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26
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.
Geometric Range Searching
, 1994
"... In geometric range searching, algorithmic problems of the following type are considered: Given an npoint set P in the plane, build a data structure so that, given a query triangle R, the number of points of P lying in R can be determined quickly. Problems of this type are of crucial importance in c ..."
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Cited by 46 (2 self)
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In geometric range searching, algorithmic problems of the following type are considered: Given an npoint set P in the plane, build a data structure so that, given a query triangle R, the number of points of P lying in R can be determined quickly. Problems of this type are of crucial importance in computational geometry, as they can be used as subroutines in many seemingly unrelated algorithms. We present a survey of results and main techniques in this area.
Constructing Levels in Arrangements and Higher Order Voronoi Diagrams
 SIAM J. COMPUT
, 1994
"... We give simple randomized incremental algorithms for computing the klevel in an arrangement of n hyperplanes in two and threedimensional space. The expected running time of our algorithms is O(nk+nff(n) log n) for the planar case, and O(nk 2 +n log 3 n) for the threedimensional case. Both bo ..."
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Cited by 42 (10 self)
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We give simple randomized incremental algorithms for computing the klevel in an arrangement of n hyperplanes in two and threedimensional space. The expected running time of our algorithms is O(nk+nff(n) log n) for the planar case, and O(nk 2 +n log 3 n) for the threedimensional case. Both bounds are optimal unless k is very small. The algorithm generalizes to computing the klevel in an arrangement of discs or xmonotone Jordan curves in the plane. Our approach can also be used to compute the klevel; this yields a randomized algorithm for computing the orderk Voronoi diagram of n points in the plane in expected time O(k(n \Gamma k) log n + n log 3 n).
Primal Dividing and Dual Pruning: OutputSensitive Construction of 4d Polytopes and 3d Voronoi Diagrams
, 1997
"... In this paper, we give an algorithm for outputsensitive construction of an fface convex hull of a set of n points in general position in E 4 . Our algorithm runs in O((n + f)log 2 f) time and uses O(n + f) space. This is the first algorithm within a polylogarithmic factor of optimal O(n log f ..."
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Cited by 31 (3 self)
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In this paper, we give an algorithm for outputsensitive construction of an fface convex hull of a set of n points in general position in E 4 . Our algorithm runs in O((n + f)log 2 f) time and uses O(n + f) space. This is the first algorithm within a polylogarithmic factor of optimal O(n log f + f) time over the whole range of f . By a standard lifting map, we obtain outputsensitive algorithms for the Voronoi diagram or Delaunay triangulation in E 3 and for the portion of a Voronoi diagram that is clipped to a convex polytope. Our approach simplifies the "ultimate convex hull algorithm" of Kirkpatrick and Seidel in E 2 and also leads to improved outputsensitive results on constructing convex hulls in E d for any even constant d ? 4. 1 Introduction Geometric structures induced by n points in Euclidean ddimensional space, such as the convex hull, Voronoi diagram, or Delaunay triangulation, can be of larger size than the point set that defines them. In many practical situat...
Randomized ExternalMemory Algorithms for Some Geometric Problems
 INTERNATIONAL JOURNAL OF COMPUTATIONAL GEOMETRY & APPLICATIONS
, 2001
"... We show that the wellknown random incremental construction of Clarkson and Shor [14] can be adapted via gradations to provide efficient externalmemory algorithms for some geometric problems. In particular, as the main result, we obtain an optimal randomized algorithm for the problem of computin ..."
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Cited by 27 (3 self)
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We show that the wellknown random incremental construction of Clarkson and Shor [14] can be adapted via gradations to provide efficient externalmemory algorithms for some geometric problems. In particular, as the main result, we obtain an optimal randomized algorithm for the problem of computing the trapezoidal decomposition determined by a set of N line segments in the plane with K pairwise intersections, that requires \Theta( log M=B N ) expected disk accesses, where M is the size of the available internal memory and B is the size of the block transfer. The approach is sufficiently general to obtain algorithms also for the problems of 3d halfspace intersections, 2d and 3d convex hulls, 2d abstract Voronoi diagrams and batched planar point location, which require an optimal expected number of disk accesses and are simpler than the ones previously known. The results extend to an externalmemory model with multiple disks. Additionally, under reasonable conditions on the parameters N;M;B, these results can be notably simplified originating practical algorithms which still achieve optimal expected bounds.
On Range Reporting, Ray Shooting and klevel Construction
"... We describe the following data structures. For halfspace range reporting, in 3space using expected preprocessing time O(n log n), worstcase storage O(n log log n) and worstcase reporting time O(log n + k), where n is the number of data points and k the number of points reported; in dspace, with ..."
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Cited by 24 (0 self)
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We describe the following data structures. For halfspace range reporting, in 3space using expected preprocessing time O(n log n), worstcase storage O(n log log n) and worstcase reporting time O(log n + k), where n is the number of data points and k the number of points reported; in dspace, with d even, using worstcase preprocessing time O(n log n), storage O(n) and reporting time O(n 1 1=bd=2c log c n + k), where c is a constant. For ray shooting in a convex polytope in dspace determined by n facets, using deterministic preprocessing time O((n= log n) bd=2c log c n) and storage O((n= log n) bd=2c 2 c log n ) and with query time O(log n). For ray shooting in arbitrary direction among n hyperplanes using preprocessing O(n d = log bd=2c n) and query time O(log n). We also describe a randomized algorithm for constructing the klevel of n planes in 3space. In the case of planes dual to points in convex position, in which the size of the klevel is O(nk), the a...
Methods for Achieving Fast Query Times in Point Location Data Structures
, 1997
"... Given a collection S of n line segments in the plane, the planar point location problem is to construct a data structure that can efficiently determine for a given query point p the first segment(s) in S intersected by vertical rays emanating out from p. It is well known that linearspace data struc ..."
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Cited by 20 (1 self)
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Given a collection S of n line segments in the plane, the planar point location problem is to construct a data structure that can efficiently determine for a given query point p the first segment(s) in S intersected by vertical rays emanating out from p. It is well known that linearspace data structures can be constructed so as to achieve O(log n) query times. But applications, such as those common in geographic information systems, motivate a reexamination of this problem with the goal of improving query times further while also simplifying the methods needed to achieve such query times. In this paper we perform such a reexamination, focusing on the issues that arise in three different classes of pointlocation query sequences: ffl sequences that are reasonably uniform spatially and temporally (in which case the constant factors in the query times become critical), ffl sequences that are nonuniform spatially or temporally (in which case one desires data structures that adapt to s...
Derandomization in Computational Geometry
, 1996
"... We survey techniques for replacing randomized algorithms in computational geometry by deterministic ones with a similar asymptotic running time. 1 Randomized algorithms and derandomization A rapid growth of knowledge about randomized algorithms stimulates research in derandomization, that is, repla ..."
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Cited by 17 (1 self)
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We survey techniques for replacing randomized algorithms in computational geometry by deterministic ones with a similar asymptotic running time. 1 Randomized algorithms and derandomization A rapid growth of knowledge about randomized algorithms stimulates research in derandomization, that is, replacing randomized algorithms by deterministic ones with as small decrease of efficiency as possible. Related to the problem of derandomization is the question of reducing the amount of random bits needed by a randomized algorithm while retaining its efficiency; the derandomization can be viewed as an ultimate case. Randomized algorithms are also related to probabilistic proofs and constructions in combinatorics (which came first historically), whose development has similarly been accompanied by the effort to replace them by explicit, nonrandom constructions whenever possible. Derandomization of algorithms can be seen as a part of an effort to map the power of randomness and explain its role. ...
Construction of 1D Lower Envelopes and Applications
"... We consider the problem of computing the lower envelope (the minimum) of n constant degree algebraic functions of one variable. The lower envelope has size O(nfi(n)) where fi(n) is a nearly constant function, and it can easily be computed in time O(nfi(n) log n) by a simple deterministic divideand ..."
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Cited by 13 (0 self)
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We consider the problem of computing the lower envelope (the minimum) of n constant degree algebraic functions of one variable. The lower envelope has size O(nfi(n)) where fi(n) is a nearly constant function, and it can easily be computed in time O(nfi(n) log n) by a simple deterministic divideandconquer algorithm [45]. We give an alternative simple (module a derandomization black box) approach using divideandconquer based on cuttings that results in a deterministic sequential algorithm that runs in the same time bound. This algorithm uses derandomization tools by now standard. This approach however allows us to obtain the following results: ffl A deterministic sequential algorithm that is output sensitive and runs in time O(n log f) if f n ffl , or O(nfi(f) log f) = O(nfi(n) log n) otherwise, where f is the size of the output; ffl a randomized parallel EREW algorithm that runs in time O(log n) and uses nearly optimal work O(nfi 2 (n) log n) with npolynomial probability...