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The Meeting problem for $k\geq 2$ searchers in a polygon $P$ (possibly withholes) consists in making the searchers move within $P$, according to adistributed algorithm, in such a way that at least two of them eventually cometo see each other, regardless of their initial positions. The polygon isinitially unknown to the searchers, and its edges obstruct both movement andvision. Depending on the shape of $P$, we minimize the number of searchers $k$for which the Meeting problem is solvable. Specifically, if $P$ has arotational symmetry of order $\sigma$ (where $\sigma=1$ corresponds to norotational symmetry), we prove that $k=\sigma+1$ searchers are sufficient, andthe bound is tight. Furthermore, we give an improved algorithm that optimallysolves the Meeting problem with $k=2$ searchers in all polygons whosebarycenter is not in a hole (which includes the polygons with no holes). Ouralgorithms can be implemented in a variety of standard models of mobile robotsoperating in Look-Compute-Move cycles. For instance, if the searchers havememory but are anonymous, asynchronous, and have no agreement on a coordinatesystem or a notion of clockwise direction, then our algorithms work even if theinitial memory contents of the searchers are arbitrary and possibly misleading.Moreover, oblivious searchers can execute our algorithms as well, encodinginformation by carefully positioning themselves within the polygon. This codeis computable with basic arithmetic operations, and each searcher cangeometrically construct its own destination point at each cycle using only acompass. We stress that such memoryless searchers may be located anywhere inthe polygon when the execution begins, and hence the information they initiallyencode is arbitrary. Our algorithms use a self-stabilizing map constructionsubroutine which is of independent interest.

Meeting in a Polygon by Anonymous Oblivious Robots