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BoundedIndependence Derandomization of Geometric Partitioning with Applications to Parallel FixedDimensional Linear Programming
"... We give fast and efficient methods for constructing... time using linear work on an EREW PRAM. ..."
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We give fast and efficient methods for constructing... time using linear work on an EREW PRAM.
Solving some discrepancy problems in NC
, 1997
"... We show that several discrepancylike problems can be solved in NC 2 nearly achieving the corresponding sequential bounds. For example, given a set system (X; S), where X is a ground set and S ` 2 X , a set R ` X can be computed in NC 2 so that, for each S 2 S, the discrepancy jjR " Sj \ ..."
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We show that several discrepancylike problems can be solved in NC 2 nearly achieving the corresponding sequential bounds. For example, given a set system (X; S), where X is a ground set and S ` 2 X , a set R ` X can be computed in NC 2 so that, for each S 2 S, the discrepancy jjR " Sj \Gamma jR " Sjj is O( p jSj log jSj). Previous NC algorithms could only achieve O( p jSj 1+ffl log jSj), while ours matches the probabilistic bound achieved sequentially by the method of conditional probabilities within a multiplicative factor 1 + o(1). Other problems whose NC solution we improve are lattice approximation, fflapproximations of range spaces of bounded VCexponent, sampling in geometric configuration spaces, and approximation of integer linear programs. 1 Introduction Problem and previous work. Discrepancy is an important concept in combinatorics, see e.g. [1, 5], and theoretical computer science, see e.g. [27, 23, 9]. It attempts to capture the idea of a good sample from ...
Geometric Set Systems
, 1998
"... Let X be a finite point set in the plane. We consider the set system on X whose sets are all intersections of X with a halfplane. Similarly one can investigate set systems defined on point sets in higherdimensional spaces by other classes of simple geometric figures (simplices, balls, ellipsoids, e ..."
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Let X be a finite point set in the plane. We consider the set system on X whose sets are all intersections of X with a halfplane. Similarly one can investigate set systems defined on point sets in higherdimensional spaces by other classes of simple geometric figures (simplices, balls, ellipsoids, etc.). It turns out that simple combinatorial properties of such set systems (most notably the VapnikChervonenkis dimension and related concepts of shatter functions) play an important role in several areas of mathematics and theoretical computer science. Here we concentrate on applications in discrepancy theory, in combinatorial geometry, in derandomization of geometric algorithms, and in geometric range searching. We believe that the described tools might be useful in other areas of mathematics too. 1 Introduction For a set system S ` 2 X on an arbitrary ground set X and for A ` X, we write Sj A = fS " A; S 2 Sg for the set system induced by S on A (or the trace of S on A). Let H den...
Decision Tree Construction in Fixed Dimensions: Being Global is Hard but Local Greed is Good
, 1995
"... We study the problem of finding optimal linear decision trees for classifying a set of points in R^d partitioned into concept classes, where d is a fixed, but arbitrary, constant. We show that optimal decision tree construction is NPcomplete, even for 3dimensional point sets. Nevertheless, we can ..."
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We study the problem of finding optimal linear decision trees for classifying a set of points in R^d partitioned into concept classes, where d is a fixed, but arbitrary, constant. We show that optimal decision tree construction is NPcomplete, even for 3dimensional point sets. Nevertheless, we can prove a number of interesting approximation bounds on the use of random sampling for finding optimal splitting hyperplanes in greedy decision tree constructions. We give experimental evidence that, while providing asymptotic guarantees on split quality, this random sampling approach behaves as good in practice as uniform randomization strategies that do not provide such guarantees. Finally, we provide experimental justification for coupling this random sampling strategy with locallygreedy "hill climbing" methods.
εNets for Halfspaces Revisited∗
, 2014
"... “It is a damn poor mind indeed which can’t think of at least two ways to spell any word.” – Andrew Jackson Given a set P of n points in R3, we show that, for any ε> 0, there exists an εnet of P for halfspace ranges, of size O(1/ε). We give five proofs of this result, which are arguably simpler t ..."
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“It is a damn poor mind indeed which can’t think of at least two ways to spell any word.” – Andrew Jackson Given a set P of n points in R3, we show that, for any ε> 0, there exists an εnet of P for halfspace ranges, of size O(1/ε). We give five proofs of this result, which are arguably simpler than previous proofs [?,?,?]. We also consider several related variants of this result, including the case of points and pseudodisks in the plane. 1
Dynamic Coresets
, 2008
"... Abstract We give a dynamic data structure that can maintain an "coreset of n points, with respect to the extent measure, in O(log n) time for any constant " ? 0 and any constant dimension. The previous method by Agarwal, HarPeled, and Varadarajan requires polylogarithmic update t ..."
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Abstract We give a dynamic data structure that can maintain an &quot;coreset of n points, with respect to the extent measure, in O(log n) time for any constant &quot; ? 0 and any constant dimension. The previous method by Agarwal, HarPeled, and Varadarajan requires polylogarithmic update time. For points with integer coordinates bounded by U, we alternatively get O(log log U) time. Numerous applications follow, for example, on dynamically approximating the width, smallest enclosing cylinder, minimum bounding box, or minimumwidth annulus. We can also use the same approach to maintain approximate kcenters in O(minflog n; log log U g) randomized amortized time for any constant k and any constant dimension. For the smallest enclosing cylinder problem, we also show that a constantfactor approximation can be maintained in O(1) randomized amortized time on the word RAM.
Computing Faces in Segment and Simplex Arrangements (Preliminary Version)
"... 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(log2 n) time using O(n logn + 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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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(log2 n) time using O(n logn + 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 inO(n2(n) logn) time, which improves on a previous O(n log2 n) time algorithm. For collections of simplices in IRd, we give methods for constructing a set ofm = O(nd1 logc 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(logn) time using O(m) work in the EREW PRAM model. The covering can be augmented to answer point location queries in O(logn) 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(log2 n) in IR3 and O(log3 n) in IRd, d> 3), and also the size and construction cost of the covering (from O(nd1+ + k)). 1
MultiPass Geometric Algorithms \Lambda
, 2006
"... Abstract We propose the study of exact geometric algorithms that require limited storage and make only a small number of passes over the input. Fundamental problems such as lowdimensional linear programming and convex hulls are considered. 1 Introduction The multipass model. Streaming algorithms t ..."
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Abstract We propose the study of exact geometric algorithms that require limited storage and make only a small number of passes over the input. Fundamental problems such as lowdimensional linear programming and convex hulls are considered. 1 Introduction The multipass model. Streaming algorithms that make a single pass over the input and work with a small amount of space have grown in popularity [31], because of the ability of such algorithms to handle massive data sets. Since only one pass over the input is required, data elements may arrive one at a time and the entire data set never needs to be physically stored. Study of geometric algorithms in the datastream model has already begun to take place in several recent papers (e.g., [3, 10, 37]). In this paper, we examine a more powerful multipass model, where algorithms are allowed to make multiple passes over the input. The input remains unchanged after each pass, and depending on the problem, the answer may be sent to a writeonly output stream. The goal is to minimize the amount of working space (measured in words, in this paper), while keeping the number of passes small. As usual, we would like to bound the total running time as well (which includes the cost of scanning and is at least the input size times the number of passes); for this purpose, we assume unitcost random access for the working space (but not the input, of course).