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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 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)
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>&#8805;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 &#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)
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 &gt; 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)
Motivated by the desire to cope with data imprecision [31], we study methods for taking advantage of preliminary information about point sets in order to speed up the computation of certain structures associated with them. In particular, we study the following problem: given a set L of n lines in the plane, we wish to preprocess L such that later, upon(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-coloringsχ 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)