Let D be a Jordan domain in the plane. We consider a pursuit-evasion, contamination clearing, or sensor sweep problem in which the pursuer at each point in time is modeled by a continuous curve, called the sensor curve. Both time and space are continuous, and the intruders are invisible to the pursuer. Given D, what is the shortest length of a sensor curve necessary to provide a sweep of domain D, so that no continuously-moving intruder in D can avoid being hit by the curve? We define this length to be the sweeping cost of D. We provide an analytic formula for the sweeping cost of any Jordan domain in terms of the geodesic Frechet distance between two curves on the boundary of D with non-equal winding numbers. As a consequence, we show that the sweeping cost of any convex domain is equal to its width, and that a convex domain of unit area with maximal sweeping cost is the equilateral triangle.
The paper @cite_2 introduces the between two polylines, a notion that is very relevant for our problem. The geodesic width between two curves @math , @math is the same as the strong geodesic Fr ' e chet distance between them (Definition ) when domain @math is chosen to be a region with boundary consisting of curves @math , @math , and the two shortest paths connecting the endpoints of @math and @math . If curves @math and @math are polylines with @math vertices in total, then @cite_2 gives an @math algorithm for computing the geodesic width between them. The goal of our paper is instead to rigorously prove an analytic formula for the sweeping cost of a domain, Theorem , that is closely related to geodesic widths. Indeed, the right hand side of in Theorem is unchanged if we replace the geodesic distance between two curves (Definition ) with the weak geodesic Fr ' e chet distance between them (Definition ). The paper @cite_2 also studies sweeps of planar domains by piecewise linear curves in which the cost of a sweep is not equal to a length, but instead to the number of vertices or joints in the curve.
@cite_2
2062556295 2407537893 2162114735 2607925462
We introduce two new related metrics, the geodesic width and the link width, for measuring the "distance" between two nonintersecting polylines in the plane. If the two polylines have n vertices in total, we present algorithms to compute the geodesic width of the two polylines in O(n 2 log n) time using O(n 2) space and the link width in O(n 3 log n) time using O(n 2) working space where n is the total number of edges of the polylines. Our computation of these metrics relies on two closely related combinatorial strutures: the shortest-path diagram and the link diagram of a simple polygon. The shortest-path (resp., link) diagram encodes the Euclidean (resp., link) shortest path distance between all pairs of points on the boundary of the polygon. We use these algorithms to solve two problems: Given a polygon @math , for two points @math and @math contained in the polygon, their is the length of the shortest @math -path within @math . A of radius @math centered at a point @math is the set of points in @math whose geodesic distance to @math is at most @math . We present a polynomial time @math -approximation algorithm for finding a densest geodesic unit disk packing in @math . Allowing arbitrary radii but constraining the number of disks to be @math , we present a @math -approximation algorithm for finding a packing in @math with @math geodesic disks whose minimum radius is maximized. We then turn our focus on of @math and present a @math -approximation algorithm for covering @math with @math geodesic disks whose maximal radius is minimized. Furthermore, we show that all these problems are @math -hard in polygons with holes. Lastly, we present a polynomial time exact algorithm which covers a polygon with two geodesic disks of minimum maximal radius. We present a heuristic algorithm to compute approximate geodesic distances on a triangular manifold S containing n vertices with partially missing data. The proposed method computes an approximation of the geodesic distance between two vertices pi and pj on S and provides an upper bound of the geodesic distance that is shown to be optimal in the worst case. This yields a relative error bound of the estimate that is worst-case optimal. The algorithm approximates the geodesic distance without trying to reconstruct the missing data by embedding the surface in a low dimensional space via multi-dimensional scaling (MDS). We derive a new heuristic method to add an object to the embedding computed via least-squares MDS. Coalescence of semi-infinite geodesics remains a central question in planar first passage percolation. In this paper we study finer properties of the coalescence structure of finite and semi-infinite geodesics for exactly solvable models of last passage percolation. Consider directed last passage percolation on @math with i.i.d. exponential weights on the vertices. Fix two points @math and @math for some @math , and consider the maximal paths @math and @math starting at @math and @math respectively to the point @math for @math . Our object of study is the point of coalescence, i.e., the point @math with smallest @math . We establish that the distance to coalescence @math scales as @math , by showing the upper tail bound @math for some @math . We also consider the problem of coalescence for semi-infinite geodesics. For the almost surely unique semi-infinite geodesics in the direction @math starting from @math and @math , we establish the optimal tail estimate @math , for the point of coalescence @math . This answers a question left open by Pimentel (Ann. Probab., 2016) who proved the corresponding lower bound.