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.
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31 Oct 2005
TL;DR: It is shown that location-dependent queries may be answered satisfactorily using locale covers, with small loss of accuracy, independent of locations and speeds of clients, and is applicable to mobile clients.
Abstract: Location-dependent data are central to many emerging applications, ranging from traffic information services to sensor networks. The standard pull- and push-based data dissemination models become unworkable since the data volumes and number of clients are high.We address this problem using locale covers, a subset of the original set of locations of interest, chosen to include at least one location in a suitably defined neighborhood of any client. Since location-dependent values are highly correlated with location, a query can be answered using a location close to the query point.We show that location-dependent queries may be answered satisfactorily using locale covers, with small loss of accuracy. Our approach is independent of locations and speeds of clients, and is applicable to mobile clients.
3 citations
19 Dec 2005
TL;DR: This paper studies the spectrum of the dilation, over all pairs of vertices of G, and presents deterministic algorithms that approximate the number of vertex pairs of dilation > κ up to an 1+η factor.
Abstract: Let G be a graph embedded in Euclidean space. For any two vertices of G their dilation denotes the length of a shortest connecting path in G, divided by their Euclidean distance. In this paper we study the spectrum of the dilation, over all pairs of vertices of G. For paths, trees, and cycles in 2D we present O(n3/2+e) randomized algorithms that compute, for a given value κ ≥ 1, the exact number of vertex pairs of dilation > κ. Then we present deterministic algorithms that approximate the number of vertex pairs of dilation > κ up to an 1+η factor. They run in time O(n log2n) for chains and cycles, and in time O(n log3n) for trees, in any constant dimension.
3 citations
Cites background from "ź-nets and simplex range queries"
...random sampling [ 8 ], m¢ • m=r with high probability, for all ¢. If this condition is not satisfled for the sample R, we pick a new random sample....
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01 Jan 1991
TL;DR: A data structure for point location in the zone of a k-flat using O preprocessing time and space and a query time of O(log2 n) is presented.
3 citations
TL;DR: Past accomplishments of Computational Geometry are reviewed and future directions adumbrated.
Abstract: Past accomplishments of Computational Geometry are reviewed and future directions adumbrated.
3 citations
14 Aug 1991
TL;DR: Using these procedures as building blocks, the algorithm can compute a Euclidean maximum spanning tree or a minimum-diameter two-partition of n points in ɛ3 in O(n4/3log7/3n) expected time.
Abstract: We present a randomized algorithm of expected time complexity O(m2/3n2/3log4/3m+mlog2m+nlog2n) for computing bi-chromatic farthest neighbors between n red points and m blue points in ɛ3. The algorithm can also be used to compute all farthest neighbors or external farthest neighbors of n points in ɛ3 in O(n4/3log4/3n) expected time. Using these procedures as building blocks, we can compute a Euclidean maximum spanning tree or a minimum-diameter two-partition of n points in ɛ3 in O(n4/3log7/3n) expected time. The previous best bound for these problems was O(n3/2log1/2n). Our algorithms can be extended to higher dimensions.
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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