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 51 st Annual Symposium on Foundations of Computer Science A non-linear lower bound for planar epsilon-nets

What is an epsilon-net? Intuitive Idea:  Given a (finite) set X of n points in ℛ 2, can we find a small(say f(n,ε)) P ⊆ X, so that any triangle T ⊆ ℛ 2 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:F or 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)

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 ε i of positive reals tending to zero, so that for every ε=ε i in the sequence and for all n > n 0 (ε i ) there exists a set Y n of n points in general position in the plane, so that the smallest possible size of an ε-net for fat lines for Y n 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 k i ∊ [k] for i ∊ I for which L is the following set of k members of [k] d : L={l 1,l 2,...,l k } where l j ={(x 1,x 2,...,x d )}: x i = k i for all i ∊ I and x i = 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 d 0 = d 0 (k,δ) so that for any d ≥ d 0, any set Y of at least δk d members of [k] d contains a combinatorial line.  Construction (for Theorem 1):  Every combinatorial line in X=[k] d is a line in ℛ d. If d = d 0 (k,1/2) and n=k d, ε=k/k d, 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 ℛ 2  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 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.

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),  D. Haussler and E.Welzl, ε-nets and simplex range queries, Discrete and Computational Geometry 2 (1987),  J. Pach and G. Woeginger, Some new bounds for ε-nets, Proc. 6-th Annual Symposium on Computational Geometry, ACM Press, New York (1990),  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