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55
RAY SHOOTING AND PARAMETRIC SEARCH
, 1993
"... Efficient algorithms for the ray shooting problem are presented: Given a collection F of objects in d, build a data structure so that, for a query ray, the first object of F hit by the ray can be quickly determined. Using the parametric search technique, this problem is reduced to the segment emptin ..."
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Cited by 125 (25 self)
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Efficient algorithms for the ray shooting problem are presented: Given a collection F of objects in d, build a data structure so that, for a query ray, the first object of F hit by the ray can be quickly determined. Using the parametric search technique, this problem is reduced to the segment emptiness problem. For various ray shooting problems, space/querytime tradeoffs of the following type are achieved: For some integer b and a parameter m (n _< m < n b) the queries are answered in time O((n/m /b) log <) n), with O(m!+) space and preprocessing time (t> 0 is arbitrarily small but fixed constant), b Ld/2J is obtained for ray shooting in a convex dpolytope defined as an intersection of n half spaces, b d for an arrangement of n hyperplanes in d, and b 3 for an arrangement of n half planes in 3. This approach also yields fast procedures for finding the first k objects hit by a query ray, for searching nearest and farthest neighbors, and for the hidden surface removal. All the data structures can be maintained dynamically in amortized time O (m + / n) per insert/delete operation.
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 81 (20 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...
Range Searching
, 1996
"... Range searching is one of the central problems in computational geometry, because it arises in many applications and a wide variety of geometric problems can be formulated as a rangesearching problem. A typical rangesearching problem has the following form. Let S be a set of n points in R d , an ..."
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Cited by 71 (1 self)
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Range searching is one of the central problems in computational geometry, because it arises in many applications and a wide variety of geometric problems can be formulated as a rangesearching problem. A typical rangesearching problem has the following form. Let S be a set of n points in R d , and let R be a family of subsets; elements of R are called ranges . We wish to preprocess S into a data structure so that for a query range R, the points in S " R can be reported or counted efficiently. Typical examples of ranges include rectangles, halfspaces, simplices, and balls. If we are only interested in answering a single query, it can be done in linear time, using linear space, by simply checking for each point p 2 S whether p lies in the query range.
OutputSensitive Results on Convex Hulls, Extreme Points, and Related Problems
, 1996
"... . We use known data structures for rayshooting and linearprogramming queries to derive new outputsensitive results on convex hulls, extreme points, and related problems. We show that the f face convex hull of an npoint set P in a fixed dimension d # 2 can be constructed in O(n log f + (nf) ..."
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Cited by 69 (13 self)
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. We use known data structures for rayshooting and linearprogramming queries to derive new outputsensitive results on convex hulls, extreme points, and related problems. We show that the f face convex hull of an npoint set P in a fixed dimension d # 2 can be constructed in O(n log f + (nf) 11/(#d/2#+1) log O(1) n) time; this is optimal if f = O(n 1/#d/2# / log K n) for some sufficiently large constant K . We also show that the h extreme points of P can be computed in O(n log O(1) h + (nh) 11/(#d/2#+1) log O(1) n) time. These results are then applied to produce an algorithm that computes the vertices of all the convex layers of P in O(n 2# ) time for any constant #<2/(#d/2# 2 + 1). Finally, we obtain improved time bounds for other problems including levels in arrangements and linear programming with few violated constraints. In all of our algorithms the input is assumed to be in general position. 1. Introduction Let P be a set of n points in ddimen...
Evaluation of Collision Detection Methods for Virtual Reality FlyThroughs
 In Canadian Conference on Computational Geometry
, 1995
"... We consider the problem of preprocessing a scene of polyhedral models in order to perform collision detection very efficiently for an object that moves amongst obstacles. This problem is of central importance in virtual reality applications, where it is necessary to check for collisions at realtime ..."
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Cited by 69 (7 self)
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We consider the problem of preprocessing a scene of polyhedral models in order to perform collision detection very efficiently for an object that moves amongst obstacles. This problem is of central importance in virtual reality applications, where it is necessary to check for collisions at realtime rates. We give an algorithm for collision detection that is based on the use of a mesh (tetrahedralization) of the free space that has (hopefully) low stabbing number. The algorithm has been implemented and tested, and we give experimental results comparing its performance against three other algorithms that we implemented, based on standard data structures. A preliminary version of this paper appeared in the proceedings of the 7 th Canad. Conf. Computat. Geometry, Qu'ebec, Aug 1013, 1995. y held@ams.sunysb.edu; Supported by NSF Grant DMS9312098. On sabbatical leave from Universitat Salzburg, Salzburg, Austria. z jklosow@ams.sunysb.edu; Supported by NSF grants ECSE8857642 and C...
Vertical decomposition of shallow levels in 3dimensional arrangements and its applications
 SIAM J. Comput
"... Let F be a collection of n bivariate algebraic functions of constant maximum degree. We show that the combinatorial complexity of the vertical decomposition of the ≤klevel of the arrangement A(F) is O(k 3+ε ψ(n/k)), for any ε> 0, where ψ(r) is the maximum complexity of the lower envelope of a su ..."
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Cited by 60 (15 self)
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Let F be a collection of n bivariate algebraic functions of constant maximum degree. We show that the combinatorial complexity of the vertical decomposition of the ≤klevel of the arrangement A(F) is O(k 3+ε ψ(n/k)), for any ε> 0, where ψ(r) is the maximum complexity of the lower envelope of a subset of at most r functions of F. This bound is nearly optimal in the worst case, and implies the existence of shallow cuttings, in the sense of [52], of small size in arrangements of bivariate algebraic functions. We also present numerous applications of these results, including: (i) data structures for several generalized threedimensional rangesearching problems; (ii) dynamic data structures for planar nearest and farthestneighbor searching under various fairly general distance functions; (iii) an improved (nearquadratic) algorithm for minimumweight bipartite Euclidean matching in the plane; and (iv) efficient algorithms for certain geometric optimization problems in static and dynamic settings.
Approximate Nearest Neighbor Queries Revisited
, 1998
"... This paper proposes new methods to answer approximate nearest neighbor queries on a set of n points in ddimensional Euclidean space. For any fixed constant d, a data structure with O(" (1\Gammad)=2 n log n) preprocessing time and O(" (1\Gammad)=2 log n) query time achieves approximat ..."
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Cited by 57 (3 self)
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This paper proposes new methods to answer approximate nearest neighbor queries on a set of n points in ddimensional Euclidean space. For any fixed constant d, a data structure with O(" (1\Gammad)=2 n log n) preprocessing time and O(" (1\Gammad)=2 log n) query time achieves approximation factor 1 + " for any given 0 ! " ! 1; a variant reduces the "dependence by a factor of " \Gamma1=2 . For any arbitrary d, a data structure with O(d 2 n log n) preprocessing time and O(d 2 log n) query time achieves approximation factor O(d 3=2 ). Applications to various proximity problems are discussed. 1 Introduction Let P be a set of n point sites in ddimensional space IR d . In the wellknown post office problem, we want to preprocess P into a data structure so that a site closest to a given query point q (called the nearest neighbor of q) can be found efficiently. Distances are measured under the Euclidean metric. The post office problem has many applications within computational...
Geometric Range Searching
, 1994
"... In geometric range searching, algorithmic problems of the following type are considered: Given an npoint set P in the plane, build a data structure so that, given a query triangle R, the number of points of P lying in R can be determined quickly. Problems of this type are of crucial importance in c ..."
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Cited by 50 (2 self)
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In geometric range searching, algorithmic problems of the following type are considered: Given an npoint set P in the plane, build a data structure so that, given a query triangle R, the number of points of P lying in R can be determined quickly. Problems of this type are of crucial importance in computational geometry, as they can be used as subroutines in many seemingly unrelated algorithms. We present a survey of results and main techniques in this area.
QuerySensitive Ray Shooting
 IN PROC. 10TH ANNU. ACM SYMPOS. COMPUT. GEOM
, 1994
"... Ray (segment) shooting is the problem of determining the first intersection between a ray (directed line segment) and a collection of polygonal or polyhedral obstacles. In order to process queries efficiently, the set of obstacle polyhedra is usually preprocessed into a data structure. In this pa ..."
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Cited by 49 (10 self)
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Ray (segment) shooting is the problem of determining the first intersection between a ray (directed line segment) and a collection of polygonal or polyhedral obstacles. In order to process queries efficiently, the set of obstacle polyhedra is usually preprocessed into a data structure. In this paper, we propose a querysensitive data structure for ray shooting, which means that the performance of our data structure depends on the "local" geometry of obstacles near the query segment. We measure the complexity of the local geometry near the segment by a parameter called the simple cover complexity , denoted by scc(s) for a segment s. Our data structure consists of a subdivision that partitions the space into a collection of polyhedral cells of O(1) complexity. We answer a segment shooting query by walking along the segment through the subdivision. Our first result is that, for any fixed dimension d, there exists a simple hierarchical subdivision in which no query segment s int...
New Lower Bounds for Hopcroft's Problem
, 1996
"... We establish new lower bounds on the complexity of the following basic geometric problem, attributed to John Hopcroft: Given a set of n points and m hyperplanes in R d , is any point contained in any hyperplane? We define a general class of partitioning algorithms, and show that in the worst cas ..."
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Cited by 33 (6 self)
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We establish new lower bounds on the complexity of the following basic geometric problem, attributed to John Hopcroft: Given a set of n points and m hyperplanes in R d , is any point contained in any hyperplane? We define a general class of partitioning algorithms, and show that in the worst case, for all m and n, any such algorithm requires time #(n log m+n 2/3 m 2/3 +m log n) in two dimensions, or #(n log m+n 5/6 m 1/2 +n 1/2 m 5/6 + m log n) in three or more dimensions. We obtain slightly higher bounds for the counting version of Hopcroft's problem in four or more dimensions. Our planar lower bound is within a factor of 2 O(log # (n+m)) of the best known upper bound, due to Matousek. Previously, the best known lower bound, in any dimension, was #(n log m + m log n). We develop our lower bounds in two stages. First we define a combinatorial representation of the relative order type of a set of points and hyperplanes, called a monochromatic cover, and derive low...