Meeting in a polygon by anonymous oblivious robots
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Meeting in a polygon by anonymous oblivious robots Giuseppe Antonio Di Luna1 · Paola Flocchini2 · Nicola Santoro3 · Giovanni Viglietta4 · Masafumi Yamashita5 Received: 5 February 2018 / Accepted: 8 September 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
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 σ (where σ = 1 corresponds to no rotational symmetry), we prove that k = σ + 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.
1 Introduction A preliminary version of this paper has appeared at the 31st International Symposium on Distributed Computing (DISC’17) [18].
1.1 Framework
B
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. The boundary of P limits both visibility and mobility, in that robots cannot move or see through the edges of P. Each robot observes the visible portion of P (taking an instantaneous snapshot of it), executes an algorithm to compute a visible destination point, and then moves to that point. Such a Look–Compute–Move cycle is repeated forever by every robot, each time taking a
Giovanni Viglietta [email protected] Giuseppe Antonio Di Luna [email protected] Paola Flocchin
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