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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 254 (41 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.
Geometric Shortest Paths and Network Optimization
 Handbook of Computational Geometry
, 1998
"... Introduction A natural and wellstudied problem in algorithmic graph theory and network optimization is that of computing a "shortest path" between two nodes, s and t, in a graph whose edges have "weights" associated with them, and we consider the "length" of a path to ..."
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Cited by 152 (13 self)
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Introduction A natural and wellstudied problem in algorithmic graph theory and network optimization is that of computing a "shortest path" between two nodes, s and t, in a graph whose edges have "weights" associated with them, and we consider the "length" of a path to be the sum of the weights of the edges that comprise it. Efficient algorithms are well known for this problem, as briefly summarized below. The shortest path problem takes on a new dimension when considered in a geometric domain. In contrast to graphs, where the encoding of edges is explicit, a geometric instance of a shortest path problem is usually specified by giving geometric objects that implicitly encode the graph and its edge weights. Our goal in devising efficient geometric algorithms is generally to avoid explicit construction of the entire underlying graph, since the full induced graph may be very large (even exponential in the input size, or infinite). Computing an optimal
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 70 (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.
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 62 (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...
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 48 (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...
Dynamic Trees and Dynamic Point Location
 In Proc. 23rd Annu. ACM Sympos. Theory Comput
, 1991
"... This paper describes new methods for maintaining a pointlocation data structure for a dynamicallychanging monotone subdivision S. The main approach is based on the maintenance of two interlaced spanning trees, one for S and one for the graphtheoretic planar dual of S. Queries are answered by using ..."
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Cited by 44 (9 self)
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This paper describes new methods for maintaining a pointlocation data structure for a dynamicallychanging monotone subdivision S. The main approach is based on the maintenance of two interlaced spanning trees, one for S and one for the graphtheoretic planar dual of S. Queries are answered by using a centroid decomposition of the dual tree to drive searches in the primal tree. These trees are maintained via the linkcut trees structure of Sleator and Tarjan, leading to a scheme that achieves vertex insertion/deletion in O(log n) time, insertion/deletion of kedge monotone chains in O(log n + k) time, and answers queries in O(log 2 n) time, with O(n) space, where n is the current size of subdivision S. The techniques described also allow for the dual operations expand and contract to be implemented in O(log n) time, leading to an improved method for spatial pointlocation in a 3dimensional convex subdivision. In addition, the interlacedtree approach is applied to online pointlo...
Planar spanners and approximate shortest path queries among obstacles
 in the plane, Proc. 4th European Sympos. Algorithms
, 1996
"... Abstract. We consider the problem of finding an obstacleavoiding path between two points s and t in the plane, amidst a set of disjoint polygonal obstacles with a total of n vertices. The length of this path should be within a small constant factor c of the length of the shortest possible obstacle ..."
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Cited by 40 (14 self)
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Abstract. We consider the problem of finding an obstacleavoiding path between two points s and t in the plane, amidst a set of disjoint polygonal obstacles with a total of n vertices. The length of this path should be within a small constant factor c of the length of the shortest possible obstacleavoiding st path measured in the Lvmetric. Such an approximate shortest path is called a cshort path, or a short path with stretch]actor c. The goal is to preprocess the obstaclescattered plane by creating an efficient data structure that enables fast reporting of a cshort path (or its length). In this paper, we give a family of algorithms for the above problem that achieve an interesting tradeoff between the stretch factor, the query time and the preprocessing bounds. Our main results are algorithms that achieve logarithmic length query time, after subquadratic time and space preprocessing. 1
Efficient PiecewiseLinear Function Approximation Using the Uniform Metric
 Discrete & Computational Geometry
, 1994
"... We give an O(n log n)time method for finding a best klink piecewiselinear function approximating an npoint planar data set using the wellknown uniform metric to measure the error, ffl 0, of the approximation. Our method is based upon new characterizations of such functions, which we exploit to ..."
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Cited by 39 (0 self)
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We give an O(n log n)time method for finding a best klink piecewiselinear function approximating an npoint planar data set using the wellknown uniform metric to measure the error, ffl 0, of the approximation. Our method is based upon new characterizations of such functions, which we exploit to design an efficient algorithm using a plane sweep in "ffl space" followed by several applications of the parametric searching technique. The previous best running time for this problem was O(n 2 ). 1 Introduction Approximating a set S = f(x 1 ; y 1 ); (x 2 ; y 2 ); : : : ; (x n ; y n )g of points in the plane by a function is a classic problem in applied mathematics. The general goals in this area of research are to find a function F belonging to a class of functions F such that each F 2 F is simple to describe, represent, and compute and such that the chosen F approximates S well. For example, one may desire that F be the class of linear or piecewiselinear functions, and, for any parti...
Efficiently approximating polygonal paths in three and higher dimensions
 Algorithmica
, 1998
"... Abstract. We present efficient algorithms for solving polygonalpath approximation problems in three and higher dimensions. Given an nvertex polygonal curve P in R d, d ≥ 3, we approximate P by another polygonal curve P ′ of m ≤ n vertices in R d such that the vertex sequence of P ′ is an ordered s ..."
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Cited by 24 (5 self)
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Abstract. We present efficient algorithms for solving polygonalpath approximation problems in three and higher dimensions. Given an nvertex polygonal curve P in R d, d ≥ 3, we approximate P by another polygonal curve P ′ of m ≤ n vertices in R d such that the vertex sequence of P ′ is an ordered subsequence of the vertices of P. The goal is either to minimize the size m of P ′ for a given error tolerance ε (called the min # problem), or to minimize the deviation error ε between P and P ′ for a given size m of P ′ (called the minε problem). Our techniques enable us to develop efficient nearquadratictime algorithms in three dimensions and subcubictime algorithms in four dimensions for solving the min # and minε problems. We discuss extensions of our solutions to ddimensional space, where d> 4, and for the L1 and L∞ metrics. Key Words. Curve approximation, Parametric searching. 1. Introduction. In
New Results on Binary Space Partitions in the Plane
 COMPUT. GEOM. THEORY APPL
, 1994
"... We prove the existence of linear size binary space partitions for sets of objects in the plane under certain conditions that are often satisfied in practical situations. In particular, we construct linear size binary space partitions for sets of fat objects, for sets of line segments where the ra ..."
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Cited by 19 (6 self)
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We prove the existence of linear size binary space partitions for sets of objects in the plane under certain conditions that are often satisfied in practical situations. In particular, we construct linear size binary space partitions for sets of fat objects, for sets of line segments where the ratio between the lengths of the longest and shortest segment is bounded by a constant, and for homothetic objects. For all cases we also show how to turn the existence proofs into efficient algorithms.