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Journal ArticleDOI

ź-nets and simplex range queries

01 Dec 1987-Discrete and Computational Geometry (Springer New York)-Vol. 2, Iss: 1, pp 127-151
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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Citations
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Proceedings ArticleDOI
08 Jun 2003
TL;DR: In this article, it was shown that every point set X in Rd admits a weak e-net of cardinality O(e -d polylog(1/e) poly( 1/e)) for any point set x in Rd with respect to convex sets.
Abstract: A finite set ? ? Rd is a weak e-net for an n -point set X ? Rd (with respect to convex sets) if N intersects every convex set K with | K n X |= en We give an alternative, and arguably simpler, proof of the fact, first shown by Chazelle et al [7], that every point set X in Rd admits a weak e-net of cardinality O (e -d polylog(1/e)) Moreover, for a number of special point sets (eg, for points on the moment curve), our method gives substantially better bounds The construction yields an algorithm to construct such weak eps-nets in time O ( n ln(1e)) We also prove, by a different method, a near-linear upper bound for points uniformly distributed on the (d--1)-dimensional sphere

11 citations


Cites background from "ź-nets and simplex range queries"

  • ...Weak ε-nets with respect to convex sets as defined in the abstract were introduced by Haussler and Welzl [10] and later found many applications in discrete geometry, most notably in the spectacular proof of the Hadwiger-Debrunner (p, q)-conjecture by Alon and Kleitman [3]....

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Proceedings Article
01 Jan 2011
TL;DR: This work presents an ecient algorithm for computing the obstacle number for a given graph drawing D with approximation ratio O(logobs(D), achieved by showing that the V-C dimension is bounded for the family of hypergraphs of the underlying transversal problem, and using results from epsilon net theory.
Abstract: An obstacle representation for a (straight-line) graph drawing consists of the positions of the graph vertices together with a set of polygonal obstacles such that every line segment between a pair of non-adjacent vertices intersects some obstacle, while the vertices and edges of the drawing avoid all the obstacles. The obstacle number obs(D) for a graph drawing D is the least number of obstacles in an obstacle representation for it. We present an ecient algorithm for computing the obstacle number for a given graph drawing D with approximation ratio O(logobs(D)). This is achieved by showing that the V-C dimension is bounded for the family of hypergraphs of the underlying transversal problem, and using results from epsilon net theory.

11 citations


Cites background from "ź-nets and simplex range queries"

  • ...Haussler and Welzl have shown [13] that if the V-C dimension of H(X,Y ) is some integer d, then for every > 0 there is an -net of size at most cd −1 ln −1, where c is a small constant....

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Journal ArticleDOI
01 May 1988
TL;DR: For a large class of spherical nonrigid objects, exact solutions of the path existence problem are developed based on decomposition techniques and graph traversal for linear/circular and polynomial paths.
Abstract: The path existence problem and the collision detection problem for time-varying objects in a geometric scene are discussed. For a large class of spherical nonrigid objects, exact solutions of the path existence problem are developed based on decomposition techniques and graph traversal. For the collision detection problem of a single moving circle in the plane, efficient data structures are presented for linear/circular and polynomial paths.

10 citations

Journal ArticleDOI
01 Oct 2010
TL;DR: This work shows that every family of graphs of bounded domino treewidth is trimmable and derives polynomial-time approximation schemes for various forms of the problem of labeling a subset of given weighted point features with nonoverlapping sliding axes-parallel rectangular labels so as to maximize the total weight of the labeled features.
Abstract: For t>0 and g≥0, a vertex-weighted graph of total weight W is (t,g)-trimmable if it contains a vertex-induced subgraph of total weight at least (1−1/t)W and with no simple path of more than g edges. A family of graphs is trimmable if for every constant t>0, there is a constant g≥0 such that every vertex-weighted graph in the family is (t,g)-trimmable. We show that every family of graphs of bounded domino treewidth is trimmable. This implies that every family of graphs of bounded degree is trimmable if the graphs in the family have bounded treewidth or are planar. We also show that every family of directed graphs of bounded layer bandwidth (a less restrictive condition than bounded directed bandwidth) is trimmable. As an application of these results, we derive polynomial-time approximation schemes for various forms of the problem of labeling a subset of given weighted point features with nonoverlapping sliding axes-parallel rectangular labels so as to maximize the total weight of the labeled features, provided that the ratios of label heights or the ratios of label lengths are bounded by a constant. This settles one of the last major open questions in the theory of map labeling.

10 citations


Cites background from "ź-nets and simplex range queries"

  • ...One may phrase the definition of a (t, g)-trimming of an unweighted n-vertex graph G = (V ,E) in the language of ε-nets [9]....

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Journal ArticleDOI
TL;DR: It is shown that F, as well as P, for any given set P∩P∣F∈F, have fractional Helly number at most k, which improves the known bounds for fractionalHelly numbers of many families.
Abstract: Let $${\mathcal {F}}$$F be a family of geometric objects in $${\mathbb {R}}^d$$Rd such that the complexity (number of faces of all dimensions on the boundary) of the union of any m of them is $$o(m^k)$$o(mk). We show that $${\mathcal {F}}$$F, as well as $$\{F \cap P \mid F \in {\mathcal {F}}\}$${F?P?F?F} for any given set $$P \in {\mathbb {R}}^d$$P?Rd, have fractional Helly number at most k. This improves the known bounds for fractional Helly numbers of many families.

10 citations


Cites background or methods from "ź-nets and simplex range queries"

  • ...In [8] it is shown that every family of bounded VC-dimension has a (p, q) theorem....

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  • ...This result in [8] in fact uses the Alon-Kleitman method in [3] together with an established fractional Helly number of families with bounded (dual) VC-dimension, and the existence of a bounded -net for families of bounded VC-dimension [5]....

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  • ...In the proof of Theorem 1 we will follow the footsteps of Matoušek in [8] where the fractional Helly number of FP is bounded in terms of the dual VC-dimension of F and more specifically, in terms of the dual shatter function of F ....

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  • ...In [8] Matoušek shows that families of sets with bounded dual VC-dimension have a (bounded) fractional Helly number (see definition below)....

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  • ...This includes in particular the theory of -nets and weak -nets [5]....

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References
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Book ChapterDOI
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 1987
TL;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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Journal ArticleDOI
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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Journal ArticleDOI
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

Journal ArticleDOI
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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