  maximum degree of G, etc.

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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

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Related Works

• Two synchronous non oblivious agentsAlpern 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

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Impossibility ResultDe Marco et al.,
06Barrière et al., 07

• Anonymous, oblivious agents
• No a priori conditions on n and k
• No knowledge

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Impossibility ResultDe Marco et al.,
06Barrière et al., 07

• Anonymous, oblivious agents
• No a priori conditions on n and k
• No knowledge

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Impossibility ResultDe Marco et al.,
06Barrière et al., 07

• Anonymous, oblivious agents
• No a priori conditions on n and k
• No knowledge

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Impossibility ResultDe Marco et al.,
06Barrière et al., 07

• Anonymous, oblivious agents
• No a priori conditions on n and k
• No knowledge

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Impossibility ResultDe Marco et al.,
06Barrière et al., 07

• 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

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Contribution

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

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Lower Bounds

• Any deterministic rendezvous algorithm must
  guarantee that at least one agent explore the
  whole graph.

a1
ua1
a2
va1
ak
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Lower Bounds

• Any deterministic rendezvous algorithm must
  guarantee that at least one agent explores the
  whole graph.

• Any deterministic rendezvous algorithm terminates
  in O(m) rounds.

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Lower Bounds

• Any deterministic graph exploration made by an
  agent a terminates at the starting node of a.

la
6
dv
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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.

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?
?
?
?
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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(dv1) log(Lmax) 1 bits.

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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 a through 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

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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

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Algorithm
HomeF Host
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HomeF Host
HomeF Host
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1
HomeF Host
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HomeF Host
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HomeF Host
1
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Algorithm
HomeT HostX
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1
HomeF Host
HomeF Host
2
1
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3
1
HomeF Host
2
1
3
HomeF Host
1
2
HomeF Host
1
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Algorithm
HomeT HostX
3
2
1
HomeF Host
HomeF Host
2
1
2
3
1
HomeF Host
2
1
3
HomeF HostX
1
2
HomeF Host
1
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Algorithm
HomeT HostX
3
2
1
HomeF HostX
HomeT HostL
2
1
2
3
1
HomeF Host
2
1
3
HomeF HostX
1
2
HomeF Host
1
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Algorithm
HomeT HostX
3
2
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HomeF HostX
HomeT HostL
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2
3
1
HomeF Host
2
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3
HomeF HostX
1
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HomeF HostL
1
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Algorithm
HomeT HostX
3
2
1
HomeF HostX
HomeT HostL
2
1
2
3
1
HomeF Host
2
1
3
HomeF HostL
1
2
HomeF HostL
1
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Algorithm
HomeF HostL
3
2
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HomeF HostX
HomeT HostL
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1
2
3
1
HomeF HostX
2
1
3
HomeF HostL
1
2
HomeF HostL
1
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Algorithm
HomeF HostL
3
2
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HomeF HostX
HomeT HostL
2
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2
3
1
HomeF HostX
2
1
3
HomeF HostL
1
2
HomeF HostL
1
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Algorithm
HomeF HostL
3
2
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HomeF HostX
HomeT HostL
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1
2
3
1
HomeF HostX
2
1
3
HomeF HostL
1
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HomeF HostL
1
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Algorithm
HomeF HostL
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2
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HomeF HostL
HomeT HostL
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3
1
HomeF HostX
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3
HomeF HostL
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HomeF HostL
1
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Algorithm
HomeF HostL
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2
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HomeF HostL
HomeT HostL
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3
1
HomeF HostL
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3
HomeF HostL
1
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HomeF HostL
1
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Algorithm
HomeF HostL
3
2
1
HomeF HostL
HomeT HostL
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1
2
3
1
HomeF HostL
2
1
3
HomeF HostL
1
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HomeF HostL
1
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Algorithm
HomeF HostL
3
2
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HomeF HostL
HomeT HostL
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2
3
1
HomeF HostL
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3
HomeF HostL
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HomeF HostL
1
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Algorithm
HomeF HostL
3
2
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HomeF HostL
HomeT HostL
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2
3
1
HomeF HostL
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HomeF HostL
1
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HomeF HostL
1
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Algorithm
HomeF HostL
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1
0
HomeF HostL
HomeT HostL
1
0
1
2
0
HomeF HostL
1
0
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HomeF HostL
0
1
HomeF HostL
0
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Algorithm

• Performs a Rendezvous in ?(m) rounds.
• 2log(dv1) log(Lmax) 1 bits on each node.

• Asymptotically optimal in time.
• Optimal in space.

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Necessary Conditions

• Labeled edges
• Labels and whiteboards
• Unique minimum label

• (Local) Determinism

• Barriere et al., 07

• Lemma 3

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Conclusion

• Time and space complexity lower bounds
• Asymptotically space and time optimal algorithm
• Future Work directed graphs

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Thank you.
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