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70
On Bregman Voronoi Diagrams
 in "Proc. 18th ACMSIAM Sympos. Discrete Algorithms
, 2007
"... The Voronoi diagram of a point set is a fundamental geometric structure that partitions the space into elementary regions of influence defining a discrete proximity graph and dually a wellshaped Delaunay triangulation. In this paper, we investigate a framework for defining and building the Voronoi ..."
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Cited by 42 (22 self)
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The Voronoi diagram of a point set is a fundamental geometric structure that partitions the space into elementary regions of influence defining a discrete proximity graph and dually a wellshaped Delaunay triangulation. In this paper, we investigate a framework for defining and building the Voronoi diagrams for a broad class of distortion measures called Bregman divergences, that includes not only the traditional (squared) Euclidean distance, but also various divergence measures based on entropic functions. As a byproduct, Bregman Voronoi diagrams allow one to define informationtheoretic Voronoi diagrams in statistical parametric spaces based on the relative entropy of distributions. We show that for a given Bregman divergence, one can define several types of Voronoi diagrams related to each other
Line Transversals of Balls and Smallest Enclosing Cylinders in Three Dimensions
 DISCRETE COMPUT. GEOM
, 1997
"... We establish a nearcubic upper bound on the complexity of the space of line transversals of a collection of n balls in three dimensions, and show that the bound is almost tight, in the worst case. We apply this bound to obtain a nearcubic algorithm for computing a smallest infinite cylinder enclos ..."
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Cited by 39 (6 self)
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We establish a nearcubic upper bound on the complexity of the space of line transversals of a collection of n balls in three dimensions, and show that the bound is almost tight, in the worst case. We apply this bound to obtain a nearcubic algorithm for computing a smallest infinite cylinder enclosing a given set of points or balls in 3space. We also present an approximation algorithm for computing a smallest enclosing cylinder.
The Union Of Convex Polyhedra In Three Dimensions
, 1997
"... . We show that the number of vertices, edges, and faces of the union of k convex polyhedra in 3space, having a total of n faces, is O(k 3 + kn log k). This bound is almost tight in the worst case, as there exist collections of polyhedra with## k 3 + kn#(k)) union complexity. We also describe a ..."
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Cited by 39 (25 self)
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. We show that the number of vertices, edges, and faces of the union of k convex polyhedra in 3space, having a total of n faces, is O(k 3 + kn log k). This bound is almost tight in the worst case, as there exist collections of polyhedra with## k 3 + kn#(k)) union complexity. We also describe a rather simple randomized incremental algorithm for computing the boundary of the union in O(k 3 + kn log k log n) expected time. Key words. combinatorial geometry, computational geometry, combinatorial complexity, convex polyhedra, geometric algorithms, randomized algorithms AMS subject classifications. 52B10, 52B55, 65Y25, 68Q25, 68U05 PII. S0097539793250755 1. Combinatorial bounds. Let P = {P 1 , . . . , P k } be a family of k convex polyhedra in 3space, let n i be the number of faces of P i , and let n = # k i=1 n i . Put U = # P. By the combinatorial complexity of a polyhedral set we mean the total number of its vertices, edges, and faces. Our main result is the followin...
New Results on Quantifier Elimination Over Real Closed Fields and Applications to Constraint Databases
 Journal of the ACM
, 1999
"... In this paper we give a new algorithm for quantifier elimination in the first order theory of real closed fields that improves the complexity of the best known algorithm for this problem till now. Unlike previously known algorithms [3, 28, 22] the combinatorial part of the complexity (the part depen ..."
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Cited by 35 (4 self)
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In this paper we give a new algorithm for quantifier elimination in the first order theory of real closed fields that improves the complexity of the best known algorithm for this problem till now. Unlike previously known algorithms [3, 28, 22] the combinatorial part of the complexity (the part depending on the number of polynomials in the input) of this new algorithm is independent of the number of free variables. Moreover, under the assumption that each polynomial in the input depend only on a constant number of the free variables, the algebraic part of the complexity (the part depending on the degrees of the input polynomials) can also be made independent of the number of free variables. This new feature of our algorithm allows us to obtain a new algorithm for a variant of the quantifier elimination problem. We give an almost optimal algorithm for this new problem, which we call the uniform quantifier elimination problem. Using the uniform quantifier elimination algorithm, we give a...
Almost tight upper bounds for vertical decompositions in four dimensions
 In Proc. 42nd IEEE Symposium on Foundations of Computer Science
, 2001
"... We show that the complexity of the vertical decomposition of an arrangement of n fixeddegree algebraic surfaces or surface patches in four dimensions is O(n 4+ε), for any ε> 0. This improves the best previously known upper bound for this problem by a nearlinear factor, and settles a major problem i ..."
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Cited by 31 (6 self)
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We show that the complexity of the vertical decomposition of an arrangement of n fixeddegree algebraic surfaces or surface patches in four dimensions is O(n 4+ε), for any ε> 0. This improves the best previously known upper bound for this problem by a nearlinear factor, and settles a major problem in the theory of arrangements of surfaces, open since 1989. The new bound can be extended to higher dimensions, yielding the bound O(n 2d−4+ε), for any ε> 0, on the complexity of vertical decompositions in dimensions d ≥ 4. We also describe the immediate algorithmic applications of these results, which include improved algorithms for point location, range searching, ray shooting, robot motion planning, and some geometric optimization problems. 1
Almost tight upper bounds for the single cell and zone problems in three dimensions
 Geom
, 1995
"... We consider the problem of bounding the combinatorial complexity of a single cell in an arrangement of n lowdegree algebraic surface patches in 3space. We show that this complexity is O(n ..."
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Cited by 30 (17 self)
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We consider the problem of bounding the combinatorial complexity of a single cell in an arrangement of n lowdegree algebraic surface patches in 3space. We show that this complexity is O(n
Kinetic Data Structures
, 1999
"... Modeling the physical world in the computer raises problems that intertwine discrete and continuous aspects. For example, physical objects move along continuous trajectories; yet every so often discrete events occur, such as collisions between objects. In a model of objects in space, there are many ..."
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Cited by 29 (1 self)
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Modeling the physical world in the computer raises problems that intertwine discrete and continuous aspects. For example, physical objects move along continuous trajectories; yet every so often discrete events occur, such as collisions between objects. In a model of objects in space, there are many discrete attributes that one may want to compute: the closest pair, the convex hull, the minimum spanning tree, etc. When the objects are in motion, the values of these attributes change over time, and it becomes necessary to keep track of them as the objects move. In this thesis, we introduce a general approach, and an analysis framework, for solving this type of problems. To keep track of a discrete attribute, we create a new type of data structure, called a kinetic data structure. A kinetic data structure is made of a proof of correctness of the attribute which is animated through time by a discrete event simulation.
Arrangements
, 1997
"... INTRODUCTION Given a finite collection S of geometric objects such as hyperplanes or spheres in R d , the arrangement A(S) is the decomposition of R d into connected open cells of dimensions 0; 1; : : :; d induced by S. Besides being interesting in their own right, arrangements of hyperplanes ..."
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Cited by 28 (13 self)
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INTRODUCTION Given a finite collection S of geometric objects such as hyperplanes or spheres in R d , the arrangement A(S) is the decomposition of R d into connected open cells of dimensions 0; 1; : : :; d induced by S. Besides being interesting in their own right, arrangements of hyperplanes have served as a unifying structure for many problems in discrete and computational geometry. With the recent advances in the study of arrangements of curved (algebraic) surfaces, arrangements have emerged as the underlying structure of geometric problems in a variety of `physical world' application domains such as robot motion planning and computer vision. This chapter is devoted to arrangements of hyperplanes and of curved surfaces in lowdimensional Euclidean space, with an emphasis on combinatorics and algorithms. In the first section we in
Geometry helps in bottleneck matching and related problems
 Algorithmica
, 2001
"... This paper is accepted for publication in ALGORITHMICA Abstract Let A and B be two sets of n objects in Rd, and let Match be a (onetoone)matching between A and B. Let min(Match), max(Match), and \Sigma (Match) denote thelength of the shortest edge, the length of the longest edge, and the sum of th ..."
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Cited by 27 (4 self)
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This paper is accepted for publication in ALGORITHMICA Abstract Let A and B be two sets of n objects in Rd, and let Match be a (onetoone)matching between A and B. Let min(Match), max(Match), and \Sigma (Match) denote thelength of the shortest edge, the length of the longest edge, and the sum of the lengths of the edges of Match respectively. Bottleneck matchinga matching that minimizesmax( Match)is suggested as a convenient way for measuring the resemblance between A and B. Several algorithms for computing, as well as approximating, this resemblanceare proposed. The running time of all the algorithms involving planar objects is roughly O(n1.5). For instance, if the objects are points in the plane, the running time of the exactalgorithm is O(n1.5 log n). A semidynamic datastructure for answering containmentproblems for a set of congruent disks in the plane is developed. This data structure may be of independent interest.Next, the problem of finding a translation of B that maximizes the resemblance to A under the bottleneck matching criterion is considered. When A and B are pointsetsin the plane, an O(n5 log n) time algorithm for determining whether for some translatedcopy the resemblance gets below a given ae is presented, thus improving the previousresult of Alt, Mehlhorn, Wagener and Welzl by a factor of almost n. This result is usedto compute the smallest such ae in time O(n5 log2 n), and an efficient approximationscheme for this problem is also given. The uniform matching problem (also called the balanced assignment problem, or thefair matching problem) is to find Match*U, a matching that minimizes max(Match)min ( Match). A minimum deviation matching Match*D is a matching that minimizes(1 /n)\Sigma (Match) min(Match). Algorithms for computing Match*U and Match*D inroughly O(n10/3) time are presented. These algorithms are more efficient than theprevious
On Translational Motion Planning Of A Convex Polyhedron In 3Space
 SIAM J. Comput
, 1997
"... . Let B be a convex polyhedron translating in 3space amidst k convex polyhedral obstacles A 1 , . . . , A k with pairwise disjoint interiors. The free configuration space (space of all collisionfree placements) of B can be represented as the complement of the union of the Minkowski sums P i = A i ..."
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Cited by 27 (14 self)
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. Let B be a convex polyhedron translating in 3space amidst k convex polyhedral obstacles A 1 , . . . , A k with pairwise disjoint interiors. The free configuration space (space of all collisionfree placements) of B can be represented as the complement of the union of the Minkowski sums P i = A i # (B), for i = 1, . . . , k. We show that the combinatorial complexity of the free configuration space of B is O(nk log k), and that it can be ## nk#(k)) in the worst case, where n is the total complexity of the individual Minkowski sums P 1 , . . . , P k . We also derive an e#cient randomized algorithm that constructs this configuration space in expected time O(nk log k log n). Key words. combinatorial geometry, computational geometry, combinatorial complexity, convex polyhedra, geometric algorithms, randomized algorithms, algorithmic motion planning AMS subject classifications. 52B10, 52B55, 65Y25, 68Q25, 68U05 PII. S0097539794266602 1. Introduction. Let A 1 , . . . , A k be k close...