246 research outputs found
On Strong Centerpoints
Let be a set of points in and be a
family of geometric objects. We call a point a strong centerpoint of
w.r.t if is contained in all that
contains more than points from , where is a fixed constant. A
strong centerpoint does not exist even when 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. We
also prove the existence of strong centerpoints for abstract set systems with
bounded intersection
Approximating Tverberg Points in Linear Time for Any Fixed Dimension
Let P be a d-dimensional n-point set. A Tverberg-partition of P is a
partition of P into r sets P_1, ..., P_r such that the convex hulls conv(P_1),
..., conv(P_r) have non-empty intersection. A point in the intersection of the
conv(P_i)'s is called a Tverberg point of depth r for P. A classic result by
Tverberg implies that there always exists a Tverberg partition of size n/(d+1),
but it is not known how to find such a partition in polynomial time. Therefore,
approximate solutions are of interest.
We describe a deterministic algorithm that finds a Tverberg partition of size
n/4(d+1)^3 in time d^{O(log d)} n. This means that for every fixed dimension we
can compute an approximate Tverberg point (and hence also an approximate
centerpoint) in linear time. Our algorithm is obtained by combining a novel
lifting approach with a recent result by Miller and Sheehy (2010).Comment: 14 pages, 2 figures. A preliminary version appeared in SoCG 2012.
This version removes an incorrect example at the end of Section 3.
Small Strong Epsilon Nets
Let P be a set of n points in . A point x is said to be a
centerpoint of P if x is contained in every convex object that contains more
than points of P. We call a point x a strong centerpoint for a
family of objects if is contained in every object that contains more than a constant fraction of points of P. A
strong centerpoint does not exist even for halfspaces in . We
prove that a strong centerpoint exists for axis-parallel boxes in
and give exact bounds. We then extend this to small strong
-nets in the plane and prove upper and lower bounds for
where is the family of axis-parallel
rectangles, halfspaces and disks. Here represents the
smallest real number in such that there exists an
-net of size i with respect to .Comment: 19 pages, 12 figure
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
No-dimensional Tverberg Partitions Revisited
\newcommand{\epsA}{\Mh{\delta}} \newcommand{\Re}{\mathbb{R}}
\newcommand{\reals}{\mathbb{R}} \newcommand{\SetX}{\mathsf{X}}
\newcommand{\diam}{\Delta} \newcommand{\Mh}[1]{#1} \newcommand{\query}{q}
\newcommand{\eps}{\varepsilon} \newcommand{\VorX}[1]{\mathcal{V} \pth{#1}}
\newcommand{\IntRange}[1]{[ #1 ]} \newcommand{\Space}{\overline{\mathsf{m}}}
\newcommand{\pth}[2][\!]{#1\left({#2}\right)}
\newcommand{\polylog}{\mathrm{polylog}} \newcommand{\N}{\mathbb N}
\newcommand{\Z}{\mathbb Z} \newcommand{\pt}{p} \newcommand{\distY}[2]{\left\|
{#1} - {#2} \right\|} \newcommand{\PP}{P} \newcommand{\ptq}{q}
\newcommand{\pts}{s} Given a set \PP \subset \Re^d of points, with
diameter \diam, and a parameter \epsA \in (0,1), it is known that there is
a partition of \PP into sets \PP_1, \ldots, \PP_t, each of size
O(1/\epsA^2), such that their convex-hulls all intersect a common ball of
radius \epsA \diam. We prove that a random partition, with a simple
alteration step, yields the desired partition, resulting in a linear time
algorithm. Previous proofs were either existential (i.e., at least exponential
time), or required much bigger sets. In addition, the algorithm and its proof
of correctness are significantly simpler than previous work, and the constants
are slightly better.
In addition, we provide a linear time algorithm for computing a ``fuzzy''
centerpoint. We also prove a no-dimensional weak \eps-net theorem with an
improved constant
- …