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Better Lower Bounds on Detecting Affine and Spherical Degeneracies
, 1995
"... We show that in the worst case,\Omega (n d ) sidedness queries are required to determine whether a set of n points in IR d is affinely degenerate, i.e., whether it contains d +1points on a common hyperplane. This matches known upper bounds. Wegive a straightforward adversary argument, b ..."
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Cited by 25 (8 self)
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We show that in the worst case,\Omega (n d ) sidedness queries are required to determine whether a set of n points in IR d is affinely degenerate, i.e., whether it contains d +1points on a common hyperplane. This matches known upper bounds. Wegive a straightforward adversary argument, based on the explicit construction of a point set containing\Omega (n d ) "collapsible" simplices, anyoneof which can be made degenerate without changing the orientation of anyother simplex. As an immediate corollary,wehavean\Omega (n d )lower bound on the number of sidedness queries required to determine the order type of a set of n points in IR d . Using similar techniques, wealsoshowthat\Omega (n d+1 ) insphere queries are required to decide the existence of spherical degeneracies in a set of n points in IR d . 1 Introduction A fundamental problem in computational geometry is determining whether a given set of points is in "general position." A simple example of ...
On the Relative Complexities of Some Geometric Problems
 In Proc. 7th Canad. Conf. Comput. Geom
, 1995
"... We consider the relative complexities of a large number of computational geometry problems whose complexities are believed to be roughly \Theta(n 4=3 ). For certain pairs of problems, we show that the complexity of one problem is asymptotically bounded by the complexity of the other. Almost all of ..."
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Cited by 21 (6 self)
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We consider the relative complexities of a large number of computational geometry problems whose complexities are believed to be roughly \Theta(n 4=3 ). For certain pairs of problems, we show that the complexity of one problem is asymptotically bounded by the complexity of the other. Almost all of the problems we consider can be solved in time O(n 4=3+ffi ) or better, and there are\Omega\Gamma n 4=3 ) lower bounds for a few of them in specialized models of computation. However, the best known lower bound in any general model of computation is only\Omega\Gamma n log n). The paper is naturally divided into two parts. In the first part, we consider a large number of problems that are harder than Hopcroft's problem. These problems include various ray shooting problems, sorting line segments in IR 3 , collision detection in IR 3 , and halfspace emptiness checking in IR 5 . In the second, we survey known reductions among problems involving lines in threespace, and among highe...
Reporting Leaders and Followers Among Trajectories of Moving Point Objects
"... Abstract. Widespread availability of location aware devices (such as GPS receivers) promotes capture of detailed movement trajectories of people, animals, vehicles and other moving objects, opening new options for a better understanding of the processes involved. In this paper we investigate spatio ..."
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Cited by 15 (4 self)
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Abstract. Widespread availability of location aware devices (such as GPS receivers) promotes capture of detailed movement trajectories of people, animals, vehicles and other moving objects, opening new options for a better understanding of the processes involved. In this paper we investigate spatiotemporal movement patterns in large tracking data sets. We present a natural definition of the pattern ‘one object is leading others’, which is based on behavioural patterns discussed in the behavioural ecology literature. Such leadership patterns can be characterised by a minimum time length for which they have to exist and by a minimum number of entities involved in the pattern. Furthermore, we distinguish two models (discrete and continuous) of the time axis for which patterns can start and end. For all variants of these leadership patterns, we describe algorithms for their detection, given the trajectories of a group of moving entities. A theoretical analysis as well as experiments show that these algorithms efficiently report leadership patterns.
Computing a Largest Empty Anchored Cylinder, and Related Problems
"... Let S be a set of n points in IR d , and let each point p of S have a positive weight w(p). We consider the problem of computing a ray R emanating from the origin (resp. a line l through the origin) such that min p2S w(p) \Delta d(p; R) (resp. min p2S w(p) \Delta d(p; l)) is maximal. If all wei ..."
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Cited by 8 (0 self)
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Let S be a set of n points in IR d , and let each point p of S have a positive weight w(p). We consider the problem of computing a ray R emanating from the origin (resp. a line l through the origin) such that min p2S w(p) \Delta d(p; R) (resp. min p2S w(p) \Delta d(p; l)) is maximal. If all weights are one, this corresponds to computing a silo emanating from the origin (resp. a cylinder whose axis contains the origin) that does not contain any point of S and whose radius is maximal. For d = 2, we show how to solve these problems in O(n log n) time, which is optimal in the algebraic computation tree model. For d = 3, we give algorithms that are based on the parametric search technique and run in O(n log 4 n) time. The previous best known algorithms for these threedimensional problems had almost quadratic running time. In the final part of the paper, we consider some related problems.
Lower Bounds for Fundamental Geometric Problems
 IN 5TH ANNUAL EUROPEAN SYMPOSIUM ON ALGORITHMS (ESA'97
, 1996
"... We develop lower bounds on the number of primitive operations required to solve several fundamental problems in computational geometry. For example, given a set of points in the plane, are any three colinear? Given a set of points and lines, does any point lie on a line? These and similar question ..."
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Cited by 7 (0 self)
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We develop lower bounds on the number of primitive operations required to solve several fundamental problems in computational geometry. For example, given a set of points in the plane, are any three colinear? Given a set of points and lines, does any point lie on a line? These and similar questions arise as subproblems or special cases of a large number of more complicated geometric problems, including point location, range searching, motion planning, collision detection, ray shooting, and hidden surface removal. Previously these problems were studied only in general models of computation, but known techniques for these models are too weak to prove useful results. Our approach is to consider, for each problem, a more specialized model of computation that is still rich enough to describe all known algorit...
FINDING POPULAR PLACES
, 2008
"... Widespread availability of location aware devices (such as GPS receivers) promotes capture of detailed movement trajectories of people, animals, vehicles and other moving objects. We investigate spatiotemporal movement patterns in large tracking data sets, i.e. in large sets of polygonal paths. Sp ..."
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Cited by 5 (2 self)
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Widespread availability of location aware devices (such as GPS receivers) promotes capture of detailed movement trajectories of people, animals, vehicles and other moving objects. We investigate spatiotemporal movement patterns in large tracking data sets, i.e. in large sets of polygonal paths. Specifically, we study socalled ‘popular places’, that is, regions that are visited by many entities. Given a set of polygonal paths with a total of ¯n vertices, we look at the problem of computing such popular places in two different settings. For the discrete model, where only the vertices of the polygonal paths are considered, we propose an O(¯nlog ¯n) algorithm; and for the continuous model, where also the straight line segments between the vertices of a polygonal path are considered, we develop an O(¯n²) algorithm. We also present lower bounds and hardness results.
Maxmin Location of an Anchored Ray in 3Space and Related Problems (Extended Abstract)
"... ) Abstract We consider the problem of locating a ray emanating from the origin of 3space such as to maximize the minimum weighted Euclidean distance to a set of weighted obstacles (points, lines or line segments). We present algorithms based on the parametric search paradigm which run in O(n log ..."
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Cited by 4 (1 self)
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) Abstract We consider the problem of locating a ray emanating from the origin of 3space such as to maximize the minimum weighted Euclidean distance to a set of weighted obstacles (points, lines or line segments). We present algorithms based on the parametric search paradigm which run in O(n log 4 n) time in the case of point obstacles, and in O(n 2 log 2 n) (O(n 2 log 2 n 2 ff(n) )) time in the case of line (segment) obstacles. We also show that for practically interesting restricted settings of the line obstacle problem, subquadratic algorithms can be obtained. Furthermore we discuss some related problems. 1 Introduction Facility location problems, such as the well known largest empty circle problem, are one of the major topics in geometric optimization. They are mostly motivated by layout problems in operations research and are thus generally formulated in a planar setting. In this abstract we discuss several location problems in 3space which arise from an entirely di...
An Improved Ray Shooting Method for Constructive Solid Geometry Models via Tree Contraction
, 1990
"... In the Constructive Solid Geometry (CSG) representation a geometric object is described as the hierarchical combination of a number of primitive shapes using the operations union, intersection, subtraction, and exclusiveunion. This hierarchical description defines an expression tree, T , called the ..."
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Cited by 4 (2 self)
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In the Constructive Solid Geometry (CSG) representation a geometric object is described as the hierarchical combination of a number of primitive shapes using the operations union, intersection, subtraction, and exclusiveunion. This hierarchical description defines an expression tree, T , called the CSG tree, with leaves associated with primitive shapes, internal nodes associated with operations, and whose "value" is the geometric object. Evaluation of CSG trees is an important computation that arises in many rendering and analysis problems for geometric models, with ray shooting (also known as "ray casting") being one of the most important. Given any CSG tree T , which may be unbalanced, we show how to convert T into a functionallyequivalent tree, D, that is balanced. We demonstrate the utility of this conversion by showing how it can be used to improve the worstcase running time for ray shooting against a CSG model from O(n 2 ) to O(n log n), which is optimal. Keywords: Boolean a...
Smallest Enclosing Cylinders (Extended Abstract)
"... This paper addresses the complexity of computing the smallestradius infinite cylinder that encloses an input set of n points in 3space. We show that the problem can be solved in time O(n 4 log O(1) n) in an algebraic complexity model. We also achieve a time of O(n 4 L \Delta ¯(L)) in a bit ..."
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This paper addresses the complexity of computing the smallestradius infinite cylinder that encloses an input set of n points in 3space. We show that the problem can be solved in time O(n 4 log O(1) n) in an algebraic complexity model. We also achieve a time of O(n 4 L \Delta ¯(L)) in a bit complexity model. These and several other results highlight a general linearization technique which transforms nonlinear problems into some higher dimensional but linear problems. The technique is reminiscent of the use of Plücker coordinates, and is used here in conjunction with Megiddo's parametric searching. We further report on experimental work comparing an exact with a numerical strategy.
How hard are n²hard problems
 SIGACT NEWS
, 1994
"... Many of the "n²hard" problems described by Gajentaan and Overmars can be solved using limited nondeterminism or other sharplybounded quanti ers. Thus we suggest that these problems are not among the hardest problems solvable in quadratic time. ..."
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Many of the "n²hard" problems described by Gajentaan and Overmars can be solved using limited nondeterminism or other sharplybounded quanti ers. Thus we suggest that these problems are not among the hardest problems solvable in quadratic time.