Results 1  10
of
14
DavenportSchinzel Sequences and Their Geometric Applications
, 1998
"... An (n; s) DavenportSchinzel sequence, for positive integers n and s, is a sequence composed of n distinct symbols with the properties that no two adjacent elements are equal, and that it does not contain, as a (possibly noncontiguous) subsequence, any alternation a \Delta \Delta \Delta b \Delta \ ..."
Abstract

Cited by 425 (121 self)
 Add to MetaCart
An (n; s) DavenportSchinzel sequence, for positive integers n and s, is a sequence composed of n distinct symbols with the properties that no two adjacent elements are equal, and that it does not contain, as a (possibly noncontiguous) subsequence, any alternation a \Delta \Delta \Delta b \Delta \Delta \Delta a \Delta \Delta \Delta b \Delta \Delta \Delta of length s + 2 between two distinct symbols a and b. The close relationship between DavenportSchinzel sequences and the combinatorial structure of lower envelopes of collections of functions make the sequences very attractive because a variety of geometric problems can be formulated in terms of lower envelopes. A nearlinear bound on the maximum length of DavenportSchinzel sequences enable us to derive sharp bounds on the combinatorial structure underlying various geometric problems, which in turn yields efficient algorithms for these problems.
Iterated Nearest Neighbors and Finding Minimal Polytopes
, 1994
"... Weintroduce a new method for finding several types of optimal kpoint sets, minimizing perimeter, diameter, circumradius, and related measures, by testing sets of the O(k) nearest neighbors to each point. We argue that this is better in a number of ways than previous algorithms, whichwere based o ..."
Abstract

Cited by 56 (6 self)
 Add to MetaCart
Weintroduce a new method for finding several types of optimal kpoint sets, minimizing perimeter, diameter, circumradius, and related measures, by testing sets of the O(k) nearest neighbors to each point. We argue that this is better in a number of ways than previous algorithms, whichwere based on high order Voronoi diagrams. Our technique allows us for the first time to efficiently maintain minimal sets as new points are inserted, to generalize our algorithms to higher dimensions, to find minimal convex kvertex polygons and polytopes, and to improvemany previous results. Weachievemany of our results via a new algorithm for finding rectilinear nearest neighbors in the plane in time O(n log n+kn). We also demonstrate a related technique for finding minimum area kpoint sets in the plane, based on testing sets of nearest vertical neighbors to each line segment determined by a pair of points. A generalization of this technique also allows us to find minimum volume and boundary measure sets in arbitrary dimensions.
Computational geometry  a survey
 IEEE TRANSACTIONS ON COMPUTERS
, 1984
"... We survey the state of the art of computational geometry, a discipline that deals with the complexity of geometric problems within the framework of the analysis ofalgorithms. This newly emerged area of activities has found numerous applications in various other disciplines, such as computeraided de ..."
Abstract

Cited by 19 (3 self)
 Add to MetaCart
We survey the state of the art of computational geometry, a discipline that deals with the complexity of geometric problems within the framework of the analysis ofalgorithms. This newly emerged area of activities has found numerous applications in various other disciplines, such as computeraided design, computer graphics, operations research, pattern recognition, robotics, and statistics. Five major problem areasconvex hulls, intersections, searching, proximity, and combinatorial optimizationsare discussed. Seven algorithmic techniques incremental construction, planesweep, locus, divideandconquer, geometric transformation, pruneandsearch, and dynamizationare each illustrated with an example.Acollection of problem transformations to establish lower bounds for geometric problems in the algebraic computation/decision model is also included.
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 ..."
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 addre ..."
Abstract

Cited by 9 (3 self)
 Add to MetaCart
. 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. Keywords: optimal polygon placement, tolerancing, robot localization, offsetting. 1 Introduction 1.1 Background and Applications In this paper we address several variants of the problem of placing an annulus defined by a given polygon such that it covers all (or a maximum number of) points of a given set of points. This problem is motivated by seve...
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 ..."
Abstract

Cited by 6 (1 self)
 Add to MetaCart
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...
Finding planar regions in a terrain: In practice and with a guarantee
 In Proceedings of the Twentieth Annual Symposium on Computational Geometry, (SCG’04
, 2004
"... We consider the problem of computing large connected regions in a triangulated terrain of size n for which the normals of the triangles deviate by at most some small fixed angle. In previous work an exact nearquadratic algorithm was presented, but only a heuristic implementation with no guarantee w ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
We consider the problem of computing large connected regions in a triangulated terrain of size n for which the normals of the triangles deviate by at most some small fixed angle. In previous work an exact nearquadratic algorithm was presented, but only a heuristic implementation with no guarantee was practicable. We present a new approximation algorithm for the problem which runs in O(n/ǫ 2) time and—apart from giving a guarantee on the quality of the produced solution—has been implemented and shows good performance on real data sets representing fracture surfaces consisting of around half a million triangles. Further we present a simple approximation algorithm for a related problem: given a set of n points in the plane, determine the placement of the unit disk which contains most points. This algorithm runs in linear time as well.
Computing A DoubleRay Center For A Planar Point Set
, 1998
"... A doubleray configuration is a configuration in the plane consisting of two rays emanating from one point. Given a set S of n points in the plane, we want to find a doubleray configuration that minimizes the Hausdorff distance from S to this configuration. We call this problem the doubleray cente ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
A doubleray configuration is a configuration in the plane consisting of two rays emanating from one point. Given a set S of n points in the plane, we want to find a doubleray configuration that minimizes the Hausdorff distance from S to this configuration. We call this problem the doubleray center problem. We present an efficient algorithm for computing the doubleray center for set S of n points in the plane which runs in time O(n 3 ff(n) log 2 n). Keywords: Algorithm, Optimization, Center, Planar Point Set, Hausdorff Distance. 1. Introduction The doubleray center problem extends a list of the following well researched center problems: ffl The point center 16 : Find the center (and the radius) of the smallest disk enclosing a set of points S. ffl The two center problem 17;8 : Find centers of two disks, whose maximum radius is minimal, which covers the set S. ffl The line center problem 15 : Given a set of points S find the line that minimizes the distance between t...
Covering Point Sets with Two Convex Objects
, 2005
"... Let P2n be a point set in the plane with n red and n blue points. Let CR and CB (SR and SB) respectively be red and blue colored and disjoint disks (axisparallel squares). In this paper we prove the following results. Finding the positions for CR and CB that maximizes the number of red points covere ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
Let P2n be a point set in the plane with n red and n blue points. Let CR and CB (SR and SB) respectively be red and blue colored and disjoint disks (axisparallel squares). In this paper we prove the following results. Finding the positions for CR and CB that maximizes the number of red points covered by CR plus the number of blue points covered by CB can be done in O(n 3 log n) time. Finding two axisparallel unitsquares with disjoint interiors that maximizes the sum of the red points covered by SR plus the number of blue points covered by SB can be done in O(n²) time.
Covering point sets with two disjoint disks or squares
, 2007
"... We study the following problem: Given a set of red points and a set of blue points on the plane, find two unit disks CR and CB with disjoint interiors such that the number of red points covered by CR plus the number of blue points covered by CB is maximized. We give an algorithm to solve this proble ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
We study the following problem: Given a set of red points and a set of blue points on the plane, find two unit disks CR and CB with disjoint interiors such that the number of red points covered by CR plus the number of blue points covered by CB is maximized. We give an algorithm to solve this problem in O(n 8/3 log² n) time, where n denotes the total number of points. We also show that the analogous problem of finding two axisaligned unit squares SR and SB instead of unit disks can be solved in O(n log n) time, which is optimal. If we do not restrict ourselves to axisaligned squares, but require that both squares have a common orientation, we give a solution using O(n³ log n) time.