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Almost Universal Anonymous Rendezvous in the Plane

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Abstract

Two mobile agents represented by points freely moving in the plane and starting at two different positions, have to meet. The meeting, called rendezvous, occurs when agents are at distance at most r of each other and never move after this time, where r is a positive real unknown to them, called the visibility radius. Agents are anonymous and execute the same deterministic algorithm. Each agent has a set of private attributes, some or all of which can differ between agents. These attributes are: the initial position of the agent, its system of coordinates (orientation and chirality), the rate of its clock, its speed when it moves, and the time of its wake-up. If all attributes (except the initial positions) are identical and agents start at distance larger than r then they can never meet, as the distance between them can never change. However, differences between attributes make it sometimes possible to break the symmetry and accomplish rendezvous. Such instances of the rendezvous problem (formalized as lists of attributes), are called feasible. Our contribution is three-fold. We first give an exact characterization of feasible instances. Thus it is natural to ask whether there exists a single algorithm that guarantees rendezvous for all these instances. We give a strong negative answer to this question: we show two sets \(S_1\) and \(S_2\) of feasible instances such that none of them admits a single rendezvous algorithm valid for all instances of the set. On the other hand, we construct a single algorithm that guarantees rendezvous for all feasible instances outside of sets \(S_1\) and \(S_2\). We observe that these exception sets \(S_1\) and \(S_2\) are geometrically very small, compared to the set of all feasible instances: they are included in low-dimension subspaces of the latter. Thus, our rendezvous algorithm handling all feasible instances other than these small sets of exceptions can be justly called almost universal.

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Notes

  1. Note that, as opposed to [20], we needed to say precisely which moves are allowed because some of our results are negative and thus we need to prove that some actions are impossible. In [20] this could be left implicit, as they only had positive results. We decided for the easiest option with segment moves but is is easy to see that all our results (also the negative ones) remain valid if circles are allowed as they were in [20].

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Authors and Affiliations

  1. MIS Lab, Université de Picardie Jules Verne, Amiens, France

    Yoann Dieudonné

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

    Andrzej Pelc

  3. CNRS, INRIA, LIP6, Sorbonne Université, 75005, Paris, France

    Franck Petit

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  1. Yoann Dieudonné
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  2. Andrzej Pelc
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  3. Franck Petit
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Correspondence to Yoann Dieudonné.

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A preliminary version of this paper appeared in [15]. One of the co-authors of the preliminary version, Sébastien Bouchard, asked to be removed from the list of authors.

Andrzej Pelc was supported in part by NSERC discovery Grant 2018-03899 and by the Research Chair in Distributed Computing of the Université du Québec en Outaouais. Performed within Project ESTATE (Ref. ANR-16-CE25-0009-03), supported by French state funds managed by the ANR (Agence Nationale de la Recherche).

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Dieudonné, Y., Pelc, A. & Petit, F. Almost Universal Anonymous Rendezvous in the Plane. Algorithmica 85, 3110–3143 (2023). https://doi.org/10.1007/s00453-023-01122-2

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