A non-linear lower bound for planar
epsilon-nets
CSCE 669
Project Presentation (Paper Reading)
Student Presenter: Praveen Tiwari
Original Author: Noga Alon, Tel Aviv University
Publication: FOCS '10 Proceedings of the 2010 IEEE 51st Annual
Symposium on Foundations of Computer Science
What is an epsilon-net?
Intuitive Idea:

Given a (finite) set X of n points in ℛ2, can we find a
small(say f(n,ε)) P⊆X, so that any triangle T⊆ ℛ2 covering
some points in X (≥εn) contains atleast one point in P?

That is, we somehow want to approximate the larger set X
by a subset P satisfying some property
What is an epsilon-net?
Formal Definition:

Range Space: S:(X,ℛ) for a (finite) set X of points
(objects) and ℛ (range) is a set of subsets of X

VC(Vapnik-Chervonenkis)-dimension: A set A⊆X is
shattered by ℛ if ∀ B⊂A, ∃ R∊ℛ s.t. R⋂A = B
VC(S) = sup {|A| | A⊆X is shattered}

ε-Net: A subset N⊂A, s.t. ∀ R∊ℛ, and 0<ε<1,
|R⋂A|≥ε|A| and R⋂N≠Ø
Bounds on epsilon-nets

Question:For a given range space S(X,ℛ) with a VC-
dimension d in a geometric scenario, what is the lower
bound on size of ε-net?

Haussler and Welzl: For any n and ε>0, any set of size n in
a range space of VC-dimension d contains an ε-net of size at
most O((d/ε)log(1/ε))

Is this bound tight?

Lower Bound: There is no natural geometric example
where size of smallest ε-net is better than trivial Ω(1/ε)

Question: Whether or not in all geometric scenarios of VCdimension d, there exists an ε-net of size O(d/ε)? (Matousek,
Siedel and Welzl)
Previous Work

Linear upper bounds have been established for special
geometric cases, like point objects and half space ranges in
2D and 3D

Pach, Woeginger: For d=2, there exist range spaces that
require nets of size Ω(1/ε log(1/ε)) (no geometric scenario)
Contributions

The linear bound on size of ε does not hold, not even in very
simple geometric situations (VC-dimension=2)

The minimum size of such an ε-net is Ω((1/ε)ω(1/ε)) where
ω is inverse Ackermann's function with respect to lines, i.e.
for VC-dimension = 2

Using VC dimension = 2:

Two theorems on strong ε-nets

One theorem on weak ε-nets
Results

Theorem 1: For every (large) positive constant C there
exist n and ε > 0 and a set X of n points in the plane, so that
the smallest possible size of an ε-net for lines of X is larger
that C·(1/ε)

Def.: A fat line in a plane is the set of all points within
distance μ from a line in the plane.

Theorem 2: For every (large) positive constant C there
exists a sequence εi of positive reals tending to zero, so that
for every ε=εi in the sequence and for all n > n0(εi) there
exists a set Yn of n points in general position in the plane, so
that the smallest possible size of an ε-net for fat lines for Yn
is larger than C·(1/ε)
Results

Weak ε-nets: A relaxation to Strong ε-nets

Def.: For a finite set of points X in ℛ, given A⊂X, a subset
N⊂R is a weak ε-net if ∀R∊ℛ, and 0<ε<1, |R⋂A| ≥ ε|A| and
R⋂N≠∅

The difference is that the set N need not be a subset of A as
earlier

Theorem 3: For every (large) positive constant C there
exist n and ε>0 and a set X of n points in the plane, so that
the smallest possible size of a weak ε-net for lines for X is
larger than C·(1/ε)
Proofs

Proofs use a strong result by Furstenberg and Katznelson,
known as the density version of Hales-Jewett Theorem.

Def.: For an integer k ≥ 2, let [k]={1,2,...,k} and let [k]d
denote the set of all vectors of length d with coordinates in
[k]. A combinatorial line is a subset L ⊂ [k]d so that there is
a set of coordinates I ⊂ [d] = {1,2,...,d}, I ≠ [d], and values
ki ∊ [k] for i ∊ I for which L is the following set of k
members of [k]d:
L={l1,l2,...,lk}
where
lj ={(x1,x2,...,xd)}: xi = ki for all i ∊ I and xi = j for all i ∊ [d]\I}
Proofs

Density Hales-Jewett Theorem (Furstenberg and
Katznelson): For any fixed integer k and any fixed δ > 0
there exists an integer d0 = d0(k,δ) so that for any d ≥ d0,
any set Y of at least δkd members of [k]d contains a
combinatorial line.

Construction (for Theorem 1):


Every combinatorial line in X=[k]d is a line in ℛd. If d =
d0(k,1/2) and n=kd, ε=k/kd, then any ε-net with respect to
lines must be of size ≥ (k/2)(1/ε)

Now for planar construction, project these combinatorial
lines randomly on ℛ2
Other two theorems use similar constructions with simple
modifications
Conclusions

The conjecture that minimum size of epsilon net is linearly
bounded by (1/ε) is not true for geometric examples in VCdimension = 2 for both strong as well as weak ε-nets.

This paper only proves that the bounds are not linear, but
whether there are natural examples for an Ω ((d/ε)log(1/ε))
lower bound for range spaces of VC dimension d, is still
open.

Recent Work: Pach and Tardos proved that there are
geometric range spaces of VC-dimension 2 in which the
minimum possible size of an ε-net is Ω ((1/ε)log(1/ε)). Their
method does not seem to provide any non-linear bounds for
weak ε-nets.
Bibliography

Alon N., A non-linear lower bound for planar epsilon-nets, FOCS 2010

Alon N., Web Seminars, Isaac Newton Institute for Mathematical
Sciences, Jan 11, 2011

H. Furstenberg and Y. Katznelson, A density version of the Hales-Jewett
theorem, J. Anal. Math. 57 (1991), 64-119

D. Haussler and E.Welzl, ε-nets and simplex range queries, Discrete and
Computational Geometry 2 (1987), 127-151

J. Pach and G. Woeginger, Some new bounds for ε-nets, Proc. 6-th Annual
Symposium on Computational Geometry, ACM Press, New York (1990),
10-15

J. Matousek, R. Seidel and E. Welzl, How to net a lot with little: Small nets for disks and halfspaces, In Proc. 6th Annu. ACM Sympos. Comput.
Geom., pages 16-22, 1990