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_3 , they investigate upper and lower bounds on the movements of two robots solving the Rendezvous problem in a polygon. The authors give a wealth of results under different assumptions, but a common handedness (i.e., a common notion of clockwise direction) and unlimited persistent memory are always assumed.
@cite_3
2048209083 1766799846 2040570464 2187411536
We consider the problem of mapping an initially unknown polygon of size n with a simple robot that moves inside the polygon along straight lines between the vertices. The robot sees distant vertices in counter-clockwise order and is able to recognize the vertex among them which it came from in its last move, i.e. the robot can look back. Other than that the robot has no means of distinguishing distant vertices. We assume that an upper bound on n is known to the robot beforehand and show that it can always uniquely reconstruct the visibility graph of the polygon. Additionally, we show that multiple identical and deterministic robots can always solve the weak rendezvous problem in which the robots need to position themselves such that all of them are mutually visible to each other. Assume that two robots are located at the centre of a unit disk. Their goal is to evacuate from the disk through an exit at an unknown location on the boundary of the disk. At any time the robots can move anywhere they choose on the disk, independently of each other, with maximum speed @math . The robots can cooperate by exchanging information whenever they meet. We study algorithms for the two robots to minimize the evacuation time: the time when both robots reach the exit. In [CGGKMP14] the authors gave an algorithm defining trajectories for the two robots yielding evacuation time at most @math and also proved that any algorithm has evacuation time at least @math . We improve both the upper and lower bounds on the evacuation time of a unit disk. Namely, we present a new non-trivial algorithm whose evacuation time is at most @math and show that any algorithm has evacuation time at least @math . To achieve the upper bound, we designed an algorithm which non-intuitively proposes a forced meeting between the two robots, even if the exit has not been found by either of them. The author considers the problem faced by two people who are placed randomly in a known search region and move about at unit speed to find each other in the least expected time. This time is called the rendezvous value of the region. It is shown how symmetries in the search region may hinder the process by preventing coordination based on concepts such as north or clockwise. A general formulation of the rendezvous search problem is given for a compact metric space endowed with a group of isometrics which represents the spatial uncertainties of the players. These concepts are illustrated by considering upper bounds for various rendezvous values for the circle and an arbitrary metric network. The discrete rendezvous problem on a cycle graph for players restricted to symmetric Markovian strategies is then solved. Finally, the author considers the problem faced by two people on an infinite line who each know the distribution of the distance but not the direction to each other. Assume that two robots are located at the centre of a unit disk. Their goal is to evacuate from the disk through an exit at an unknown location on the boundary of the disk. At any time the robots can move anywhere they choose on the disk, independently of each other, with maximum speed @math . The robots can cooperate by exchanging information whenever they meet. We study algorithms for the two robots to minimize the evacuation time: the time when both robots reach the exit. In [9] the authors gave an algorithm defining trajectories for the two robots yielding evacuation time at most @math and also proved that any algorithm has evacuation time at least @math . We improve both the upper and lower bounds on the evacuation time of a unit disk. Namely, we present a new non-trivial algorithm whose evacuation time is at most @math and show that any algorithm has evacuation time at least @math . To achieve the upper bound, we designed an algorithm which non-intuitively proposes a forced meeting between the two robots, even if the exit has not been found by either of them.