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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Dissertation•
19 Dec 2013
TL;DR: This thesis will be on the combinatorial and algorithmic aspects of approximations of point-set data P in d-dimensional Euclidean space, and presents a polynomial-time approximation scheme for computing hitting-sets for disks in the plane.
Abstract: At the core of successful manipulation and computation over large geometric data is the notion of approximation, both structural and computational. The focus of this thesis will be on the combinatorial and algorithmic aspects of approximations of point-set data P in d-dimensional Euclidean space. It starts with a study of geometric data depth where the goal is to compute a point which is the 'combinatorial center' of P. Over the past 50 years several such measures of combinatorial centers have been proposed, and we will re-examine several of them: Tukey depth, Simplicial depth, Oja depth and Ray-Shooting depth. This can be generalized to approximations with a subset, leading to the notion of epsilon-nets. There we will study the problem of approximations with respect to convexity. Along the way, this requires re-visiting and generalizing some basic theorems of convex geometry, such as the Caratheodory's theorem. Finally we will turn to the algorithmic aspects of these problems. We present a polynomial-time approximation scheme for computing hitting-sets for disks in the plane. Of separate interest is the technique, an analysis of local-search via locality graphs. A further application of this technique is then presented in computing independent sets in intersection graphs of rectangles in the plane.
6 citations
29 Jun 2008
TL;DR: Simulation results show that compared with the distributed randomized k-coverage algorithm, DPCCM significantly maintain coverage in probabilistic model and long the network lifetime in some sense.
Abstract: The fundamental issue in sensor networks is providing a certaindegree of coverage and maintaining connectivity under the energyconstraint. In this paper, the connected k-coverageproblem is investigated under the probabilistic sensing andcommunication models, which are more realistic than deterministicmodels. Furthermore, different weights for nodes are added in orderto estimate the real power consumption. Because the problem isNP-hard, a distributedprobabilisticcoverageandconnectivitymaintenancealgorithm(DPCCM) for dense sensornetworks is proposed. DPCCM converts task requirement into twoparameters by using the consequence of Chebyshev's inequality, thenactivate sensors based on the properties of weightede-net. It is proved that the sensors chosen byDPCCM have (θ,k)-coverage andα-connectivity. And the time and communicationcomplexities are theoretically analyzed. Simulation results showthat compared with the distributed randomized k-coverage algorithm,DPCCM significantly maintain coverage in probabilistic model andprolong the network lifetime in some sense.
6 citations
Dissertation•
01 Jan 2006
TL;DR: This thesis develops data streaming algorithms, which process the data sequencially without storing, and introduces data stream algorithms to approximately count minors like triangles or bipartite cliques in huge graphs given as data streams that can be used to obtain structural information about sozial graphs and the webgraph.
Abstract: The increasing interconnectivity of modern computer systems generates huge amounts of data, which cannot be stored. In this thesis we develop data streaming algorithms, which process the data sequencially without storing. We only store a small sketch or summary of the data, and use it to approximately answer queries about the content of the data. In particular we look at the model of dynamic geometric data streams. Here we are given a data stream consisting of m insert and delete operations of points from a ddimensional space into a set M . We develop data stream algorithms, which use at most poly(log m) bits of memory: First we show how to efficiently maintain a sample of M in dynamic data streams. Then we develop a coreset technique, which can be used to maintain (1+ )-approximative k-Median, k-Means, and MaxCut clusterings of the dynamic point set M . We furthermore show many more applications of the coreset technique and use it to accelerate iterative clustering algorithms in practice. In the last chapter we introduce data stream algorithms to approximately count minors like triangles or bipartite cliques in huge graphs given as data streams. They can be used to obtain structural information about sozial graphs and the webgraph.
6 citations
Cites background or methods from "ź-nets and simplex range queries"
...Theorem 7 [66] Let (X, R) be a range space of VC-dimension D, let A be a finite subset of X and suppose 0 < , δ < 1....
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...Having a random sample of points from the technique developed in the last chapter, the algorithms to maintain -nets and -approximations will follow relatively easily from [66] and [119]....
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TL;DR: The Crossing Lemma for simple Jordan curves has been studied in this paper, where it is shown that the number of intersection points between two simple closed (i.e., Jordan) curves is at least (1 − o ( 1 ) ) n 2.
6 citations
TL;DR: This paper develops some new data structures for storing a set of disks that can answer different types of intersection queries efficiency and presents a linear size data structure for ray shooting queries.
Abstract: In this paper we develop some new data structures for storing a set of disks that can answer different types of intersection queries efficiency. If the disks are non-intersecting we obtain a linear size data structure that can report allk disks intersecting a query line segment in timeO(n
β+e
+k), wheren is the number of disks,β=log2(1+√5)−1 ≈ 0.695, and e is an arbitrarily small positive constant. If the segment is a full line, the query time becomesO(n
β
+k). For intersecting disks we obtain anO(n logn) size data structure that can answer an intersection query in timeO(n
2/3 log2
n+k). We also present a linear size data structure for ray shooting queries, whose query time isO(n
β
).
6 citations
Cites methods from "ź-nets and simplex range queries"
...The preprocessing often depends on the construction of primary structure, and we refer to known results (randomized or deterministic) on this topic [3, 6, 9, 11, 12, 13 ]....
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...To this end we replace the conjugation tree by a structure based on e-nets as described by Haussler and Welzl [ 13 ]....
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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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