Results 1  10
of
59
Geometric Range Searching and Its Relatives
 CONTEMPORARY MATHEMATICS
"... ... process a set S of points in so that the points of S lying inside a query R region can be reported or counted quickly. Wesurvey the known techniques and data structures for range searching and describe their application to other related searching problems. ..."
Abstract

Cited by 267 (43 self)
 Add to MetaCart
... process a set S of points in so that the points of S lying inside a query R region can be reported or counted quickly. Wesurvey the known techniques and data structures for range searching and describe their application to other related searching problems.
Efficient algorithms for geometric optimization
 ACM Comput. Surv
, 1998
"... We review the recent progress in the design of efficient algorithms for various problems in geometric optimization. We present several techniques used to attack these problems, such as parametric searching, geometric alternatives to parametric searching, pruneandsearch techniques for linear progra ..."
Abstract

Cited by 115 (12 self)
 Add to MetaCart
We review the recent progress in the design of efficient algorithms for various problems in geometric optimization. We present several techniques used to attack these problems, such as parametric searching, geometric alternatives to parametric searching, pruneandsearch techniques for linear programming and related problems, and LPtype problems and their efficient solution. We then describe a variety of applications of these and other techniques to numerous problems in geometric optimization, including facility location, proximity problems, statistical estimators and metrology, placement and intersection of polygons and polyhedra, and ray shooting and other querytype problems.
Arrangements and Their Applications
 Handbook of Computational Geometry
, 1998
"... The arrangement of a finite collection of geometric objects is the decomposition of the space into connected cells induced by them. We survey combinatorial and algorithmic properties of arrangements of arcs in the plane and of surface patches in higher dimensions. We present many applications of arr ..."
Abstract

Cited by 88 (20 self)
 Add to MetaCart
(Show Context)
The arrangement of a finite collection of geometric objects is the decomposition of the space into connected cells induced by them. We survey combinatorial and algorithmic properties of arrangements of arcs in the plane and of surface patches in higher dimensions. We present many applications of arrangements to problems in motion planning, visualization, range searching, molecular modeling, and geometric optimization. Some results involving planar arrangements of arcs have been presented in a companion chapter in this book, and are extended in this chapter to higher dimensions. Work by P.A. was supported by Army Research Office MURI grant DAAH049610013, by a Sloan fellowship, by an NYI award, and by a grant from the U.S.Israeli Binational Science Foundation. Work by M.S. was supported by NSF Grants CCR9122103 and CCR9311127, by a MaxPlanck Research Award, and by grants from the U.S.Israeli Binational Science Foundation, the Israel Science Fund administered by the Israeli Ac...
Raising Roofs, Crashing Cycles, and Playing Pool: Applications of a Data Structure for Finding Pairwise Interactions
 In Proc. 14th Annu. ACM Sympos. Comput. Geom
, 1998
"... The straight skeleton of a polygon is a variant of the medial axis, introduced by Aichholzer et al., defined by a shrinking process in which each edge of the polygon moves inward at a fixed rate. We construct the straight skeleton of an ngon with r reflex vertices in time O(n 1+" +n 8=11+& ..."
Abstract

Cited by 56 (1 self)
 Add to MetaCart
(Show Context)
The straight skeleton of a polygon is a variant of the medial axis, introduced by Aichholzer et al., defined by a shrinking process in which each edge of the polygon moves inward at a fixed rate. We construct the straight skeleton of an ngon with r reflex vertices in time O(n 1+" +n 8=11+" r 9=11+" ), for any fixed " ? 0, improving the previous best upper bound of O(nr log n). Our algorithm simulates the sequence of collisions between edges and vertices during the shrinking process, using a technique of Eppstein for maintaining extrema of binary functions to reduce the problem of finding successive interactions to two dynamic range query problems: (1) maintain a changing set of triangles in IR 3 and answer queries asking which triangle would be first hit by a query ray, and (2) maintain a changing set of rays in IR 3 and answer queries asking for the lowest intersection of any ray with a query triangle. We also exploit a novel characterization of the straight skeleton as a ...
LowDimensional Linear Programming with Violations
 In Proc. 43th Annu. IEEE Sympos. Found. Comput. Sci
, 2002
"... Two decades ago, Megiddo and Dyer showed that linear programming in 2 and 3 dimensions (and subsequently, any constant number of dimensions) can be solved in linear time. In this paper, we consider linear programming with at most k violations: finding a point inside all but at most k of n given half ..."
Abstract

Cited by 45 (3 self)
 Add to MetaCart
(Show Context)
Two decades ago, Megiddo and Dyer showed that linear programming in 2 and 3 dimensions (and subsequently, any constant number of dimensions) can be solved in linear time. In this paper, we consider linear programming with at most k violations: finding a point inside all but at most k of n given halfspaces. We give a simple algorithm in 2d that runs in O((n + k ) log n) expected time; this is faster than earlier algorithms by Everett, Robert, and van Kreveld (1993) and Matousek (1994) and is probably nearoptimal for all k n=2. A (theoretical) extension of our algorithm in 3d runs in near O(n + k ) expected time. Interestingly, the idea is based on concavechain decompositions (or covers) of the ( k)level, previously used in proving combinatorial klevel bounds.
On approximate range counting and depth
 In Proc. 23rd Annu. ACM Sympos. Comput. Geom
, 2007
"... ABSTRACT We improve the previous results by Aronov and HarPeled (SODA'05) and Kaplan and Sharir (SODA'06) and present a randomized data structure of O(n) expected size which can answer 3D approximate halfspace range counting queries in O(log n k) expected time, where k is the actual value ..."
Abstract

Cited by 24 (1 self)
 Add to MetaCart
(Show Context)
ABSTRACT We improve the previous results by Aronov and HarPeled (SODA'05) and Kaplan and Sharir (SODA'06) and present a randomized data structure of O(n) expected size which can answer 3D approximate halfspace range counting queries in O(log n k) expected time, where k is the actual value of the count. This is the first optimal method for the problem in the standard decision tree model; moreover, unlike previous methods, the new method is Las Vegas instead of Monte Carlo. In addition, we describe new results for several related problems, including approximate Tukey depth queries in 3D, approximate regression depth queries in 2D, and approximate linear programming with violations in low dimensions. Categories and Subject Descriptors F.2.2 [Analysis of Algorithms and Problem Complexity]: Nonnumerical Algorithms and Problemsgeometrical problems and computations
Approximation and Exact Algorithms for MinimumWidth Annuli and Shells
 Discrete Comput. Geom
, 1999
"... Let S be a set of n points in R d . The "roundness" of S can be measured by computing the width ! = ! (S) of the thinnest spherical shell (or annulus in R 2 ) that contains S. This paper contains three main results related to computing ! : (i) For d = 2, we can compute in O(n ..."
Abstract

Cited by 23 (14 self)
 Add to MetaCart
(Show Context)
Let S be a set of n points in R d . The "roundness" of S can be measured by computing the width ! = ! (S) of the thinnest spherical shell (or annulus in R 2 ) that contains S. This paper contains three main results related to computing ! : (i) For d = 2, we can compute in O(n log n) time an annulus containing S whose width is at most 2! (S). We extend this algorithm, so that for any given parameter " ? 0, an annulus containing S whose width is at most (1 + ")! , is computed in time O(n log n + n=" 2 ). (ii) For d 3, given a parameter " ? 0, we can compute a shell containing S of width at most (1+ ")! either in time O \Gamma n " d log( \Delta ! " ) \Delta or in time O \Gamma n " d\Gamma2 \Gamma log n + 1 " \Delta log \Gamma \Delta ! " \Delta\Delta . Work by P.A. was supported by Army Research Office MURI grant DAAH049610013, by a Sloan fellowship, by NSF grants EIA9870724, and CCR9732787, by an NYI award, and by a grant from ...
Computing the maximum detour and spanning ratio of planar chains, trees and cycles
 In Proc. 19th Internat. Symp. Theor. Aspects of C.Sc., LNCS 2285:250–261
, 2002
"... Let G = (V, E) be an embedded connected graph with n vertices and m edges. Specifically, the vertex set V consists of points in R 2, and E consists ..."
Abstract

Cited by 21 (1 self)
 Add to MetaCart
(Show Context)
Let G = (V, E) be an embedded connected graph with n vertices and m edges. Specifically, the vertex set V consists of points in R 2, and E consists