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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 ⊆ X, with the property that each range R ∈ 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)

We present upper and lower bounds for the number of iterations performed by the Iterative Closest Point (ICP) algorithm. This algorithm has been proposed by Besl and McKay [4] as a successful heuristics for pattern matching under translation, where the input consists of two point sets in <i>d</i>-space, for <i>d</i>≥1, but so far it seems not to have… (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, for any fixed Δ > 0, the combinatorial complexity of the union of <i>n</i> triangles in the plane, each of whose angles is at least Δ, is <i>O</i>(<i>n</i>2<sup>α(<i>n</i>)</sup> log* <i>n</i>), with the constant of proportionality depending on Δ. This considerably improves the twenty-year-old bound <i>O</i>(<i>n</i>… (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)