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.
If searchers are unable to move, then the Meeting problem becomes equivalent to asking what is the maximum number of points that can be placed in a polygon in such a way that no two of them see each other. This is called the , and has been studied in @cite_23 , where tight bounds are given in terms of the number of vertices of the polygon. The situation with mobile searchers is of course radically different.
@cite_23
2962739409 1982770663 32433637 2033649214
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 ( ) (where ( =1 ) corresponds to no rotational symmetry), we prove that (k= +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. The problem of searching for mobile intruders in a polygonal region by mobile searchers is considered. A searcher can move continuously inside a polygon holding a flashlight that emits a single ray of light whose direction can be changed continuously. The vision of a searcher at any time instant is limited to the points on the ray. The intruders can move continuously with unbounded speed. We denote by ps(P) the polygon search number of a simple polygon P , which is the number of searchers necessary and sufficient to search P . Let n , r , b , and g be the number of edges, the number of reflex vertices, the bushiness, and the size of a minimum guard set of P , respectively. In this paper we present matching upper and (worst case) lower bounds of 1 + log 3 (2b+1) on ps(P) . Also upper bounds on ps(P) in terms of n,r , and g are presented;ps(P) ≤ 1 + log 3 (n-3) ,ps(P) ≤ 1 + log 3 r , and ps(P) ≤ 2 + log 2 g . These upper bounds are tight or almost tight in the worst case, since we show that for any natural number s 2 , there is a polygon P such that ps(P) = log 3 (n+1) = log 3 (2r+3) = 1 + log 3 (2g-1) = s . In this paper we consider the group search problem, or evacu- ation problem, in which k mobile entities ( ( M E )s) located on the line perform search for a specific destination. The ( M E )s are initially placed at the same origin on the line L and the target is located at an unknown distance d, either to the left or to the right from the origin. All ( M E )s must simultaneously occupy the destination, and the goal is to minimize the time necessary for this to happen. The problem with k = 1 is known as the cow-path problem, and the time required for this problem is known to be 9d − o(d) in the worst case (when the cow moves at unit speed); it is also known that this is the case for k ≥ 1 unit-speed ( M E )s. In this paper we present a clear argument for this claim by showing a rather counter-intuitive result. Namely, independent of the number of ( M E )s, group search cannot be performed faster than in time 9d − o(d). We also examine the case of k = 2 ( M E )s with different speeds, showing a surprising result that the bound of 9d can be achieved when one ( M E ) has unit speed, and the other ( M E ) moves with speed at least 1 3. The problem of searching for a mobile intruder in a simple polygon by a single mobile searcher is considered. This paper investigates the capabilities of searchers having different degrees of visibility by introducing the searcher having k flashlights whose visibility is limited to k rays emanating from his position, and the searcher having a point light source who can see in all directions simultaneously. This paper presents necessary and sufficient conditions for a polygon to be searchable by various searchers. The paper also introduces a class of polygons for which the searcher having two flashlights is as capable as the searcher having a point light source, and it gives a simple necessary and sufficient condition for such polygons to be searchable by the searcher having two flashlights. The complexity of generating a search schedule under some of these conditions is also discussed. Many of the results are proved using chord systems that represent the visibility relations among the vertices and edges o...