On the union of Jordan regions and collision-free translational motion amidst polygonal obstacles
Discrete & Computational Geometry
Applications of random sampling in computational geometry, II
Discrete & Computational Geometry - Selected papers from the fourth ACM symposium on computational geometry, Univ. of Illinois, Urbana-Champaign, June 6 8, 1988
How to net a lot with little: small &egr;-nets for disks and halfspaces
SCG '90 Proceedings of the sixth annual symposium on Computational geometry
Reporting points in halfspaces
Computational Geometry: Theory and Applications
Decision theoretic generalizations of the PAC model for neural net and other learning applications
Information and Computation
On the union of fat wedges and separating a collection of segments by a line
Computational Geometry: Theory and Applications
Fat Triangles Determine Linearly Many Holes
SIAM Journal on Computing
Improved bounds on the sample complexity of learning
Journal of Computer and System Sciences
Improved Approximation Algorithms for Geometric Set Cover
Discrete & Computational Geometry
Proceedings of the twenty-fourth annual symposium on Computational geometry
On Approximating the Depth and Related Problems
SIAM Journal on Computing
Epsilon nets and union complexity
Proceedings of the twenty-fifth annual symposium on Computational geometry
A constructive proof of the general lovász local lemma
Journal of the ACM (JACM)
Small-Size $\eps$-Nets for Axis-Parallel Rectangles and Boxes
SIAM Journal on Computing
Relative (p,ε)-Approximations in Geometry
Discrete & Computational Geometry
Tight lower bounds for the size of epsilon-nets
Proceedings of the twenty-seventh annual symposium on Computational geometry
Geometric Approximation Algorithms
Geometric Approximation Algorithms
Improved bound for the union of fat triangles
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
Geometric packing under non-uniform constraints
Proceedings of the twenty-eighth annual symposium on Computational geometry
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We present improved upper bounds for the size of relative (p,ε)-approximation for range spaces with the following property: For any (finite) range space projected onto (that is, restricted to) a ground set of size n and for any parameter 1 ≤ k ≤ n, the number of ranges of size at most k is only nearly-linear in n and polynomial in k. Such range spaces are called "well behaved". Our bound is an improvement over the bound O(log(1/p)/ε2 p) introduced by Li et. al. [17] for the general case (where this bound has been shown to be tight in the worst case), when p l ε. We also show that such small size relative (p,ε)-approximations can be constructed in expected polynomial time. Our bound also has an interesting interpretation in the context of "p-nets": As observed by Har-Peled and Sharir [13], p-nets are special cases of relative (p,ε)-approximations. Specifically, when ε is a constant smaller than 1, the analysis in [13, 17] implies that there are p-nets of size O(log{(1/p)}/p) that are also relative approximations. In this context our construction significantly improves this bound for well-behaved range spaces. Despite the progress in the theory of p-nets and the existence of improved bounds corresponding to the cases that we study, these bounds do not necessarily guarantee a bounded relative error. Lastly, we present several geometric scenarios of well-behaved range spaces, and show the resulting bound for each of these cases obtained as a consequence of our analysis. In particular, when ε is a constant smaller than 1, our bound for points and axis-parallel boxes in two and three dimensions, as well as points and "fat" triangles in the plane, matches the optimal bound for p-nets introduced in [3, 25].