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- Boris Aronov, Esther Ezra, Micha Sharir
- 2008

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)

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)

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 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)

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)

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)

The [Leu]enkephalin-containing peptides derived from prodynorphin have been quantitated in porcine neurointermediate lobe. Dynorphin-17, beta-neo-endorphin, [Leu]enkephalin, rimorphin and dynorphin-24 were found in highest concentrations (200-700 pmol/gm). No detectable amounts of the carboxyl terminal-extended rimorphin sequence (rimorphin 1-29) were… (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)