Meeting in a Polygon by Anonymous Oblivious Robots

The Meeting problem for $k\geq 2$ 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, 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.Comment: 37 pages, 9 figure

Positional Encoding by Robots with Non-Rigid Movements

Consider a set of autonomous computational entities, called \emph{robots}, operating inside a polygonal enclosure (possibly with holes), that have to perform some collaborative tasks. The boundary of the polygon obstructs both visibility and mobility of a robot. Since the polygon is initially unknown to the robots, the natural approach is to first explore and construct a map of the polygon. For this, the robots need an unlimited amount of persistent memory to store the snapshots taken from different points inside the polygon. However, it has been shown by Di Luna et al. [DISC 2017] that map construction can be done even by oblivious robots by employing a positional encoding strategy where a robot carefully positions itself inside the polygon to encode information in the binary representation of its distance from the closest polygon vertex. Of course, to execute this strategy, it is crucial for the robots to make accurate movements. In this paper, we address the question whether this technique can be implemented even when the movements of the robots are unpredictable in the sense that the robot can be stopped by the adversary during its movement before reaching its destination. However, there exists a constant $\delta > 0$, unknown to the robot, such that the robot can always reach its destination if it has to move by no more than $\delta$ amount. This model is known in literature as \emph{non-rigid} movement. We give a partial answer to the question in the affirmative by presenting a map construction algorithm for robots with non-rigid movement, but having $O(1)$ bits of persistent memory and ability to make circular moves

Representing convex geometries by almost-circles

Finite convex geometries are combinatorial structures. It follows from a recent result of M.\ Richter and L.G.\ Rogers that there is an infinite set $T_{rr}$ of planar convex polygons such that $T_{rr}$ with respect to geometric convex hulls is a locally convex geometry and every finite convex geometry can be represented by restricting the structure of $T_{rr}$ to a finite subset in a natural way. An \emph{almost-circle of accuracy} $1-\epsilon$ is a differentiable convex simple closed curve $S$ in the plane having an inscribed circle of radius $r_1>0$ and a circumscribed circle of radius $r_2$ such that the ratio $r_1/r_2$ is at least $1-\epsilon$. % Motivated by Richter and Rogers' result, we construct a set $T_{new}$ such that (1) $T_{new}$ contains all points of the plane as degenerate singleton circles and all of its non-singleton members are differentiable convex simple closed planar curves; (2) $T_{new}$ with respect to the geometric convex hull operator is a locally convex geometry; (3) as opposed to $T_{rr}$, $T_{new}$ is closed with respect to non-degenerate affine transformations; and (4) for every (small) positive $\epsilon\in\real$ and for every finite convex geometry, there are continuum many pairwise affine-disjoint finite subsets $E$ of $T_{new}$ such that each $E$ consists of almost-circles of accuracy $1-\epsilon$ and the convex geometry in question is represented by restricting the convex hull operator to $E$. The affine-disjointness of $E_1$ and $E_2$ means that, in addition to $E_1\cap E_2=\emptyset$, even $\psi(E_1)$ is disjoint from $E_2$ for every non-degenerate affine transformation $\psi$.Comment: 18 pages, 6 figure