Meeting in a polygon by anonymous oblivious robots

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.

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Notes

  1. 1.

    The typical assumption in this model is that a searcher’s local reference frame retains its orientation, scale, and handedness after each move. We will make this assumption as well, although it is not strictly needed by our algorithms (see the footnote in Sect. 2).

  2. 2.

    In Sect. 3 we will drop this assumption in order to give a cleaner exposition of our algorithms. In Sect. 4 we will restore the assumption and show how to extend our algorithms to oblivious searchers.

  3. 3.

    For this reason, there is no distributed algorithm that, for every polygon P, allows a team of memoryless searchers to either solve the Meeting problem in P or terminate if the problem is unsolvable in P.

  4. 4.

    A real number is said to be algebraic if it is a root of a polynomial with integer coefficients [8].

  5. 5.

    Collectively, these polygons constitute a subset of measure 0 of the set of all polygons with holes.

  6. 6.

    Here and throughout the paper, we will refer to the barycenter only because it is a well-defined point in every polygon. However, this choice is not essential: equivalently, we could take the center of symmetry if the polygon has one, or any point otherwise.

  7. 7.

    The fact that a searcher retains its local reference frame’s orientation, scale, and handedness is common in most of the related literature. However, in this work we will not strictly need it: in our algorithms, a searcher will always move (close) to a vertex of P. Hence, after a move, it will always be able to correctly match its new view with the previous one. This is possible even if its reference frame has reflected after the move: it is sufficient to make the searcher stop close enough to the angle bisector stemming from the destination vertex, but not exactly on it. On the next turn, the searcher will be able to tell its new reference frame’s handedness based on whether it is located to the left or the right of the angle bisector (cf. Sect. 4.2).

  8. 8.

    As an example, we show how to do it when P has no holes. Extending this method to the general case is just slightly more complicated, but the principles are the same. Pick the (unique) circle of smallest radius that contains all the vertices of P, and let r be its radius. Name the vertices of P \(v_0\), \(v_1\), ..., \(v_{n-1}\) in clockwise order. Pick any vertex \(v_i\), and construct the right-handed coordinate system having origin in \(v_i\), unit r, and x axis oriented like \(\overrightarrow{v_iv_{i+1}}\) (indices are always taken modulo n). Give a representation of P in this coordinate system, i.e., the ordered list of the x and y coordinates of \(v_{i+1}\), ..., \(v_n\), \(v_1\), ..., \(v_{i-1}\). Then construct another representation in the same coordinate system, but taking the vertices in the reverse order, i.e., \(v_{i-1}\), ..., \(v_1\), \(v_n\), ..., \(v_{i+1}\). Pick the lexicographically smaller of these two representations (if they are equal, pick any of them), and call it \(R_i\). Repeating the same construction with all the \(v_i\)’s yields the representations \(R_0\), \(R_1\), ..., \(R_{n-1}\): let \(R_m\) be the lexicographically smallest among them. Now, pick all vertices \(v_i\) such that \(R_i=R_m\): these constitute a rotation class of P chosen in a similarity-invariant way. Indeed, no matter how we rotate, translate, uniformly scale by a non-zero factor, or reflect P, we will always pick the same set of vertices.

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Acknowledgements

The authors wish to thank Francesco Veneziano for a clarifying discussion. This research has been supported in part by the Natural Sciences and Engineering Research Council of Canada under the Discovery Grant program and by Prof. Flocchini’s University Research Chair.

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Correspondence to Giovanni Viglietta.

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A preliminary version of this paper has appeared at the 31st International Symposium on Distributed Computing (DISC’17) [18].

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Di Luna, G.A., Flocchini, P., Santoro, N. et al. Meeting in a polygon by anonymous oblivious robots. Distrib. Comput. 33, 445–469 (2020). https://doi.org/10.1007/s00446-019-00362-2

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