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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Home=FHost=
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Algorithm
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Home=FHost=
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Algorithm
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Home=FHost=
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Algorithm
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Home=THost=L
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Algorithm
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Algorithm
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Algorithm
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Algorithm
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Algorithm
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Algorithm
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Algorithm
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Algorithm
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Algorithm
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Algorithm
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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.
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