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Time versus space trade-offs for rendezvous in trees

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Abstract

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 \(k \ge c\log n\), 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.

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Notes

  1. A result from [21] implies that rendezvous with simultaneous start from arbitrary non-symmetric initial positions in a \(n\)-node line is possible with \(O(\log \log n)\) bits of memory. This result from [21] is untrue, although it holds in the model with adversarial port labelings, cf. the corrected version of that paper [22].

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Authors and Affiliations

  1. Département d’informatique, Université du Québec en Outaouais, Gatineau, QC, J8X 3X7, Canada

    Jurek Czyzowicz & Andrzej Pelc

  2. CEPAGE Project, Inria Bordeaux Sud-Ouest, 33400, Talence, France

    Adrian Kosowski

Authors
  1. Jurek Czyzowicz
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  2. Adrian Kosowski
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  3. Andrzej Pelc
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Correspondence to Adrian Kosowski.

Additional information

A preliminary version of this paper appeared in the Proceedings of 24th ACM Symposium on Parallelism in Algorithms and Architectures (SPAA 2012), June 2012, Pittsburgh, USA, 1-10.

Jurek Czyzowicz was supported in part by NSERC discovery grant.

This research was partially done while Adrian Kosowski was working at the Department of Algorithms and System Modeling of the Gdańsk University of Technology and during his visit to the Research Chair in Distributed Computing of the Université du Québec en Outaouais. Supported in part by ANR project DISPLEXITY and by NCN under contract DEC-2011/02/A/ST6/00201.

Andrzej Pelc was supported in part by NSERC discovery grant and by the Research Chair in Distributed Computing of the Université du Québec en Outaouais.

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Czyzowicz, J., Kosowski, A. & Pelc, A. Time versus space trade-offs for rendezvous in trees. Distrib. Comput. 27, 95–109 (2014). https://doi.org/10.1007/s00446-013-0201-4

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