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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)

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)

- Nir Ailon, Ron Begleiter, Esther Ezra
- COLT
- 2012

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)

- Esther Ezra, Micha Sharir, Alon Efrat
- Symposium on Computational Geometry
- 2006

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)

- Pankaj K. Agarwal, Esther Ezra, Micha Sharir
- Algorithmica
- 2009

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)

- Esther Ezra
- SIAM J. Comput.
- 2014

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)

- Kunal Dutta, Esther Ezra, Arijit Ghosh
- Symposium on Computational Geometry
- 2015

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)

- Esther Ezra, Boris Aronov, Micha Sharir
- SODA
- 2011

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)

- Boris Aronov, Esther Ezra, Micha Sharir
- STOC
- 2009

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 reals<sup>3</sup> and axis-parallel boxes; these are the first known non-trivial bounds for these range spaces. Our technique… (More)

- Esther Ezra, Micha Sharir
- 48th Annual IEEE Symposium on Foundations of…
- 2007

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<sup>2+epsiv</sup>),for any epsiv > 0: the bound is almost tight in the worst case, thus almost settling a conjecture of Pach el al. [24]. Our result extends, in… (More)