A non-linear lower bound for planar epsilon-nets

1. 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 A non-linear lower bound for planar epsilon-nets

2. 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

3. 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

4. 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 VC-dimension d, there exists an ε-net of size O(d/ε)? (Matousek, Siedel and Welzl)

5. 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)

6. 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 ωis 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

7. 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)

8. 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)

9. 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 = kifor all i ∊ I and xi = j for all i ∊[d]\I}

10. 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

11. Conclusions The conjecture that minimum size of epsilon net is linearly bounded by (1/ε) is not true for geometric examples in VC-dimension = 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.

12. 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