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.
Exact coverings of @math points in the plane with two Euclidean disks of minimum maximal radius, commonly referred to as the problem, has been heavily studied. The best deterministic algorithm runs in @math time @cite_37 and in @cite_6 an expected @math time algorithm is presented. For polygons, in @cite_39 a @math time algorithm for covering a convex polygon with two Euclidean disks of minimum maximal radius is presented.
@cite_37 @cite_6 @cite_39
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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. Abstract This paper considers the planar Euclidean two-center problem: given a planar n-point set S, find two congruent circular disks of the smallest radius covering S. The main result is a deterministic algorithm with running time O(nlog2nlog2logn), improving the previous O(nlog9n) bound of Sharir and almost matching the randomized O(nlog2n) bound of Eppstein. If a point in the intersection of the two disks is given, then we can solve the problem in O(nlogn) time with high probability. Given a set @math of m unit disks and a set @math of n points in the plane, the discrete unit disk cover problem is to select a minimum cardinality subset @math to cover @math . This problem is NP-hard [14] and the best previous practical solution is a 38-approximation algorithm by [5]. We first consider the line-separable discrete unit disk cover problem (the set of disk centers can be separated from the set of points by a line) for which we present an O(n(log n + m))-time algorithm that finds an exact solution. Combining our line-separable algorithm with techniques from the algorithm of [5] results in an O(m2n4) time 22-approximate solution to the discrete unit disk cover problem. For a set Pof npoints in i¾?2, the Euclidean 2-center problem computes a pair of congruent disks of the minimal radius that cover P. We extend this to the (2,k)-center problem where we compute the minimal radius pair of congruent disks to cover ni¾? kpoints of P. We present a randomized algorithm with O(nk7log3n) expected running time for the (2,k)-center problem. We also study the (p,k)-center problem in i¾?2under the i¾? i¾? -metric. We give solutions for p= 4 in O(kO(1)nlogn) time and for p= 5 in O(kO(1)nlog5n) time.