Results 1 
5 of
5
Small weak epsilonnets
, 2008
"... Given a set P of points in the plane, a set of points Q is a weak εnet with respect to a family of sets S (e.g., rectangles, disks, or convex sets) if every set of S containing εP  points contains a point of Q. In this paper, we determine bounds on εS i, the smallest epsilon that can be guarante ..."
Abstract

Cited by 13 (1 self)
 Add to MetaCart
Given a set P of points in the plane, a set of points Q is a weak εnet with respect to a family of sets S (e.g., rectangles, disks, or convex sets) if every set of S containing εP  points contains a point of Q. In this paper, we determine bounds on εS i, the smallest epsilon that can be guaranteed for any P when Q  = i, for small values of i.
Weak ɛnets and interval chains
, 2007
"... We construct weak ɛnets of almost linear size for certain types of point sets. Specifically, for planar point sets in convex position we construct weak 1 rnets of size O(rα(r)), where α(r) denotes the inverse Ackermann function. For point sets along the moment curve in Rd we construct weak 1 rnet ..."
Abstract

Cited by 4 (1 self)
 Add to MetaCart
We construct weak ɛnets of almost linear size for certain types of point sets. Specifically, for planar point sets in convex position we construct weak 1 rnets of size O(rα(r)), where α(r) denotes the inverse Ackermann function. For point sets along the moment curve in Rd we construct weak 1 rnets of size r · 2poly(α(r)) , where the degree of the polynomial in the exponent depends (quadratically) on d. Our constructions result from a reduction to a new problem, which we call stabbing interval chains with jtuples. Given the range of integers N = [1, n], an interval chain of length k is a sequence of k consecutive, disjoint, nonempty intervals contained in N. A jtuple p = (p1,..., pj) is said to stab an interval chain C = I1 · · · Ik if each pi falls on a different interval of C. The problem is to construct a smallsize family Z of jtuples that stabs all kinterval chains in N. Let z (j)
Small Strong Epsilon Nets
"... In this paper, we initiate the study of small strong ϵnets and prove bounds for axisparallel rectangles, half spaces, strips and wedges. We also give some improved bounds for small weak ϵnets. 1 ..."
Abstract

Cited by 3 (3 self)
 Add to MetaCart
(Show Context)
In this paper, we initiate the study of small strong ϵnets and prove bounds for axisparallel rectangles, half spaces, strips and wedges. We also give some improved bounds for small weak ϵnets. 1
On Strong Centerpoints
"... Let P be a set of n points in Rd and F be a family of geometric objects. We call a point x ∈ P a strong centerpoint of P w.r.t F if x is contained in all F ∈ F that contains more than cn points from P, where c is a fixed constant. A strong centerpoint does not exist even when F is the family of half ..."
Abstract
 Add to MetaCart
(Show Context)
Let P be a set of n points in Rd and F be a family of geometric objects. We call a point x ∈ P a strong centerpoint of P w.r.t F if x is contained in all F ∈ F that contains more than cn points from P, where c is a fixed constant. A strong centerpoint does not exist even when F is the family of halfspaces in the plane. We prove the existence of strong centerpoints with exact constants for convex polytopes defined by a fixed set of orientations and hyperplanes in Rd. We also prove the existence of strong centerpoints for abstract set systems with bounded intersection. 1