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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. ..."
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Cited by 256 (40 self)
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... 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.
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 ..."
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Cited by 78 (22 self)
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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+" r ..."
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Cited by 46 (0 self)
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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 ...
An Optimal Randomized Algorithm for Maximum Tukey Depth
, 2004
"... We present the first optimal algorithm to compute the maximum Tukey depth (also known as location or halfspace depth) for a nondegenerate point set in the plane. The algorithm is randomized and requires O(n log n) expected time for n data points. In a higher fixed dimension d 3, the expected tim ..."
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Cited by 46 (4 self)
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We present the first optimal algorithm to compute the maximum Tukey depth (also known as location or halfspace depth) for a nondegenerate point set in the plane. The algorithm is randomized and requires O(n log n) expected time for n data points. In a higher fixed dimension d 3, the expected time bound is O(n ), which is probably optimal as well. The result is obtained using an interesting variant of the author's randomized optimization technique, capable of solving "implicit" linearprogrammingtype problems; some other applications of this technique are briefly mentioned.
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 ..."
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Cited by 46 (3 self)
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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.
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 log n) tim ..."
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Cited by 22 (14 self)
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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 ..."
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Cited by 21 (1 self)
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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
Computing the Detour of Polygonal Curves
, 2002
"... Let P be a simple polygonal chain in E with n edges. The detour of P between two points, x and y, is the ratio between the length of P between x any y and their Euclidean distance. The detour ..."
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Cited by 12 (3 self)
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Let P be a simple polygonal chain in E with n edges. The detour of P between two points, x and y, is the ratio between the length of P between x any y and their Euclidean distance. The detour
Parametric search made practical
 SoCG: 18th Symposium on Computational Geometry
, 2002
"... In this paper we show that in sortingbased applications of parametric search, Quicksort can replace the parallel sorting algorithms that are usually advocated, and we argue that Cole’s optimization of certain parametricsearch algorithms may be unnecessary under realistic assumptions about the inpu ..."
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Cited by 10 (1 self)
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In this paper we show that in sortingbased applications of parametric search, Quicksort can replace the parallel sorting algorithms that are usually advocated, and we argue that Cole’s optimization of certain parametricsearch algorithms may be unnecessary under realistic assumptions about the input. Furthermore, we present a generic, flexible, and easytouse framework that greatly simplifies the implementation of algorithms based on parametric search. We use our framework to implement an algorithm that solves the Fréchetdistance problem. The implementation based on parametric search is faster than the binarysearch approach that is often suggested as a practical replacement for the parametricsearch technique.