1,072 research outputs found
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
On interference among moving sensors and related problems
We show that for any set of points moving along "simple" trajectories
(i.e., each coordinate is described with a polynomial of bounded degree) in
and any parameter , one can select a fixed non-empty
subset of the points of size , such that the Voronoi diagram of
this subset is "balanced" at any given time (i.e., it contains points
per cell). We also show that the bound is near optimal even for
the one dimensional case in which points move linearly in time. As
applications, we show that one can assign communication radii to the sensors of
a network of moving sensors so that at any given time their interference is
. We also show some results in kinetic approximate range
counting and kinetic discrepancy. In order to obtain these results, we extend
well-known results from -net theory to kinetic environments
Selection Lemmas for various geometric objects
Selection lemmas are classical results in discrete geometry that have been
well studied and have applications in many geometric problems like weak epsilon
nets and slimming Delaunay triangulations. Selection lemma type results
typically show that there exists a point that is contained in many objects that
are induced (spanned) by an underlying point set.
In the first selection lemma, we consider the set of all the objects induced
(spanned) by a point set . This question has been widely explored for
simplices in , with tight bounds in . In our paper,
we prove first selection lemma for other classes of geometric objects. We also
consider the strong variant of this problem where we add the constraint that
the piercing point comes from . We prove an exact result on the strong and
the weak variant of the first selection lemma for axis-parallel rectangles,
special subclasses of axis-parallel rectangles like quadrants and slabs, disks
(for centrally symmetric point sets). We also show non-trivial bounds on the
first selection lemma for axis-parallel boxes and hyperspheres in
.
In the second selection lemma, we consider an arbitrary sized subset of
the set of all objects induced by . We study this problem for axis-parallel
rectangles and show that there exists an point in the plane that is contained
in rectangles. This is an improvement over the previous
bound by Smorodinsky and Sharir when is almost quadratic
Polychromatic Coloring for Half-Planes
We prove that for every integer , every finite set of points in the plane
can be -colored so that every half-plane that contains at least
points, also contains at least one point from every color class. We also show
that the bound is best possible. This improves the best previously known
lower and upper bounds of and respectively. We also show
that every finite set of half-planes can be colored so that if a point
belongs to a subset of at least of the half-planes then
contains a half-plane from every color class. This improves the best previously
known upper bound of . Another corollary of our first result is a new
proof of the existence of small size \eps-nets for points in the plane with
respect to half-planes.Comment: 11 pages, 5 figure
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