Tell me where I am so I can meet you sooner: (Asynchronous rendezvous with location information)

Andrew Collins, Jurek Czyzowicz, Leszek Ga̧sieniec, Arnaud Labourel

Research output: Chapter in Book/Report/Conference proceedingConference contribution

33 Scopus citations

Abstract

In this paper we study efficient rendezvous of two mobile agents moving asynchronously in the Euclidean 2d-space. Each agent has limited visibility, permitting it to see its neighborhood at unit range from its current location. Moreover, it is assumed that each agent knows its own initial position in the plane given by its coordinates. The agents, however, are not aware of each others position. The agents possess coherent compasses and the same unit of length, which permit them to consider their current positions within the same system of coordinates. The cost of the rendezvous algorithm is the sum of lengths of the trajectories of both agents. This cost is taken as the maximum over all possible asynchronous movements of the agents, controlled by the adversary. We propose an algorithm that allows the agents to meet in a local neighborhood of diameter O(d), where d is the original distance between the agents. This seems rather surprising since each agent is unaware of the possible location of the other agent. In fact, the cost of our algorithm is O(d 2+ε ), for any constant ε>0. This is almost optimal, since a lower bound of Ω(d 2) is straightforward. The only up to date paper [12] on asynchronous rendezvous of bounded-visibility agents in the plane provides the feasibility proof for rendezvous, proposing a solution exponential in the distance d and in the labels of the agents. In contrast, we show here that, when the identity of the agent is based solely on its original location, an almost optimal solution is possible. An integral component of our solution is the construction of a novel type of non-simple space-filling curves that preserve locality. An infinite curve of this type visits specific grid points in the plane and provides a route that can be adopted by the mobile agents in search for one another. This new concept may also appear counter-intuitive in view of the result from [22] stating that for any simple space-filling curve, there always exists a pair of close points in the plane, such that their distance along the space-filling curve is arbitrarily large.

Original languageEnglish (US)
Title of host publicationAutomata, Languages and Programming - 37th International Colloquium, ICALP 2010, Proceedings
Pages502-514
Number of pages13
EditionPART 2
DOIs
StatePublished - Aug 12 2010
Externally publishedYes
Event37th International Colloquium on Automata, Languages and Programming, ICALP 2010 - Bordeaux, France
Duration: Jul 6 2010Jul 10 2010

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
NumberPART 2
Volume6199 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Conference

Conference37th International Colloquium on Automata, Languages and Programming, ICALP 2010
CountryFrance
CityBordeaux
Period7/6/107/10/10

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science(all)

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  • Cite this

    Collins, A., Czyzowicz, J., Ga̧sieniec, L., & Labourel, A. (2010). Tell me where I am so I can meet you sooner: (Asynchronous rendezvous with location information). In Automata, Languages and Programming - 37th International Colloquium, ICALP 2010, Proceedings (PART 2 ed., pp. 502-514). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 6199 LNCS, No. PART 2). https://doi.org/10.1007/978-3-642-14162-1_42