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We show the existence of ε-nets of size O 1 ε log log 1 ε for planar point sets and axis-parallel rectangular ranges. The same bound holds for points in the plane with " fat " triangular ranges, and for point sets in R 3 and axis-parallel boxes; these are the first known non-trivial bounds for these range spaces. Our technique also yields improved bounds on(More)
The disagreement coefficient of Hanneke has become a central concept in proving active learning rates. It has been shown in various ways that a concept class with low complexity together with a bound on the disagreement coefficient at an optimal solution allows active learning rates that are superior to passive learning ones. We present a different tool for(More)
We refine the bound on the packing number, originally shown by Haussler, for shallow geometric set systems. Specifically, let V be a finite set system defined over an n-point set X; we view V as a set of indicator vectors over the n-dimensional unit cube. A δ-separated set of V is a subcollection W, s.t. the Hamming distance between each pair u, v ∈ W is(More)
We study the problem of covering a two-dimensional spatial region P , cluttered with occluders, by sensors. A sensor placed at a location p covers a point x in P if x lies within sensing radius r from p and x is visible from p, i.e., the segment px does not intersect any occluder. The goal is to compute a placement of the minimum number of sensors that(More)
Given a set system (X,R), the <i>hitting set</i> problem is to find a smallest-cardinality subset H &#8838; X, with the property that each range R &#8712; R has a non-empty intersection with H. We present near-linear time approximation algorithms for the hitting set problem, under the following geometric settings: (i) R is a set of planar regions with small(More)
Let (X, S) be a set system on an n-point set X. The discrepancy of S is defined as the minimum of the largest deviation from an even split, over all subsets of S ∈ S and two-colorings χ on X. We consider the scenario where, for any subset X ′ ⊆ X of size m ≤ n and for any parameter 1 ≤ k ≤ m, the number of restrictions of the sets of S to X ′ of size at(More)
We show that the combinatorial complexity of the union of n " fat " tetrahedra in 3-space (i.e., tetrahedra all of whose solid angles are at least some fixed constant) of arbitrary sizes, is O(n 2+ε), for any ε > 0; the bound is almost tight in the worst case, thus almost settling a conjecture of Pach et al. [23]. Our result extends, in a significant way,(More)