Derandomization in Computational Geometry

Matoušek, Jiřı́

doi:10.1006/jagm.1996.0027

Cited by 28 publications

(25 citation statements)

References 123 publications

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“…The general VC-dimension theory only guarantees the existence of 1/r-nets of size O(r log r), where the constant factor is a fast-growing function of the dimension (see e.g. [15,10,11]). …”

Section: Lemma 2 Let a Be An -Approximation To X Let B Be A δ-Appromentioning

confidence: 99%

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Parallel Convex Hull Computation by Generalised Regular Sampling

Tiskin

2002

Euro-Par 2002 Parallel Processing

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Abstract. The model of bulk-synchronous parallel (BSP) computation is an emerging paradigm of general-purpose parallel computing. We propose the first optimal deterministic BSP algorithm for computing the convex hull of a set of points in three-dimensional Euclidean space. Our algorithm is based on known fundamental results from combinatorial geometry, concerning small-sized, efficiently constructible -nets and -approximations of a given point set. The algorithm generalises the technique of regular sampling, used previously for sorting and twodimensional convex hull computation. The cost of the simple algorithm is optimal only for extremely large inputs; we show how to reduce the required input size by applying regular sampling in a multi-level fashion.

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“…The general VC-dimension theory only guarantees the existence of 1/r-nets of size O(r log r), where the constant factor is a fast-growing function of the dimension (see e.g. [15,10,11]). …”

Section: Lemma 2 Let a Be An -Approximation To X Let B Be A δ-Appromentioning

confidence: 99%

“…A suitable concept here is that of -net (see e.g. [15,10,11]). As before, let X be a finite set of points in R d , and an arbitrary real number, 0 ≤ ≤ 1.…”

Section: Theorem 1 Let X Be a Set Of N Points Inmentioning

confidence: 99%

Parallel Convex Hull Computation by Generalised Regular Sampling

Tiskin

2002

Euro-Par 2002 Parallel Processing

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“…A final technicality is the proof of appropriate sampling bounds for the sizes of the chain-conflict lists of our conformal decomposition: such bounds are known under locality or monotonicity properties that our decomposition does not satisfy [1,9,11,22,24]. Fortunately, we can prove appropriate bounds using the fact that, although the faces in the decomposition do not satisfy a locality property, they are chosen from a relatively small "pool" of candidates that satisfy a locality property.…”

Section: O U R R E S U L T Smentioning

confidence: 99%

“…(2). Such a bound can be proved in the framework of configuration spaces, when certain locality [9,24] or monotonicity [11,1] properties hold for the decomposition induced by the sample (see [22] for a survey), but neither of these properties hold for our chain-trapezoidation. Fortunately, we can prove a weaker bound that is only a factor O(f(log A)) larger, and that suffices to verify that our algorithm has expected linear running time.…”

Section: Sampling Boundsmentioning

confidence: 99%

Linear-time triangulation of a simple polygon made easier via randomization

Amato

Goodrich

Ramos

2000

Proceedings of the Sixteenth Annual Symposium on Computational Geometry

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We describe a randomized algorithm for computing the trapezoidal decomposition of a simple polygon. Its expected running time is linear in the size of the polygon. By a well-known and simple linear time reduction, this implies a linear time algorithm for triangulating a simple polygon. Our algorithm is considerably simpler than Chazelle's (1991) celebrated optimal deterministic algorithm and, hence, positively answers his question of whether a simpler randomized algorithm for the problem exists. The new algorithm can be viewed as a combination of Chazelle's algorithm and of non-optimal randomized algorithms due to Clarkson et al. (1991) and to Seidel (1991), with the essential innovation that sampling is performed on subchains of the initial polygonal chain, rather than on its edges. It is also essential, as in Chazelle's algorithm, to include a bottom-up preprocessing phase previous to the top-down construction phase.

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“…We also give a randomized algorithm that computes, for an input of n disjoint disks, a 1 r -cutting of size O(r 2 ) in polynomial expected time. It can be derandomized with standard techniques [23].…”

Section: Introductionmentioning

confidence: 99%

Cuttings for Disks and Axis-Aligned Rectangles in Three-Space

Rafalin

Souvaine

Tóth

2009

Discrete Comput Geom

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We present new asymptotically tight bounds on cuttings, a fundamental data structure in computational geometry. For n objects in space and a parameter r ∈ N, a 1 r -cutting is a covering of the space with simplices such that the interior of each simplex intersects at most n/r objects. For n pairwise disjoint disks in R 3 and a parameter r ∈ N, we construct a 1 r -cutting of size O(r 2 ). For n axis-aligned rectangles in R 3 , we construct a 1 r -cutting of size O(r 3/2 ). As an application related to multi-point location in three-space, we present tight bounds on the cost of spanning trees across barriers. Given n points and a finite set of disjoint disk barriers in R 3 , the points can be connected with a straight line spanning tree such that every disk is stabbed by at most O( √ n) edges of the tree. If the barriers are axis-aligned rectangles, then there is a straight line spanning tree such that every rectangle is stabbed by O(n 1/3 ) edges. Both bounds are best possible.

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Derandomization in Computational Geometry

Cited by 28 publications

References 123 publications

Parallel Convex Hull Computation by Generalised Regular Sampling

Parallel Convex Hull Computation by Generalised Regular Sampling

Linear-time triangulation of a simple polygon made easier via randomization

Cuttings for Disks and Axis-Aligned Rectangles in Three-Space

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