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_6: Consider a set of @math identical mobile computational entities in the plane, called robots, operating in Look-Compute-Move cycles, without any means of direct communication. The Gathering Problem is the primitive task of all entities gathering in finite time at a point not fixed in advance, without any external control. The problem has been extensively studied in the literature under a variety of strong assumptions (e.g., synchronicity of the cycles, instantaneous movements, complete memory of the past, common coordinate system, etc.). In this paper we consider the setting without those assumptions, that is, when the entities are oblivious (i.e., they do not remember results and observations from previous cycles), disoriented (i.e., have no common coordinate system), and fully asynchronous (i.e., no assumptions exist on timing of cycles and activities within a cycle). The existing algorithmic contributions for such robots are limited to solutions for @math or for restricted sets of initial configura...
Here we focus on Gathering in geometric spaces, since it is related to our setting. In the Look-Compute-Move model, asynchronous oblivious robots with unlimited visibility can solve the Gathering problem without additional assumptions @cite_6 .