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18
Efficient algorithms for geometric optimization
 ACM Comput. Surv
, 1998
"... We review the recent progress in the design of efficient algorithms for various problems in geometric optimization. We present several techniques used to attack these problems, such as parametric searching, geometric alternatives to parametric searching, pruneandsearch techniques for linear progra ..."
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Cited by 94 (12 self)
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We review the recent progress in the design of efficient algorithms for various problems in geometric optimization. We present several techniques used to attack these problems, such as parametric searching, geometric alternatives to parametric searching, pruneandsearch techniques for linear programming and related problems, and LPtype problems and their efficient solution. We then describe a variety of applications of these and other techniques to numerous problems in geometric optimization, including facility location, proximity problems, statistical estimators and metrology, placement and intersection of polygons and polyhedra, and ray shooting and other querytype problems.
Parallel Algorithms for HigherDimensional Convex Hulls
"... We give fast randomized and deterministic parallel methods for constructing convex hulls in R^d, for any fixed d. Our methods are for the weakest sharedmemory model,the EREW PRAM, and have optimal work bounds (with high probability for the randomized methods). In particular, we show that the convex ..."
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Cited by 44 (14 self)
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We give fast randomized and deterministic parallel methods for constructing convex hulls in R^d, for any fixed d. Our methods are for the weakest sharedmemory model,the EREW PRAM, and have optimal work bounds (with high probability for the randomized methods). In particular, we show that the convex hull of n points in R^d can be constructed in O(log n) time using O(n log n + nbd=2c) work, with high probability. We also show that it can be constructed deterministically in O(log² n) time using O(n log n) work for d = 3 and in O(log n) time using O(nbd=2c logc(dd=2e\Gamma bd=2c) n) work, for d * 4, where c? 0is a constant, which is optimal for even d * 4. We also showhow to make our 3dimensional methods outputsensitive with only a small increase in running time.These methods can be applied to other problems as well. A variation of the convex hull algorithm for even dimensions deterministically constructs a (1=r)cutting of n hyperplanes in IR d in O(log n) time using optimal O(nrd\Gamma 1) work; when r = n, we obtain their arrangement and a pointlocation data structure for it. With appropriate modifications, our deterministic 3dimensional convex hull algorithmcan be used to compute, in the same resource bounds, the intersection of n balls of equal radius in R³. This leads to asequential algorithm for computing the diameter of a point set in IR3 with running time O(n log³ n), which is arguably simpler than an algorithm with the same running time by Brönnimann et al.
An ExpanderBased Approach to Geometric Optimization
 IN PROC. 9TH ANNU. ACM SYMPOS. COMPUT. GEOM
, 1993
"... We present a new approach to problems in geometric optimization that are traditionally solved using the parametric searching technique of Megiddo [34]. Our new approach ..."
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Cited by 40 (16 self)
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We present a new approach to problems in geometric optimization that are traditionally solved using the parametric searching technique of Megiddo [34]. Our new approach
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. ...
Computing faces in segment and simplex arrangements
 In Proc. 27th Annu. ACM Sympos. Theory Comput
, 1995
"... For a set S of n line segments in the plane, we give the first workoptimal deterministic parallel algorithm for constructing their arrangement. It runs in O(log 2 n) time using O(n log n + k) work in the EREW PRAM model, where k is the number of intersecting line segment pairs, and provides a fairl ..."
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Cited by 17 (11 self)
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For a set S of n line segments in the plane, we give the first workoptimal deterministic parallel algorithm for constructing their arrangement. It runs in O(log 2 n) time using O(n log n + k) work in the EREW PRAM model, where k is the number of intersecting line segment pairs, and provides a fairly simple divideandconquer alternative to the optimal sequential “planesweep ” algorithm of Chazelle and Edelsbrunner. Moreover, our method can be used to output all k intersecting pairs while using only O(n) working space, which solves an open problem posed by Chazelle and Edelsbrunner. We also describe a sequential algorithm for computing a single face in an arrangement of n line segments that runs in O(n 2 (n) log n) time, which improves on a previous O(n log 2 n) time algorithm. For collections of simplices in IR d, we give methods for constructing a set of m = O(n d,1 log c n+k) cells of constant descriptive complexity that covers their arrangement, where c> 1 is a constant and k is the number of faces in the arrangement. The construction is performed sequentially in O(m) time, or in O(log n) time using O(m) work in the EREW PRAM model. The covering can be augmented to answer point location queries in O(log n) time. In addition to supplying the first parallel methods for these problems, we improve on the previous best sequential methods by reducing the query times (from O(log 2 n) in IR 3 and O(log 3 n) in IR d, d>3), and also the size and construction cost of the covering (from O(n d,1+ + k)). 1
On computing Voronoi diagrams by dividepruneandconquer
 IN PROC. 12TH ANNUAL ACM SYMPOS. COMPUT. GEOM
, 1996
"... Using a divide, prune, and conquer approach based on geometric partitioning, we obtain: (1) An output sensitive algorithm for computing a weighted Voronoi diagram in R 4 (the projection of certain polyhedra in R 5) that runs in time O((n+f) log³ f) where n is the number of sites and f is the number ..."
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Cited by 14 (3 self)
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Using a divide, prune, and conquer approach based on geometric partitioning, we obtain: (1) An output sensitive algorithm for computing a weighted Voronoi diagram in R 4 (the projection of certain polyhedra in R 5) that runs in time O((n+f) log³ f) where n is the number of sites and f is the number of output cells; and (2) a deterministic parallel algorithm in the EREW PRAM model for computing an algebraic planar Voronoi diagram (in which bisectors between sites are simple curves consisting of a constant number of algebraic pieces of constant degree) that runs in time O(log² n) using optimal O(n log n) work. The first result implies an algorithm for the problems of computing the convex hull of a point set and the intersection of a set of halfspaces in R 5, and computing the Euclidean Voronoi diagram in R 4. The second result implies both sequential and parallel workoptimal deterministic algorithms for a number of Voronoi diagram problems (including line segments in the plane), and other nonVoronoi diagram problems that can fit in the framework (including the intersection of equal radius balls in R³ and some lower envelope problems in R³).
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...