Abstract

The Meeting problem for $$k\ge 2$$ 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 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 (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 and a straightedge. 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.

Highlights

  • Meeting problem Consider a set of k ≥ 2 autonomous mobile robots, modeled as geometric points located in a polygonal enclosure P, which may contain holes

  • In this paper we study the Meeting problem, which prescribes the k robots to move in such a way that eventually at least two of them come to see each other and become “mutually aware”

  • Statement of results We prove that the Meeting problem in a polygon P can be solved by k = σ + 1 searchers, where σ is the order of the rotation group of P

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Summary

Introduction

Meeting problem Consider a set of k ≥ 2 autonomous mobile robots, modeled as geometric points located in a polygonal enclosure P, which may contain holes. Each robot observes the visible portion of P (taking an instantaneous snapshot of it), executes an algorithm to compute a visible destination point, and moves to that point. Such a Look–Compute–Move cycle is repeated forever by every robot, each time taking a Searchers’ limitations Our searchers are severely limited, which makes the Meeting problem harder to solve. – They are asynchronous, in the sense that we make no assumptions on how fast each searcher completes a Look–Compute–Move cycle compared to the others These parameters are dynamic and are controlled by an adversarial scheduler

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