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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12 Nov 2000
TL;DR: This work describes algorithms that yield provable guarantees for a particular problem of this type: detecting a network failure, and establishes a connection between graph separators and the notion of VC-dimension, using techniques based on matchings and disjoint paths.
Abstract: Measuring the properties of a large, unstructured network can be difficult: one may not have full knowledge of the network topology, and detailed global measurements may be infeasible. A valuable approach to such problems is to take measurements from selected locations within the network and then aggregate them to infer large-scale properties. One sees this notion applied in settings that range from Internet topology discovery tools to remote software agents that estimate the download times of popular Web pages. Some of the most basic questions about this type of approach, however, are largely unresolved at an analytical level. How reliable are the results? How much does the choice of measurement locations affect the aggregate information one infers about the network? We describe algorithms that yield provable guarantees for a particular problem of this type: detecting a network failure. Suppose we want to detect events of the following form: an adversary destroys up to k nodes or edges, after which two subsets of the nodes, each at least an /spl epsi/ fraction of the network, are disconnected from one another. We call such an event an (/spl epsi/,k) partition. One method for detecting such events would be to place "agents" at a set D of nodes, and record a fault whenever two of them become separated from each other. To be a good detection set, D should become disconnected whenever there is an (/spl epsi/,k)-partition; in this way, it "witnesses" all such events. We show that every graph has a detection set of size polynomial in k and /spl epsi//sup -1/, and independent of the size of the graph itself. Moreover, random sampling provides an effective way to construct such a set. Our analysis establishes a connection between graph separators and the notion of VC-dimension, using techniques based on matchings and disjoint paths.
24 citations
01 Aug 1997
TL;DR: This work describes the first known algorithm for efficiently maintaining a Binary Space Partition (BSP) for n continuously moving segments in the plane, whose interiors remain disjoint throughout the motion.
Abstract: We describe the first known algorithm for efficiently maintaining a Binary Space Partition (BSP) for n continuously moving segments in the plane, whose interiors remain disjoint throughout the motion. Under reasonable assumptions on the motion, we show that the total number of times this BSP changes is O.n 2 / ,a nd that we can update the BSP in O.logn/ expected time per change. Throughout the motion, the expected size of the BSP is O.n logn/. We also consider the problem of constructing a BSP for n static triangles with pairwise-disjoint interiors in R 3 . We present a randomized algorithm that constructs a BSP of size O.n 2 / in O.n 2 log 2 n/ expected time. We also describe a deterministic algorithm that constructs a BSP of size O..nCk/log 2 n/ and height O.logn/ in O..nCk/log 3 n/ time, where k is the number of intersection points between the edges of the projections of the triangles onto the xy-plane. This is the first known algorithm that constructs a BSP of O.logn/ height for disjoint triangles inR 3 . © 2000 Elsevier Science B.V. All rights reserved.
24 citations
TL;DR: The δ-relativeε-approximation method, developed for the CRCW variant of the PRAM parallel computation model, can be easily implemented to run in $O(\log n(\log\log n)^{d-1})$ time using linear work on an EREW PRAM.
Abstract: We give fast and efficient methods for constructing e-nets and e-approximations for range spaces with bounded VC-exponent. These combinatorial structures have wide applicability to geometric partitioning problems, which are often used in divide-and-conquer constructions in computational geometry algorithms. In addition, we introduce a new deterministic set approximation for range spaces with bounded VC-exponent, which we call the δ-relativee-approximation, and we show how such approximations can be efficiently constructed in parallel. To demonstrate the utility of these constructions we show how they can be used to solve the linear programming problem in \({\Bbb R}^d\) deterministically in \(O((\log\log n)^d)\) time using linear work in the PRAM model of computation, for any fixed constant d. Our method is developed for the CRCW variant of the PRAM parallel computation model, and can be easily implemented to run in \(O(\log n(\log\log n)^{d-1})\) time using linear work on an EREW PRAM.
24 citations
Cites background or result from "ź-nets and simplex range queries"
...Relaxing this requirement a bit, Y is said to be an"-net [ 36 ], [51] of.X;R/ if Y\ R6D; for each R2R such thatjRj >" j Xj. This is clearly a weaker notion than that of an "-approximation, for any "-approximation is automatically an "-net, but the converse need not be true....
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...The study of random sampling in the design of efficient computational geometry methods really began in earnest with some outstanding early work of Clarkson [20], Haussler and Welzl [ 36 ], and Clarkson and Shor [22]....
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...Random sampling can be applied to construct such a partitioning so that each cell intersects at most "n hyperplanes, for " D log r=r [22], [ 36 ]....
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...Interestingly, most of the combinatorial properties needed by geometric random samples can be characterized by two notions—the "-approximation [51], [68] and the"-net [ 36 ], [51]....
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TL;DR: It is shown that the well-known random incremental construction of Clarkson and Shor18 can be adapted to provide efficient external-memory algorithms for some geometric problems.
Abstract: We show that the well-known random incremental construction of Clarkson and Shor18 can be adapted to provide efficient external-memory algorithms for some geometric problems. In particular, as the ...
23 citations
04 Jun 2009
TL;DR: It is proved that one can compute a set L of O (*** log*** log(1/*** )) landmarks so that if a set S of sensors covers L, then S covers at least (1 *** *** )-fraction of P .
Abstract: We study the problem of covering a two-dimensional spatial region P , cluttered with occluders, by sensors. A sensor placed at a location p covers a point x in P if x lies within sensing radius r from p and x is visible from p , i.e., the segment px does not intersect any occluder. The goal is to compute a placement of the minimum number of sensors that cover P . We propose a landmark-based approach for covering P . Suppose P has *** holes, and it can be covered by h sensors. Given a small parameter *** > 0, let *** : = *** (h ,*** ) = (h /*** )(1 + ln (1 + *** )). We prove that one can compute a set L of O (*** log*** log(1/*** )) landmarks so that if a set S of sensors covers L , then S covers at least (1 *** *** )-fraction of P . It is surprising that so few landmarks are needed, and that the number of landmarks depends only on h , and does not directly depend on the number of vertices in P . We then present efficient randomized algorithms, based on the greedy approach, that, with high probability, compute $O(\tilde{h}\log \lambda)$ sensor locations to cover L ; here $\tilde{h} \le h$ is the number sensors needed to cover L . We propose various extensions of our approach, including: (i) a weight function over P is given and S should cover at least (1 *** *** ) of the weighted area of P , and (ii) each point of P is covered by at least t sensors, for a given parameter t *** 1.
23 citations
Additional excerpts
...A seminal result by Haussler and Welzl [16] shows that if VC-dim(Σ) = d, then a random subsetN ⊆ X of sizeO((d/ε) log(d/εδ)) is an ε-net of Σ with probability at least 1 − δ....
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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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