ź-nets and simplex range queries

doi:10.1007/BF02187876

Home
/
Papers
/
ź-nets and simplex range queries

Journal Article•DOI•

ź-nets and simplex range queries

David Haussler¹, Emo Welzl²•Institutions (2)

University of California, Santa Cruz¹, University of Graz²

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 Ã‚-net with any desired probability are given.

read less

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.

...read moreread less

Content maybe subject to copyright Report

Citations

PDF

Open Access

More filters

Proceedings Article•DOI•

Small-size ε-nets for axis-parallel rectangles and boxes

[...]

Boris Aronov¹, Esther Ezra², Micha Shair³•Institutions (3)

New York University¹, Duke University², Tel Aviv University³

31 May 2009

TL;DR: Improved approximation factors are obtained for the hitting set or the set cover problems associated with the corresponding range spaces for ε-nets of size O(1/ε log log log 1/ε) for planar point sets and axis-parallel rectangular ranges.

...read moreread less

Abstract: We show the existence of e-nets of size O(1/e log log 1/e) for planar point sets and axis-parallel rectangular ranges. The same bound holds for points in the plane with "fat" triangular ranges, and for point sets in reals3 and axis-parallel boxes; these are the first known non-trivial bounds for these range spaces. Our technique also yields improved bounds on the size of e-nets in the more general context considered by Clarkson and Varadarajan. For example, we show the existence of e-nets of sizeO(1/e log log log 1/e) for the dual range space of "fat" regions and planar point sets (where the regions are the ground objects and the ranges are subsets stabbed by points). Plugging our bounds into the technique of Bronnimann and Goodrich, we obtain improved approximation factors (computable in randomized polynomial time) for the hitting set or the set cover problems associated with the corresponding range spaces.

...read moreread less

87 citations

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

...of [HW87]....
[...]
...By the standard ε-net theory of [HW87], with high probability each rectangle of M contains at most O ( n s log s )...
[...]
...Of particular interest are range spaces of finite VC-dimension; the reader is referred to [HW87] for the exact definition....
[...]
..., the deterministic algorithm of Matoušek [Mat95] (or a straightforward random sampling mechanism [HW87])....
[...]
...Since their introduction in 1987 by Haussler and Welzl [HW87] (see also Clarkson [Cla87] and Clarkson and Shor [CS89] for related techniques), ε-nets have become one of the central concepts in computational and combinatorial geometry, and have been used in a variety of applications, such as range searching, geometric partitions, and bounds on curve-point incidences, to name a few; see, e....
[...]

Journal Article•DOI•

Triangles in space or building (and analyzing) castles in the air

[...]

Boris Aronov¹, Micha Sharir¹, Micha Sharir²•Institutions (2)

Courant Institute of Mathematical Sciences¹, Tel Aviv University²

01 Jun 1990-Combinatorica

TL;DR: It is shown that the total combinatorial complexity of all non-convex cells in an arrangement ofn (possibly intersecting) triangles in 3-space isO(n7/3 logn) and that this bound is almost tight in the worst case.

...read moreread less

Abstract: We show that the total combinatorial complexity of all non-convex cells in an arrangement ofn (possibly intersecting) triangles in 3-space isO(n 7/3 logn) and that this bound is almost tight in the worst case. Our bound significantly improves a previous nearly cubic bound of Pach and Sharir. We also present a (nearly) worst-case optimal randomized algorithm for calculating a single cell of the arrangement and an alternative less efficient, but still subcubic algorithm for calculating all non-convex cells, analyze some special cases of the problem where improved bounds (and faster algorithms) can be obtained, and describe applications of our results to translational motion planning for polyhedra in 3-space.

...read moreread less

85 citations

Journal Article•DOI•

Catching elephants with mice: Sparse sampling for monitoring sensor networks

[...]

Sorabh Gandhi¹, Subhash Suri¹, Emo Welzl²•Institutions (2)

University of California, Santa Barbara¹, ETH Zurich²

05 Jan 2010-ACM Transactions on Sensor Networks

TL;DR: The overall simplicity and generality of the technique suggests that it is well suited for a wide class of sensornet applications, including monitoring of physical environments, network anomalies, network security, or any abstract binary event that affects a significant number of nodes in the network.

...read moreread less

Abstract: We propose a scalably efficient scheme for detecting large-scale physically correlated events in sensor networks. Specifically, we show that in a network of n sensors arbitrarily distributed in the plane, a sample of O(1/ϵ log 1/ϵ) sensor nodes (mice) is sufficient to catch any, and only those, events that affect Ω (ϵn) nodes (elephants), for any 0

...read moreread less

85 citations

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

...We formalize the niceness of event shapes using the concept of Vapnik-Chervonenkis (VC) dimension from computational geometry and statistics [Haussler and Welzl 1987; Vapnik and Chervonenkis 1971]....
[...]
...The e-net theorem of Haussler-Welzl [Haussler and Welzl 1987] shows that if we draw m independent random draws from X , where 8d 8d 42 m = max log ,log , (1) e eed then this random sample is an e-net with probability at least (1 -d)....
[...]
...There is a classical strengthening of the e-net, called esample (also known as e-approximation), which remedies both these problems, but unfortunately requires much larger sample size [ 10 , 17, 26]....
[...]
...VC dimension, which arose in statistics [26] and developed further in computation geometry and learning theory [ 10 , 2], gives a framework to formalize our informal intuition that a circle is a simpler shape than an axis-parallel rectangle, which in turn is simpler than a triangle....
[...]
...difference ranges of dimension d . By the result of [ 10 ], such a net can be constructed with high probability by taking O( d e log d e ) random samples from S....
[...]

Journal Article•DOI•

On ray shooting in convex polytopes

[...]

Jiří Matoušek¹, Otfried Schwarzkopf²•Institutions (2)

Free University of Berlin¹, Utrecht University²

01 Aug 1993-Discrete and Computational Geometry

TL;DR: It is shown that some specific data structures for the membership problem can be used for ray shooting in a more direct way, reducing the overhead in the query time and eliminating the use of parametric search.

...read moreread less

Abstract: LetP be a convex polytope withn facets in the Euclidean space of a (small) fixed dimensiond. We consider themembership problem forP (given a query point, decide whether it lies inP) and theray shooting problem inP (given a query ray originating insideP, determine the first facet ofP hit by it). It was shown in [AM2] that a data structure for the membership problem satisfying certain mild assumptions can also be used for the ray shooting problem, with a logarithmic overhead in query time. Here we show that some specific data structures for the membership problem can be used for ray shooting in a more direct way, reducing the overhead in the query time and eliminating the use of parametric search. We also describe an improved static solution for the membership problem, approaching the conjectured lower bounds more tightly.

...read moreread less

84 citations

Journal Article•DOI•

Ray shooting and other applications of spanning trees with low stabbing number

[...]

Pankaj K. Agarwal¹•Institutions (1)

Courant Institute of Mathematical Sciences¹

01 Jun 1992-SIAM Journal on Computing

TL;DR: An algorithm is presented that preprocesses G, a set of n (possibly intersecting) line segments in the plane, into a data structure of size O(n\alpha (n))\log ^4 n) so that for a query ray $\rho $, $\Phi (\mathcal{G},\rho )$ can be computed in time.

...read moreread less

Abstract: This paper considers the following problem: Given a set $\mathcal{G}$ of n (possibly intersecting) line segments in the plane, preprocess it so that, given a query ray $\rho $ emanating from a point p, one can quickly compute the intersection point $\Phi (\mathcal{G},\rho )$ of $\rho $ with a segment of $\mathcal{G}$ that lies nearest to p. The paper presents an algorithm that preprocesses $\mathcal{G}$, in time $O(n^{3/ 2} \log^\omega n)$, into a data structure of size $O(n\alpha (n)\log ^4 n)$, so that for a query ray $\rho $, $\Phi (\mathcal{G},\rho )$ can be computed in time $O(\sqrt {n\alpha (n)} \log ^2 n)$, where $\omega $ is a constant $ < 4.33$ and $\alpha (n)$ is a functional inverse of Ackermann’s function. If the given segments are nonintersecting, the storage goes down to $O(n\log ^3 n)$ and the query time becomes $O(\sqrt n \log ^2 n)$. The main tool used is spanning trees (on the set of segment endpoints) with low stabbing number, i.e., with the property that no line intersects more than $O...

...read moreread less

84 citations

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
…
17
18
19
20
21
22
23
…
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160

Collapse

References

PDF

Open Access

More filters

Book Chapter•DOI•

On the Uniform Convergence of Relative Frequencies of Events to Their Probabilities

[...]

Vladimir Vapnik, A. Ya. Chervonenkis

01 Jan 1971-Theory of Probability and Its Applications

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.

...read moreread less

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).

...read moreread less

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•

Algorithms in Combinatorial Geometry

[...]

Herbert Edelsbrunner¹•Institutions (1)

University of Illinois at Urbana–Champaign¹

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.

...read moreread less

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.

...read moreread less

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 Article•DOI•

On the density of families of sets

[...]

Norbert Sauer¹•Institutions (1)

University of Calgary¹

01 Jul 1972-Journal of Combinatorial Theory, Series A

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.

...read moreread less

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 Article•DOI•

Central Limit Theorems for Empirical Measures

[...]

Richard M. Dudley

01 Dec 1978-Annals of Probability

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.

...read moreread less

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 $r, \sigma_m(P(C_m)(1 - P(C_m)))^r < \infty$.

...read moreread less

555 citations

Journal Article•DOI•

The power of geometric duality

[...]

Bernard Chazelle¹, Leonidas J. Guibas², Der-Tsai Lee³•Institutions (3)

Brown University¹, PARC², Northwestern University³

01 Jun 1985-Bit Numerical Mathematics

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.

...read moreread less

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.

...read moreread less

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....
[...]