ź-nets and simplex range queries
TL;DR: The concept of an ɛ-net of a set of points for an abstract set of ranges is introduced and sufficient conditions that a random sample is an Â-net with any desired probability are given.
Abstract: We demonstrate the existence of data structures for half-space and simplex range queries on finite point sets ind-dimensional space,dÂ?2, with linear storage andO(nÂ?) query time, $$\alpha = \frac{{d(d - 1)}}{{d(d - 1) + 1}} + \gamma for all \gamma > 0$$ .
These bounds are better than those previously published for alldÂ?2. Based on ideas due to Vapnik and Chervonenkis, we introduce the concept of an Â?-net of a set of points for an abstract set of ranges and give sufficient conditions that a random sample is an Â?-net with any desired probability. Using these results, we demonstrate how random samples can be used to build a partition-tree structure that achieves the above query time.
Citations
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20 Jun 1995
TL;DR: A variety of combinatorial and computational results are derived on the VC (Vapnik-Chervonenkis) dimension of set systems induced by special graph properties like clique, connectedness, path, star, tree, etc.
Abstract: We study set systems over the vertex set (or edge set) of some graph that are induced by special graph properties like clique, connectedness, path, star, tree, etc. We derive a variety of combinatorial and computational results on the VC (Vapnik-Chervonenkis) dimension of these set systems.
3 citations
Cites background from "ź-nets and simplex range queries"
...The notion of VC-dimension for neighborhoods was introduced by ttaussler and Welzl [ 6 ]....
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...Welzl [ 6 ] introduced the VC-dimension of a graph as an example in their study of simplex-range queries with epsilon nets....
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06 Jul 2016
TL;DR: This paper proposes a new algorithm termed a weighted VBGMM via Coreset, a new coreset construction method is first proposed to sample the data which is used to fit the model, and results show that the proposed algorithm is much faster comparing to classic V BGMM while maintaining the similar performance on whole dataset.
Abstract: Variational Bayesian Gaussian Mixture Model is a popular clustering algorithm with a reliable performance. However, it is noted that the model fitting process takes long time, especially when dealing with large scale data, since it utilizes the whole dataset. To address this issue, in paper we propose a new algorithm termed a weighted VBGMM via Coreset. Specifically, a new coreset construction method is first proposed to sample the data which is used to fit the model. To evaluate the algorithm, two datasets are used: 1) six rat kidney images datasets 2) three human kidney images datasets. The results show that our proposed algorithm is much faster (∼ 20 times) comparing to classic VBGMM while maintaining the similar performance on whole dataset.
3 citations
TL;DR: An optimal parallel randomized algorithm for the Voronoi diagram of a set of n nonintersecting line segments in the plane with high probability using O(n) processors on a CRCW PRAM is presented.
Abstract: . We present an optimal parallel randomized algorithm for the Voronoi diagram of a set of n nonintersecting (except possibly at endpoints) line segments in the plane. Our algorithm runs in O(log n) time with high probability using O(n) processors on a CRCW PRAM. This algorithm is optimal in terms of work done since the sequential time bound for this problem is Ω(n log n) . Our algorithm improves by an O(log n) factor the previously best known deterministic parallel algorithm, given by Goodrich, O'Dunlaing, and Yap, which runs in O( log 2
n) time using O(n) processors. We obtain this result by using a new ``two-stage'' random sampling technique. By choosing large samples in the first stage of the algorithm, we avoid the hurdle of problem-size ``blow-up'' that is typical in recursive parallel geometric algorithms. We combine the two-stage sampling technique with efficient search and merge procedures to obtain an optimal algorithm. This technique gives an alternative optimal algorithm for the Voronoi diagram of points as well (all other optimal parallel algorithms for this problem use the transformation to three-dimensional half-space intersection).
3 citations
Cites methods from "ź-nets and simplex range queries"
...Random sampling is used to obtain faster algorithms for various geometric problems such as higher-dimensional convex hulls, half-space range reporting, segment intersections, and linear programming; see, for example, [10], [28], and [20]....
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Posted Content•
TL;DR: It is shown that for any set of n moving points in R d and any parameter 1 < k < n, one can select a non-empty subset of the points of size O(k logk) such that each cell of the Voronoi diagram of this subset is \balanced" at any given time.
Abstract: We study geometric hypergraphs in a kinetic setting and show that for many of the static cases where the VC-dimension of the hypergraph is bounded the kinetic counterpart also has bounded VC-dimension. Among other results we show that for any set of n moving points in R d and any parameter 1 < k < n, one can select a non-empty subset of the points of size O(k logk) such that each cell of the Voronoi diagram of this subset is \balanced" at any given time (i.e., it contains O(n=k) of the other points). We also show that the bound is near optimal even for the onedimensional case in which points move linearly.
3 citations
TL;DR: In this article, a data structure for point location in the zone of a k-flat is proposed, which uses O(n ⌊ d 2 ⌋+e +n k+e ) preprocessing time and space and query time of O(log2 n).
Abstract: Let A(H) be the arrangement of a set H of n hyperplanes in d-space. A k-flat is a k-dimensional affine subspace of d-space. The zone of a k-flat f with respect to H is the set of all faces in A(H) that intersect f. this paper we study some problems on zones of k-flats. Our most important result is a data structure for point location in the zone of a k-flat. This structure uses O (n ⌊ d 2 ⌋+e +n k+e ) preprocessing time and space and has a query time of O(log2 n). We also show how to test efficiently whether two flats are visible from each other with respect to a set of hyperplanes. Then point location in m faces in arrangements is studied. Our data structure for this problem has size O (n ⌊ d 2 ⌋+e m ⌈ d 2 ⌉ d ) and the query time is O(log2 n).
3 citations
References
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TL;DR: This chapter reproduces the English translation by B. Seckler of the paper by Vapnik and Chervonenkis in which they gave proofs for the innovative results they had obtained in a draft form in July 1966 and announced in 1968 in their note in Soviet Mathematics Doklady.
Abstract: This chapter reproduces the English translation by B. Seckler of the paper by Vapnik and Chervonenkis in which they gave proofs for the innovative results they had obtained in a draft form in July 1966 and announced in 1968 in their note in Soviet Mathematics Doklady. The paper was first published in Russian as Вапник В. Н. and Червоненкис А. Я. О равномерноЙ сходимости частот появления событиЙ к их вероятностям. Теория вероятностеЙ и ее применения 16(2), 264–279 (1971).
3,939 citations
"ź-nets and simplex range queries" refers background or methods or result in this paper
...The drawback is that the constants, if deri~,ed from the results in [ 17 ], can be quite large....
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...More generally, we characterize the classes of ranges for which there exists a function f(E) for e S0 such that any finite point set A has an e-net of size f(e), independently of the size of A. These are precisely the classes of ranges with finite Vapnik-Chervonenkis dimension, known as Vapnik-Chervonenkis classes [ 17 ], [9], [19], [1]....
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...The key concepts and proof techniques of this section are based on the pioneering work of Vapnik and Chervonenkis [ 17 ]....
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...Example 5. Let A be a set of n points in E 2. Since the dimension of (E 2, H~-) is 2, the results in [ 17, Theorem 2 ] show that there exists a 0.01-approximation V of A for positive half-planes (and thus for all half-planes) with I VI = 2,525,039....
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...Using the related notion of an e-approxirnation (directly from [ 17 ]), we also point out trivial data structures of constant size that give approximate solutions to the counting problem for halfspaces in constant time (compare [13])....
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Book•
01 Jan 1987TL;DR: This book offers a modern approach to computational geo- metry, an area thatstudies the computational complexity of geometric problems with an important role in this study.
Abstract: This book offers a modern approach to computational geo- metry, an area thatstudies the computational complexity of geometric problems. Combinatorial investigations play an important role in this study.
2,284 citations
"ź-nets and simplex range queries" refers background in this paper
...We conclude this section by examining the relationship between the notion of an e-net and the established notion of a centerpoint [21], [11] in combinatorial geometry....
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..., [11] for a general treatment of arrangements....
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TL;DR: This paper will answer the question in the affirmative by determining the exact upper bound of T if T is a family of subsets of some infinite set S then either there exists to each number n a set A ⊂ S with |A| = n such that |T ∩ A| = 2n or there exists some number N such that •A| c for each A⩾ N and some constant c.
1,029 citations
"ź-nets and simplex range queries" refers background in this paper
...Now the assertion can be seen as the dual formulation of Caratheodry's theorem (see [ 15 ], Theorem 2.3.5), which states that if a point x is in the convex hull of a set A in E d, then there exists a subset A' of A such that JA'I -< d + 1 and x is in the convex hull of A'. []...
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TL;DR: In this article, the convergence of a stochastic process indexed by a Gaussian process to a certain Gaussian processes indexed by the supremum norm was studied in a Donsker class.
Abstract: Let $(X, \mathscr{A}, P)$ be a probability space. Let $X_1, X_2,\cdots,$ be independent $X$-valued random variables with distribution $P$. Let $P_n := n^{-1}(\delta_{X_1} + \cdots + \delta_{X_n})$ be the empirical measure and let $
u_n := n^\frac{1}{2}(P_n - P)$. Given a class $\mathscr{C} \subset \mathscr{a}$, we study the convergence in law of $
u_n$, as a stochastic process indexed by $\mathscr{C}$, to a certain Gaussian process indexed by $\mathscr{C}$. If convergence holds with respect to the supremum norm $\sup_{C \in \mathscr{C}}|f(C)|$, in a suitable (usually nonseparable) function space, we call $\mathscr{C}$ a Donsker class. For measurability, $X$ may be a complete separable metric space, $\mathscr{a} =$ Borel sets, and $\mathscr{C}$ a suitable collection of closed sets or open sets. Then for the Donsker property it suffices that for some $m$, and every set $F \subset X$ with $m$ elements, $\mathscr{C}$ does not cut all subsets of $F$ (Vapnik-Cervonenkis classes). Another sufficient condition is based on metric entropy with inclusion. If $\mathscr{C}$ is a sequence $\{C_m\}$ independent for $P$, then $\mathscr{C}$ is a Donsker class if and only if for some $r, \sigma_m(P(C_m)(1 - P(C_m)))^r < \infty$.
555 citations
TL;DR: A new formulation of the notion of duality that allows the unified treatment of a number of geometric problems is used, to solve two long-standing problems of computational geometry and to obtain a quadratic algorithm for computing the minimum-area triangle with vertices chosen amongn points in the plane.
Abstract: This paper uses a new formulation of the notion of duality that allows the unified treatment of a number of geometric problems. In particular, we are able to apply our approach to solve two long-standing problems of computational geometry: one is to obtain a quadratic algorithm for computing the minimum-area triangle with vertices chosen amongn points in the plane; the other is to produce an optimal algorithm for the half-plane range query problem. This problem is to preprocessn points in the plane, so that given a test half-plane, one can efficiently determine all points lying in the half-plane. We describe an optimalO(k + logn) time algorithm for answering such queries, wherek is the number of points to be reported. The algorithm requiresO(n) space andO(n logn) preprocessing time. Both of these results represent significant improvements over the best methods previously known. In addition, we give a number of new combinatorial results related to the computation of line arrangements.
286 citations
"ź-nets and simplex range queries" refers methods in this paper
...It should be noted that better bounds are possible for reporting in two dimensions (specifically O(log n + t) time, where t is the number of points reported [3]), but these techniques only work for half-planes....
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