scispace - formally typeset
Search or ask a question
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.

Content maybe subject to copyright    Report

Citations
More filters
Proceedings ArticleDOI
08 Jun 2003
TL;DR: This paper studies coloring problems related to frequency assignment problems in cellular networks and refers to such a coloring as a conflict-free coloring of conflict-free coloring of(P,R) (CF-coloring in short).
Abstract: In this paper, we study coloring problems related to frequency assignment problems in cellular networks. In abstract setting, the problems are of the following two types:CF-coloring of regions: Given a finite family S of n regions of some fixed type (such as discs, pseudo-discs, axis-parallel rectangles, etc.), what is the minimum integer k, such that one can assign a color to each region of S, using a total of at most k colors, such that the resulting coloring has the following property: For each point p ∈b∈S b there is at least one region b∈S that contains p in its interior, whose color is unique among all regions in S that contain p in their interior (in this case we say that p is being `served' by that color). We refer to such a coloring as a conflict-free coloring of S (CF-coloring in short).CF-coloring of a range space: Given a set P of n points in Rd and a set R of ranges (for example, the set of all discs in the plane), what is the minimum integer k, such that one can color the points of P by k colors, so that for any r ∈ R with P∩r∈≠O, there is at least one point q ∈ P ∩ r that is assigned a unique color among all colors assigned to points of P ∩ r (in this case we say that r is 'served' by that color). We refer to such a coloring as a conflict-free coloring of (P,R) (CF-coloring in short).

52 citations


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

  • ...By ε-net theorem [12], P1 is an ε-net for rectangles inside the unit square under the measure of area, with high probability....

    [...]

Proceedings ArticleDOI
01 Jul 1992
TL;DR: The paper investigates the complexity of halfspace range searching and establishes a tradeoff between the storage and the worst-case query time in the Fredman/Yao arithmetic model of computation, establishing the first nontrivial lower bound for half space range searching.
Abstract: We investigate the complexity of halfspace range searching: Given n points in d-space, build a data structure that allows us to determine efficiently how many points lie in a query halfspace. We establish a tradeoff between the storage m and the worst-case query time t in the Fredman/Yao arithmetic model of computation. We show that t must be at least on the order of (n/log n)1-((d-1)/(d(d+1))m1/d.To our knowledge, this is the first nontrivial lower bound for halfspace range searching. Although the bound is unlikely to be optimal, it falls reasonably close to the recent O(n(log m/n)d+1/m1/d) upper bound established by Matouscek. We also show that it is possible to devise a sequence of n inserts and halfspace range queries that require a total time of n2-t(1/d). Our results imply nontrivial lower bounds for spherical range searching in any fixed dimension. For example they show that, with linear storage, circular range queries in the plane require O(n1/3) time (modulo a logarithmic factor).

51 citations

Journal ArticleDOI
TL;DR: This work develops low-storage data structures to maintain e-nets and e-approximations of range spaces of P with small VC-dimension and maintain a (1 + e)-approximation of the weight of the Euclidean minimum spanning tree of P.
Abstract: A dynamic geometric data stream is a sequence of m ADD/REMOVE operations of points from a discrete geometric space {1,…, Δ}d ?. ADD (p) inserts a point p from {1,…, Δ}d into the current point set P, REMOVE(p) deletes p from P. We develop low-storage data structures to (i) maintain e-nets and e-approximations of range spaces of P with small VC-dimension and (ii) maintain a (1 + e)-approximation of the weight of the Euclidean minimum spanning tree of P. Our data structure for e-nets uses bits of memory and returns with probability 1 – δ a set of points that is an e-net for an arbitrary fixed finite range space with VC-dimension . Our data structure for e-approximations uses bits of memory and returns with probability 1 – δ a set of points that is an e-approximation for an arbitrary fixed finite range space with VC-dimension . The data structure for the approximation of the weight of a Euclidean minimum spanning tree uses O(log(1/δ)(log Δ/e)O(d)) space and is correct with probability at least 1 – δ. Our results are based on a new data structure that maintains a set of elements chosen (almost) uniformly at random from P.

51 citations

Proceedings ArticleDOI
06 Jun 2007
TL;DR: A randomized data structure of O(n) expected sizewhich can answer 3D approximate halfspace range counting queries in O(log n/k) expected time is presented, which is the first optimal method for the problem in the standard decision tree model.
Abstract: We improve the previous results by Aronov and Har-Peled (SODA'05) and Kaplan and Sharir (SODA'06) and present a randomized data structure of O(n) expected sizewhich can answer 3D approximate halfspace range counting queries in O(log n/k) expected time, where k is the actual value of the count. This is the first optimal method for the problem in the standard decision tree model; moreover, unlike previous methods, the new method is Las Vegas instead of Monte Carlo.In addition, we describe new results for several related problems, includingapproximate Tukey depth queries in 3D, approximate regression depthqueries in 2D, and approximate linear programming with violations inlow dimensions.

51 citations


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

  • ...If an additive error of en is tolerable, then the problem can be solved with constant space and query time in any fixed dimension by simply working with a sample (so-called e-approximation) of constant size [ 21 , 30]....

    [...]

References
More filters
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....

    [...]

  • ...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]....

    [...]

  • ...The key concepts and proof techniques of this section are based on the pioneering work of Vapnik and Chervonenkis [ 17 ]....

    [...]

  • ...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....

    [...]

  • ...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])....

    [...]

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....

    [...]

  • ..., [11] for a general treatment of arrangements....

    [...]

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'. []...

    [...]

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....

    [...]