The Meeting problem for @math searchers in a polygon @math (possibly with holes) consists in making the searchers move within @math , 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 @math , we minimize the number of searchers @math for which the Meeting problem is solvable. Specifically, if @math has a rotational symmetry of order @math (where @math corresponds to no rotational symmetry), we prove that @math searchers are sufficient, and the bound is tight. Furthermore, we give an improved algorithm that optimally solves the Meeting problem with @math 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 Look-Compute-Move 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, 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 self-stabilizing map construction subroutine which is of independent interest.
@cite_23: A hidden set is a set of points such that no two points in the set are visible to each other. A hidden guard set is a hidden set which is also a guard set. In this paper we consider the problem for finding hidden sets and hidden guard sets in and around polygons. In particular, we establish bounds on the maximum size of hidden sets, and show that the problem of finding a maximum hidden set is NP-hard. For minimum hidden guard sets, similar results are obtained.
If searchers are unable to move, then the Meeting problem becomes equivalent to asking what is the maximum number of points that can be placed in a polygon in such a way that no two of them see each other. This is called the , and has been studied in @cite_23 , where tight bounds are given in terms of the number of vertices of the polygon. The situation with mobile searchers is of course radically different.