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68
Combinatorial Geometry
, 1995
"... Abstract. Let P be a set of n points in ~d (where d is a small fixed positive integer), and let F be a collection of subsets of ~d, each of which is defined by a constant number of bounded degree polynomial inequalities. We consider the following Frange searching problem: Given P, build a data stru ..."
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Cited by 164 (26 self)
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Abstract. Let P be a set of n points in ~d (where d is a small fixed positive integer), and let F be a collection of subsets of ~d, each of which is defined by a constant number of bounded degree polynomial inequalities. We consider the following Frange searching problem: Given P, build a data structure for efficient answering of queries of the form, "Given a 7 ~ F, count (or report) the points of P lying in 7." Generalizing the simplex range searching techniques, we give a solution with nearly linear space and preprocessing time and with O(n 1 x/b+~) query time, where d < b < 2d 3 and ~> 0 is an arbitrarily small constant. The acutal value of b is related to the problem of partitioning arrangements of algebraic surfaces into cells with a constant description complexity. We present some of the applications of Frange searching problem, including improved ray shooting among triangles in ~3 1.
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 be the sum of the weights of t ..."
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Cited by 149 (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
On Range Searching with Semialgebraic Sets
 DISCRETE COMPUT. GEOM
, 1994
"... Let P be a set of n points in R d (where d is a small fixed positive integer), and let \Gamma be a collection of subsets of R d , each of which is defined by a constant number of bounded degree polynomials. We consider the following \Gammarange searching problem: Given P , build a data structur ..."
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Cited by 80 (22 self)
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Let P be a set of n points in R d (where d is a small fixed positive integer), and let \Gamma be a collection of subsets of R d , each of which is defined by a constant number of bounded degree polynomials. We consider the following \Gammarange searching problem: Given P , build a data structure for efficient answering of queries of the form `Given a fl 2 \Gamma, count (or report) the points of P lying in fl'. Generalizing the simplex range searching techniques, we give a solution with nearly linear space and preprocessing time and with O(n 1\Gamma1=b+ffi ) query time, where d b 2d \Gamma 3 and ffi ? 0 is an arbitrarily small constant. The actual value of b is related to the problem of partitioning arrangements of algebraic surfaces into constantcomplexity cells. We present some of the applications of \Gammarange searching problem, including improved ray shooting among triangles in R³.
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 69 (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.
Approximation Algorithms for Geometric Tour and Network Design Problems (Extended Abstract)
"... ..."
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...
Optimal OutputSensitive Convex Hull Algorithms in Two and Three Dimensions
, 1996
"... We present simple outputsensitive algorithms that construct the convex hull of a set of n points in two or three dimensions in worstcase optimal O(n log h) time and O(n) space, where h denotes the number of vertices of the convex hull. ..."
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Cited by 47 (6 self)
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We present simple outputsensitive algorithms that construct the convex hull of a set of n points in two or three dimensions in worstcase optimal O(n log h) time and O(n) space, where h denotes the number of vertices of the convex hull.
Dynamic Ray Shooting and Shortest Paths in Planar Subdivisions via Balanced Geodesic Triangulations
 J. Algorithms
, 1997
"... We give new methods for maintaining a data structure that supports ray shooting and shortest path queries in a dynamicallychanging connected planar subdivision S. Our approach is based on a new dynamic method for maintaining a balanced decomposition of a simple polygon via geodesic triangles. We ma ..."
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Cited by 38 (3 self)
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We give new methods for maintaining a data structure that supports ray shooting and shortest path queries in a dynamicallychanging connected planar subdivision S. Our approach is based on a new dynamic method for maintaining a balanced decomposition of a simple polygon via geodesic triangles. We maintain such triangulations by viewing their dual trees as balanced trees. We show that rotations in these trees can be implemented via a simple "diagonal swapping" operation performed on the corresponding geodesic triangles, and that edge insertion and deletion can be implemented on these trees using operations akin to the standard split and splice operations. We also maintain a dynamic point location structure on the geodesic triangulation, so that we may implement ray shooting queries by first locating the ray's endpoint and then walking along the ray from geodesic triangle to geodesic triangle until we hit the boundary of some region of S. The shortest path between two points in the same ...
Touring a Sequence of Polygons
 In Proc. 35th Annu. ACM Sympos. Theory Comput
, 2003
"... Given a sequence of k polygons in the plane, a start point s, and a target point, t, we seek a shortest path that starts at s, visits in order each of the polygons, and ends at t. If the polygons are disjoint and convex, we give an algorithm running in time O(kn log(n/k)), where n is the total numbe ..."
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Cited by 31 (4 self)
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Given a sequence of k polygons in the plane, a start point s, and a target point, t, we seek a shortest path that starts at s, visits in order each of the polygons, and ends at t. If the polygons are disjoint and convex, we give an algorithm running in time O(kn log(n/k)), where n is the total number of vertices specifying the polygons. We also extend our results to a case in which the convex polygons are arbitrarily intersecting and the subpath between any two consecutive polygons is constrained to lie within a simply connected region; the algorithm uses O(nk log n) time. Our methods are simple and allow shortest path queries from s to a query point t to be answered in time O(k log n + m), where m is the combinatorial path length. We show that for nonconvex polygons this "touring polygons" problem is NPhard.