Abstract
The Meeting problem for k>=2 searchers in a polygon P (possibly with holes) consists in making the searchers move within P, according to a distributed algorithm, in such a way that at least two of them eventually come to see each other, regardless of their initial positions. The polygon is initially unknown to the searchers, and its edges obstruct both movement and vision. Depending on the shape of P, we minimize the number of searchers k for which the Meeting problem is solvable. Specifically, if P has a rotational symmetry of order sigma (where sigma=1 corresponds to no rotational symmetry), we prove that k=sigma+1 searchers are sufficient, and the bound is tight. Furthermore, we give an improved algorithm that optimally solves the Meeting problem with k=2 searchers in all polygons whose barycenter is not in a hole (which includes the polygons with no holes). Our algorithms can be implemented in a variety of standard models of mobile robots operating in LookComputeMove cycles. For instance, if the searchers have memory but are anonymous, asynchronous, and have no agreement on a coordinate system or a notion of clockwise direction, then our algorithms work even if the initial memory contents of the searchers are arbitrary and possibly misleading. Moreover, oblivious searchers can execute our algorithms as well, encoding information by carefully positioning themselves within the polygon. This code is computable with basic arithmetic operations (provided that the coordinates of the polygon's vertices are algebraic real numbers in some global coordinate system), and each searcher can geometrically construct its own destination point at each cycle using only a compass. We stress that such memoryless searchers may be located anywhere in the polygon when the execution begins, and hence the information they initially encode is arbitrary. Our algorithms use a selfstabilizing map construction subroutine which is of independent interest.
BibTeX  Entry
@InProceedings{diluna_et_al:LIPIcs:2017:7983,
author = {Giuseppe A. Di Luna and Paola Flocchini and Nicola Santoro and Giovanni Viglietta and Masafumi Yamashita},
title = {{Meeting in a Polygon by Anonymous Oblivious Robots}},
booktitle = {31st International Symposium on Distributed Computing (DISC 2017)},
pages = {14:114:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770538},
ISSN = {18688969},
year = {2017},
volume = {91},
editor = {Andr{\'e}a W. Richa},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2017/7983},
URN = {urn:nbn:de:0030drops79833},
doi = {10.4230/LIPIcs.DISC.2017.14},
annote = {Keywords: Meeting problem, Oblivious robots, Polygon, Selfstabilization}
}
Keywords: 

Meeting problem, Oblivious robots, Polygon, Selfstabilization 
Collection: 

31st International Symposium on Distributed Computing (DISC 2017) 
Issue Date: 

2017 
Date of publication: 

12.10.2017 