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33
Applications of parametric searching in geometric optimization
 J. Algorithms
, 1994
"... z Sivan Toledo x ..."
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Vertical decomposition of shallow levels in 3dimensional arrangements and its applications
, 2006
"... ..."
New Bounds for Lower Envelopes in Three Dimensions, with Applications to Visibility in Terrains
 Geom
, 1997
"... We consider the problem of bounding the complexity of the lower envelope of n surface patches in 3space, all algebraic of constant maximum degree, and bounded by algebraic arcs of constant maximum degree, with the additional property that the interiors of any triple of these surfaces intersect i ..."
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Cited by 57 (22 self)
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We consider the problem of bounding the complexity of the lower envelope of n surface patches in 3space, all algebraic of constant maximum degree, and bounded by algebraic arcs of constant maximum degree, with the additional property that the interiors of any triple of these surfaces intersect in at most two points. We show that the number of vertices on the lower envelope of n such surface patches is O(n 2 \Delta 2 c p log n ), for some constant c depending on the shape and degree of the surface patches. We apply this result to obtain an upper bound on the combinatorial complexity of the `lower envelope' of the space of all rays in 3space that lie above a given polyhedral terrain K with n edges. This envelope consists of all rays that touch the terrain (but otherwise lie above it). We show that the combinatorial complexity of this rayenvelope is O(n 3 \Delta 2 c p log n ) for some constant c; in particular, there are at most that many rays that pass above th...
Efficient Randomized Algorithms for Some Geometric Optimization Problems
 DISCRETE COMPUT. GEOM
, 1995
"... In this paper we first prove the following combinatorial bound, concerning the complexity of the vertical decomposition of the minimization diagram of trivariate functions: Let F be a collection of n totally or partially defined algebraic trivariate functions of constant maximum degree, with the add ..."
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Cited by 41 (14 self)
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In this paper we first prove the following combinatorial bound, concerning the complexity of the vertical decomposition of the minimization diagram of trivariate functions: Let F be a collection of n totally or partially defined algebraic trivariate functions of constant maximum degree, with the additional property that, for a given pair of functions f; f 0 2 F , the surface f(x; y; z) = f 0 (x; y; z) is xymonotone (actually, we need a somewhat weaker propertysee below). We show that the vertical decomposition of the minimization diagram of F consists of O(n 3+" ) cells (each of constant complexity), for any " ? 0. In the second part of the paper we present a general technique that yields faster randomized algorithms for solving a number of geometric optimization problems, including (i) computing the width of a point set in 3space, (ii) computing the minimumwidth annulus enclosing a set of n points in the plane, and (iii) computing the `biggest stick' inside a simple polyg...
A NearLinear Algorithm for the Planar Segment Center Problem
, 1996
"... Let P be a set of n points in the plane and let e be a segment of fixed length. The segment center problem is to find a placement of e (allowing translation and rotation) which minimizes the maximum euclidean distance from e to the points of P. We present an algorithm that solves the problem in tim ..."
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Cited by 17 (8 self)
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Let P be a set of n points in the plane and let e be a segment of fixed length. The segment center problem is to find a placement of e (allowing translation and rotation) which minimizes the maximum euclidean distance from e to the points of P. We present an algorithm that solves the problem in time O(n1+&quot;), for any &quot; ? 0, improving the previous solution of Agarwal et al. [3] by nearly a factor of O(n).
The Overlay of Lower Envelopes and Its Applications
, 1996
"... Let F and G be two collections of a total of n (possibly partially defined) bivariate, algebraic functions of constant maximum degree. The minimization diagrams of F, G are the planar maps obtained by the xyprojections of the lower envelopes of F, G, respectively. We show that the combinatorial c ..."
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Cited by 15 (4 self)
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Let F and G be two collections of a total of n (possibly partially defined) bivariate, algebraic functions of constant maximum degree. The minimization diagrams of F, G are the planar maps obtained by the xyprojections of the lower envelopes of F, G, respectively. We show that the combinatorial complexity of the overlay of the minimization diagrams of F and of G is O(n 2+ε), for any ε>0. This result has several applications: (i) a nearquadratic upper bound on the complexity of the region in 3space enclosed between the lower envelope of one such collection of functions and the upper envelope of another collection; (ii) an efficient and simple divideandconquer algorithm for constructing lower envelopes in three dimensions; and (iii) a nearquadratic upper bound on the complexity of the space of all plane transversals of a collection of simply shaped convex sets in three
SINR diagrams: Towards algorithmically usable SINR models of wireless networks
 In Proc. 28th Symp. on Principles of Distrib. Computing
, 2009
"... The rules governing the availability and quality of connections in a wireless network are described by physical models such as the signaltointerference & noise ratio (SINR) model. For a collection of simultaneously transmitting stations in the plane, it is possible to identify a reception zone ..."
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Cited by 15 (5 self)
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The rules governing the availability and quality of connections in a wireless network are described by physical models such as the signaltointerference & noise ratio (SINR) model. For a collection of simultaneously transmitting stations in the plane, it is possible to identify a reception zone for each station, consisting of the points where its transmission is received correctly. The resulting SINR diagram partitions the plane into a reception zone per station and the remaining plane where no station can be heard. SINR diagrams appear to be fundamental to understanding the behavior of wireless networks, and may play a key role in the development of suitable algorithms for such networks, analogous perhaps to the role played by Voronoi diagrams in the study of proximity queries and related issues in computational geometry. So far, however, the properties of SINR diagrams have not been studied systematically, and most algorithmic studies in wireless networking rely on simplified graphbased models such as the unit disk graph (UDG) model, which conveniently abstract away interferencerelated complications, and make it easier to handle algorithmic issues, but consequently fail to capture accurately some important aspects of wireless networks.
Vertical decomposition of a single cell in a threedimensional arrangement of surfaces and its applications
 Geom
, 1997
"... Let \Sigma be a collection of n algebraic surface patches of ..."
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Cited by 11 (3 self)
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Let \Sigma be a collection of n algebraic surface patches of
Ray Shooting Amidst Spheres in Three Dimensions and Related Problems
 SIAM J. Comput
, 1997
"... We consider the problem of ray shooting amidst spheres in 3space: given n arbitrary (possibly intersecting) spheres in 3space and any " ? 0, we show how to preprocess the spheres in time O(n 3+" ), into a data structure of size O(n 3+" ), so that any rayshooting query can be ..."
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Cited by 11 (4 self)
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We consider the problem of ray shooting amidst spheres in 3space: given n arbitrary (possibly intersecting) spheres in 3space and any " ? 0, we show how to preprocess the spheres in time O(n 3+" ), into a data structure of size O(n 3+" ), so that any rayshooting query can be answered in time O(n " ). Our result improves previous techniques (see [3, 5]), where roughly O(n 4 ) storage was required to support fast queries. Our result shows that ray shooting amidst spheres has complexity comparable with that of ray shooting amidst planes in 3space. Our technique applies to more general (convex) objects in 3space, and we also discuss these extensions. 1 Introduction The ray shooting problem can be defined as follows: Given a collection S of n objects in IR d , preprocess S into a data structure, so that one can quickly determine the first object of S intersected by a query ray. The ray shooting problem has received considerable attention in the past few years beca...
Algebraic Geometry and Computer Vision: Polynomial Systems, Real and Complex Roots
, 1999
"... We review the different techniques known for doing exact computations on polynomial systems. Some are based on the use of Gröbner bases and linear algebra, others on the more classical resultants and its modern counterparts. Many theoretical examples of the use of these techniques are given. Further ..."
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Cited by 11 (0 self)
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We review the different techniques known for doing exact computations on polynomial systems. Some are based on the use of Gröbner bases and linear algebra, others on the more classical resultants and its modern counterparts. Many theoretical examples of the use of these techniques are given. Furthermore, a full set of examples of applications in the domain of artificial vision, where many constraints boil down to polynomial systems, are presented. Emphasis is also put on very recent methods for determining the number of (isolated) real and complex roots of such systems.