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Translating a planar object to maximize point containment
 In Proc. 10th Annu. European Sympos. Algorithms, Lecture Notes Comput. Sci
, 2002
"... Abstract. Let C be a compact set in R ..."
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OffsetPolygon Annulus Placement Problems
, 1997
"... In this paper we address several variants of the polygon annulus placement problem: given an input polygon P and a set S of points, find an optimal placement of P that maximizes the number of points in S that fall in a certain annulus region defined by P and some offset distance ffi ? 0. We address ..."
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In this paper we address several variants of the polygon annulus placement problem: given an input polygon P and a set S of points, find an optimal placement of P that maximizes the number of points in S that fall in a certain annulus region defined by P and some offset distance ffi ? 0. We address the following variants of the problem: placement of a convex polygon as well as a simple polygon; placement by translation only, or by a translation and a rotation; offline and online versions of the corresponding decision problems; and decision as well as optimization versions of the problems. We present efficient algorithms in each case.
Optimal Placement of Convex Polygons to Maximize Point Containment
, 1996
"... Given a convex polygon P with m vertices and a set S of n points in the plane, we consider the problem of finding a placement of P that contains the maximum number of points in S. We allow both translation and rotation. Our algorithm is selfcontained and utilizes the geometric properties of the co ..."
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Cited by 8 (1 self)
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Given a convex polygon P with m vertices and a set S of n points in the plane, we consider the problem of finding a placement of P that contains the maximum number of points in S. We allow both translation and rotation. Our algorithm is selfcontained and utilizes the geometric properties of the containing regions in the parameter space of transformations. The algorithm requires O(nk 2 m 2 log(mk)) time and O(n +m) space, where k is the maximum number of points contained. This provides a linear improvement over the best previously known algorithm when k is large (\Theta(n)) and a cubic improvement when k is small. We also show that the algorithm can be extended to solve bichromatic and general weighted variants of the problem. 1 Introduction A planar rigid motion ae is an affine transformation of the plane that preserves distance (and therefore angles and area also). We say that a polygon P contains a set S of points if every point in S lies on P or in the interior of P . In th...
FORMULATIONS AND EXACT SOLUTION METHODS FOR A CLASS OF NEW CONTINUOUS COVERING PROBLEMS
, 2009
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Discretization of Planar Geometric Cover Problems
, 2014
"... We consider discretization of the ‘geometric cover problem’ in the plane: Given a set P of n points in the plane and a compact planar object T0, find a minimum cardinality collection of planar translates of T0 such that the union of the translates in the collection contains all the points in P. We s ..."
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We consider discretization of the ‘geometric cover problem’ in the plane: Given a set P of n points in the plane and a compact planar object T0, find a minimum cardinality collection of planar translates of T0 such that the union of the translates in the collection contains all the points in P. We show that the geometric cover problem can be converted to a form of the geometric set cover, which has a given finitesize 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/nonconvex polygons (including holes), and planar objects consisting of finite Jordan curves.
The TranslationScaleRotation Diagram for PointContaining Placements of a Convex Polygon ∗
"... We present a diagram that captures containment information for scalable rotated and translated versions of a convex polygon. For a given polygon P and a contact point q in a point set S, the diagram parameterizes possible translations, rotations, and scales of the polygon in order to represent conta ..."
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We present a diagram that captures containment information for scalable rotated and translated versions of a convex polygon. For a given polygon P and a contact point q in a point set S, the diagram parameterizes possible translations, rotations, and scales of the polygon in order to represent containment regions for each additional point v in S. We present geometric and combinatorial properties for this diagram, and describe how it can be computed and used in the solution of several geometric problems. 1
Annulus Placement Problems for Scaled and Offset Polygons (Extended Abstract)
"... A polygon annulus is the region contained in some outer polygon but not contained in an inner polygon. Typically, the outer and inner polygons are concentric scaled or offset copies of the same polygon. Offset polygons and offset polygon annulus placement problems have recently been studied in [BDG] ..."
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A polygon annulus is the region contained in some outer polygon but not contained in an inner polygon. Typically, the outer and inner polygons are concentric scaled or offset copies of the same polygon. Offset polygons and offset polygon annulus placement problems have recently been studied in [BDG] and [BBDG]. In this paper we study two related annulus placement problems: the first variant fixes the outer containing polygon and aims to maximize the inner polygon; the second variant fixes the inner polygon and aims to minimize the outer polygon.
Maximal Covering by Two Isothetic Unit Squares
"... Let P be the point set in two dimensional plane. this paper, we consider the problem of locating two isothetic unit squares such that together they cover maximum number of points of P. In case of overlapping, the points in their common zone are counted once. To solve the problem, we propose an algor ..."
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Let P be the point set in two dimensional plane. this paper, we consider the problem of locating two isothetic unit squares such that together they cover maximum number of points of P. In case of overlapping, the points in their common zone are counted once. To solve the problem, we propose an algorithm that runs in O(n 2 log 2 n) time using O(n log n) space. 1