32 research outputs found
A Center Transversal Theorem for Hyperplanes and Applications to Graph Drawing
Motivated by an open problem from graph drawing, we study several
partitioning problems for line and hyperplane arrangements. We prove a
ham-sandwich cut theorem: given two sets of n lines in R^2, there is a line l
such that in both line sets, for both halfplanes delimited by l, there are
n^{1/2} lines which pairwise intersect in that halfplane, and this bound is
tight; a centerpoint theorem: for any set of n lines there is a point such that
for any halfplane containing that point there are (n/3)^{1/2} of the lines
which pairwise intersect in that halfplane. We generalize those results in
higher dimension and obtain a center transversal theorem, a same-type lemma,
and a positive portion Erdos-Szekeres theorem for hyperplane arrangements. This
is done by formulating a generalization of the center transversal theorem which
applies to set functions that are much more general than measures. Back to
Graph Drawing (and in the plane), we completely solve the open problem that
motivated our search: there is no set of n labelled lines that are universal
for all n-vertex labelled planar graphs. As a side note, we prove that every
set of n (unlabelled) lines is universal for all n-vertex (unlabelled) planar
graphs
An upper bound on the k-modem illumination problem
A variation on the classical polygon illumination problem was introduced in
[Aichholzer et. al. EuroCG'09]. In this variant light sources are replaced by
wireless devices called k-modems, which can penetrate a fixed number k, of
"walls". A point in the interior of a polygon is "illuminated" by a k-modem if
the line segment joining them intersects at most k edges of the polygon. It is
easy to construct polygons of n vertices where the number of k-modems required
to illuminate all interior points is Omega(n/k). However, no non-trivial upper
bound is known. In this paper we prove that the number of k-modems required to
illuminate any polygon of n vertices is at most O(n/k). For the cases of
illuminating an orthogonal polygon or a set of disjoint orthogonal segments, we
give a tighter bound of 6n/k + 1. Moreover, we present an O(n log n) time
algorithm to achieve this bound.Comment: 9 pages, 4 figure