Time versus space trade-offs for rendezvous in trees

Jurek Czyzowicz (1) , Adrian Kosowski (2, 3) , Andrzej Pelc (1)

1 DII - Département d'Informatique et d'Ingénierie
2 GANG - Networks, Graphs and Algorithms
3 CEPAGE - Algorithmics for computationally intensive applications over wide scale distributed platforms

Two identical (anonymous) mobile agents start from arbitrary nodes of an unknown tree and have to meet at some node. Agents move in synchronous rounds: in each round an agent can either stay at the current node or move to one of its neighbors. We consider deterministic algorithms for this rendezvous task. The main result of this paper is a tight trade-off between the optimal time of completing rendezvous and the size of memory of the agents. For agents with $k$ memory bits, we show that optimal rendezvous time is $\Theta(n+n^2/k)$ in $n$-node trees. More precisely, if $k \geq c\log n$, for some constant $c$, we design agents accomplishing rendezvous in arbitrary trees of size $n$ {(unknown to the agents)} in time $O(n+n^2/k)$, starting with arbitrary delay. We also show that no pair of agents can accomplish rendezvous in time $o(n+n^2/k)$, even in the class of lines of known length and even with simultaneous start. Finally, we prove that at least logarithmic memory is necessary for rendezvous, even for agents starting simultaneously in a $n$-node line.

Mots clés: rendezvous; anonymous agents; time; memory space

Domaines: Algorithme et structure de données [cs.DS]

Jurek Czyzowicz, Adrian Kosowski, Andrzej Pelc. Time versus space trade-offs for rendezvous in trees. Distributed Computing, 2014, 27 (2), pp.95-109. ⟨10.1007/s00446-013-0201-4⟩. ⟨hal-00646912v2⟩