Space-Optimal Deterministic Rendezvous

Space-Optimal Deterministic Rendezvous

Stéphane DevismesVERIMAG

UJF, Grenoble I

Joint work withFabienne 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

Set of k mobile agents

Nodes are anonymous

Agents are autonomous and oblivious

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.

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

Related Works Two synchronous 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]

Anonymous, oblivious agents

No a priori conditions on n and k

No knowledge

Impossibility Result[De Marco et al., 06][Barrière et al., 07]

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Impossibility Result[De Marco et al., 06][Barrière et al., 07]

Anonymous, oblivious agents

No a priori conditions on n and k

No knowledge

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Impossibility Result[De Marco et al., 06][Barrière et al., 07]

Anonymous, oblivious agents

No a priori conditions on n and k

No knowledge

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Anonymous, oblivious agents

No a priori conditions on n and k

No knowledge

Impossibility Result[De Marco et al., 06][Barrière et al., 07]

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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

Impossibility Result[De Marco et al., 06][Barrière et al., 07]

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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

Gb

b1ub1

vb1

Lower Bounds

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

Ga

a1a2

ak

b2

bk'

ua1

va1

Lower Bounds

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

Any deterministic rendezvous algorithm terminates in Ω(m) rounds.

v

Lower Bounds

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

v'

la

la

v

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.

lbla

✓ ✓ ✓ ✓ ✓ ✓

la la lblb

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(v+1) + log(Lmax) + 1 bits.

Algorithm 3 variables on the whiteboard of each

node v: Currentv ∈ {0,…,v-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,…,v-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 v) + 1

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

Algorithm

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Algorithm Performs a Rendezvous in θ(m) rounds. 2log(v+1) + log(Lmax) + 1 bits on each node.

Asymptotically optimal in time. Optimal in space.

Necessary Conditions

Labeled edges

Labels and whiteboards

Unique minimum label

[Barriere et al., 07]

Lemma 3

(Local) Determinism

Conclusion

Time and space complexity lower bounds

Asymptotically space and time optimal algorithm

Future Work : directed graphs

Thank you.