We consider discretization of the 'geometric cover problem' in the plane: Given a set @math of @math points in the plane and a compact planar object @math , find a minimum cardinality collection of planar translates of @math such that the union of the translates in the collection contains all the points in @math . We show that the geometric cover problem can be converted to a form of the geometric set cover, which has a given finite-size collection of translates rather than the infinite continuous solution space of the former. We propose a reduced finite solution space that consists of distinct canonical translates and present polynomial algorithms to find the reduce solution space for disks, convex non-convex polygons (including holes), and planar objects consisting of finite Jordan curves.
Given a set @math of @math points in the plane, the optimal placement problem is to compute a translate that contains the maximum number of points in @math ; note that in our problem, the distinct canonical translates contain the points of @math in a locally maximal manner (see Definition ). There are some optimal placement algorithms for basic planar objects: Chazelle and Lee @cite_2 presented @math time algorithm for disks, and Eppstein and Erickson @cite_27 proposed @math time algorithm for rectangles; for convex polygons of @math vertices, @cite_15 devised @math time algorithm, and @cite_17 improved the time complexity to @math , where @math is the maximum number of points in @math that can be contained in a translate.
@cite_27 @cite_15 @cite_17 @cite_2
2007046733 2045156582 77353308
Multiple objects in the plane are called pseudodisks if the boundaries of any two of them intersect transversely at most twice. Given a set of @math (possibly intersecting) convex pseudodisks of @math complexity each and two points @math and @math in the plane, we present an efficient algorithm for computing a shortest @math -to- @math path avoiding the pseudodisks. After the union of the pseudodisks is computed, which can be done in @math randomized time or @math deterministic time, our algorithm runs in @math deterministic time, where @math is the size of the extended visibility graph of the union of the pseudodisks. Note that @math in the worst case. In over two decades, the previously best algorithms for this problem have not improved on the bound of @math time, even when all the pseudodisks are pairwise disjoint disks. Our technique is also applicable to a motion planning problem of finding a shortest path to translate a convex object in the plane from one location to anot... 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. Motivated by a frequency assignment problem in cellular networks, we introduce and study a new coloring problem that we call minimum conflict-free coloring (min-CF-coloring). In its general form, the input of the min-CF-coloring problem is a set system @math , where each @math is a subset of X. The output is a coloring @math of the sets in @math that satisfies the following constraint: for every @math there exists a color @math and a unique set @math such that @math and @math . The goal is to minimize the number of colors used by the coloring @math . Min-CF-coloring of general set systems is not easier than the classic graph coloring problem. However, in view of our motivation, we consider set systems induced by simple geometric regions in the plane. In particular, we study disks (both congruent and noncongruent), axis-parallel rectangles (with a constant ratio between the smallest and largest rectangle), regular hexagons (with a constant ratio between the smallest and largest hexagon), and general congruent centrally symmetric convex regions in the plane. In all cases we have coloring algorithms that use O(log n) colors (where n is the number of regions). Tightness is demonstrated by showing that even in the case of unit disks, @math colors may be necessary. For rectangles and hexagons we also obtain a constant-ratio approximation algorithm when the ratio between the largest and smallest rectangle (hexagon) is a constant. We also consider a dual problem of CF-coloring points with respect to sets. Given a set system @math , the goal in the dual problem is to color the elements in X with a minimum number of colors so that every set @math contains a point whose color appears only once in We investigate the minimization of Newton's functional for the problem of the body of minimal resistance with maximal height @math cite butt in the class of convex developable functions defined in a disc. This class is a natural candidate to find a (non-radial) minimizer in accordance with the results of cite lrp2. We prove that the minimizer in this class has a minimal set in the form of a regular polygon with @math sides centered in the disc, and numerical experiments indicate that the natural number @math is a non-decreasing function of @math . The corresponding functions all achieve a lower value of the functional than the optimal radially symmetric function with the same height @math .