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

Journal Article•DOI•

A Deterministic Linear Time Algorithm For Geometric Separators And Its Applications

[...]

David Eppstein¹, Gary L. Miller², Shang-Hua Teng³•Institutions (3)

University of California, Irvine¹, Carnegie Mellon University², Massachusetts Institute of Technology³

01 Dec 1995-Fundamenta Informaticae

TL;DR: An O(nlogn) time algorithm for constructing a linear space, O(logn) query time search structure for a geometric query problem associated with k-ply neighborhood systems, and an algorithm for approximating the value of k within a constant factor in time O(NLogn).

...read moreread less

Abstract: We give a deterministic linear time algorithm for finding a “good” sphere separator of a k-ply neighborhood system Φ in any fixed dimension, where a k-ply neighborhood system in $$\IR$$d is a collection of n balls such that no points in the space is covered by more than k balls. The separating sphere intersects at most O (k1/dn1−1/d) balls of Φ and divides the remaining of Φ into two parts: those in the interior and those in the exterior of the sphere, respectively, so that the larger part contains at most δn balls ((d + 1)/(d + 2) < δ < 1). This result improves the O(n2) time deterministic algorithm of Miller and Teng [30] and answers a major algorithmic open question posed by Miller, Teng, Thurston, and Vavasis [23, 26]. The deterministic algorithm hinges on the use of a new method for deriving the separator property of neighborhood systems. Using this algorithm, we devise an O(kn+nlogn) time deterministic algorithm for computing the intersection graph of a k-ply neighborhood system. We give an O(nlogn) time algorithm for constructing a linear space, O(logn) query time search structure for a geometric query problem associated with k-ply neighborhood systems, and we use this data structure in an algorithm for approximating the value of k within a constant factor in time O(nlogn). We also develop a deterministic linear time algorithm for finding an O (k1/dn1−1/d)-separator for a k-nearest neighborhood graph in d dimensions.

...read moreread less

34 citations

Journal Article•DOI•

Guarding galleries where every point sees a large area

[...]

Gil Kalai¹, Gil Kalai², Jiří Matoušek³•Institutions (3)

Princeton University¹, Hebrew University of Jerusalem², Charles University in Prague³

01 Dec 1997-Israel Journal of Mathematics

TL;DR: In this paper, it was shown that if a compact simply connected set in the plane of Lebesgue measure 1, such that any pointx ∈ X sees a part of X of measure at least ǫ, then one can choose a setG of at mostconst1/ǫ log 1/Ǫ points in X such that each point of X is seen by some point of G. And if for anyk points inX there is a point seeing at least 3 of them, then all points ofX can be seen from at most O(k

...read moreread less

Abstract: We prove a conjecture of Kavraki, Latombe, Motwani and Raghavan that ifX is a compact simply connected set in the plane of Lebesgue measure 1, such that any pointx∈X sees a part ofX of measure at least ɛ, then one can choose a setG of at mostconst1/ɛ log 1/ɛ points inX such that any point ofX is seen by some point ofG. More generally, if for anyk points inX there is a point seeing at least 3 of them, then all points ofX can be seen from at mostO(k3 logk) points.

...read moreread less

34 citations

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

...First we recall definitions and results from [17], [ 6 ]....
[...]
...Let us remark that [ 6 ] proves this result for the special case when X is finite...
[...]
...Haussler and Welzl [ 6 ], extending ideas of Vapnik and Chervonenkis [17], proved the following:...
[...]
...Haussler and Welzl [ 6 ] (VC-dimension and e-nets)....
[...]

Proceedings Article•DOI•

Product range spaces, sensitive sampling, and derandomization

[...]

Hervé Brönnimann¹, Bernard Chazelle¹, Jirí Matousek•Institutions (1)

Princeton University¹

03 Nov 1993

TL;DR: A simpler optimal deterministic convex hull algorithm is derived, and by extending the method to the intersection of a set of balls with the same radius, an O(nlog/sup 3/ n) deterministic algorithm for computing the diameter of an n-point set in 3-dimensional space is obtained.

...read moreread less

Abstract: We introduce the concept of a sensitive /spl epsi/-approximation, and use it to derive a more efficient algorithm for computing /spl epsi/-nets. We define and investigate product range spaces, for which we establish sampling theorems analogous to the standard finite VC-dimensional case. This generalizes and simplifies results from previous works. We derive a simpler optimal deterministic convex hull algorithm, and by extending the method to the intersection of a set of balls with the same radius, we obtain an O(nlog/sup 3/ n) deterministic algorithm for computing the diameter of an n-point set in 3-dimensional space. >

...read moreread less

34 citations

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

...Recall that, given a set system (called a range space) C = (X, R), an &-approximation for C [ 14 ] is a subset A of X such that...
[...]
...lInformally, the VC-dimension denotes the combinatorial complexity of a range space: it is a classical result that a standard range space can be sampled efficiently if and only if its VCdimension is finite [ 14 ]....
[...]

Book•

Red-blue intersection detection algorithms, with applications to motion planning and collision detection

[...]

Pankaj K. Agarwal¹, Micha Sharir²•Institutions (2)

Courant Institute of Mathematical Sciences¹, Tel Aviv University²

22 Oct 2010

TL;DR: For several special but useful cases, in which many faces in the arrangements of Γ and Γ' can be computed efficiently, the authors obtain randomized algorithms which are better than the general algorithm.

...read moreread less

Abstract: Let G be a collection of n (possibly intersecting) “red” Jordan arcs of some simple shape in the plane and let G' be a similar collection of m “blue” arcs. We present several efficient algorithms for detecting an intersection between an arc of G and an arc of G'. (i) If the arcs of G' form the boundary of a simply connected region, then we can detect a “red-blue” intersection in time O (ls(m)log2m + (la(m) + n)log(n + m)) where ls(m) is the (almost-linear) maximum length of (m, s) Davenport-Schinzel sequences, and where s is a fixed parameter, depending on the shape of the given arcs. Another case where we can detect an intersection in close to linear time is when the union of the arcs of G and the union of the arcs of G' are both connected. (ii) In the most general case, we can detect an intersection in time O ((m√ls(n) + n√ls(m))log1.5(m+n)). For several special but useful cases, in which many faces in the arrangements of G and G' can be computed efficiently, we obtain randomized algorithms which are better than the general algorithm. In particular when all arcs in G and G' are line segments, we obtain a randomized O((m+n)4/3+c) intersection detection algorithm.We apply the algorithm in (i) to obtain an O(ls(n) log2n) algorithm (for some small s > 0) for planning the motion of an n-sided simple polygon around a right-angle corner in a corridor, improving a previous O(n2) algorithm of [MY86], and to derive an efficient technique for fast collision detection for a simple polygon moving (translating and rotating) in the plane along a prescribed path.

...read moreread less

33 citations

Proceedings Article•DOI•

Improved bound for the union of fat triangles

[...]

Esther Ezra¹, Boris Aronov¹, Micha Sharir¹•Institutions (1)

New York University¹

23 Jan 2011

TL;DR: It is shown that, for any fixed Δ > 0, the combinatorial complexity of the union of triangles in the plane, each of whose angles is at least Δ, is O(n), with the constant of proportionality depending on Δ.

...read moreread less

Abstract: We show that, for any fixed δ > 0, the combinatorial complexity of the union of n triangles in the plane, each of whose angles is at least δ, is O(n2α(n) log* n), with the constant of proportionality depending on δ. This considerably improves the twenty-year-old bound O(n log log n), due to Matousek et al. [24, 25].

...read moreread less

33 citations

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

...[25] J. Matou.sek, J. Pach, M. Sharir, S. Sifrony, and E. Welzl, Fat triangles determine linearly many holes, SIAM J. Comput., 23:154 169, 1994....
[...]
...[24] J. Matou.sek, N. Miller, J. Pach, M. Sharir, S. Sifrony, and E. Welzl, Fat triangles determine linearly many holes, Proc....
[...]
...The classical result ofHaussler andWelzl[21], specialized to this context, assertsthat,if(P,R) has so-called .nite VC-dimension, then it admits dual e-nets of the above kind of size O((1/e)log(1/e)), where the constant of proportionality depends on the VC-dimension....
[...]
...The classical result of Haussler and Welzl [21], specialized to this context, asserts that, if (P,R) has so-called finite VC-dimension, then it admits dual ε-nets of the above kind of size O((1/ε) log(1/ε)), where the constant of proportionality depends on the VC-dimension....
[...]
...[21] D. Haussler and E. Welzl, e-nets and simplex range queries, Discrete Comput....
[...]

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....
[...]