3,777 research outputs found
Querying for the Largest Empty Geometric Object in a Desired Location
We study new types of geometric query problems defined as follows: given a
geometric set , preprocess it such that given a query point , the
location of the largest circle that does not contain any member of , but
contains can be reported efficiently. The geometric sets we consider for
are boundaries of convex and simple polygons, and point sets. While we
primarily focus on circles as the desired shape, we also briefly discuss empty
rectangles in the context of point sets.Comment: This version is a significant update of our earlier submission
arXiv:1004.0558v1. Apart from new variants studied in Sections 3 and 4, the
results have been improved in Section 5.Please note that the change in title
and abstract indicate that we have expanded the scope of the problems we
stud
On finding widest empty curved corridors
Open archive-ElsevierAn α-siphon of width w is the locus of points in the plane that are at the same distance w from a 1-corner polygonal chain C
such that α is the interior angle of C. Given a set P of n points in the plane and a fixed angle α, we want to compute the widest
empty α-siphon that splits P into two non-empty sets.We present an efficient O(n log3 n)-time algorithm for computing the widest
oriented α-siphon through P such that the orientation of a half-line of C is known.We also propose an O(n3 log2 n)-time algorithm
for the widest arbitrarily-oriented version and an (nlog n)-time algorithm for the widest arbitrarily-oriented α-siphon anchored
at a given point
Extensions of the Maximum Bichromatic Separating Rectangle Problem
In this paper, we study two extensions of the maximum bichromatic separating
rectangle (MBSR) problem introduced in \cite{Armaselu-CCCG, Armaselu-arXiv}.
One of the extensions, introduced in \cite{Armaselu-FWCG}, is called
\textit{MBSR with outliers} or MBSR-O, and is a more general version of the
MBSR problem in which the optimal rectangle is allowed to contain up to
outliers, where is given as part of the input. For MBSR-O, we improve the
previous known running time bounds of to . The other extension is called \textit{MBSR among circles} or
MBSR-C and asks for the largest axis-aligned rectangle separating red points
from blue unit circles. For MBSR-C, we provide an algorithm that runs in time.Comment: 14 pages, 14 figures, full version of CCCG pape
Motion planning in 2D and 3D with rotation
Imperial Users onl
Conflict-Free Coloring Made Stronger
In FOCS 2002, Even et al. showed that any set of discs in the plane can
be Conflict-Free colored with a total of at most colors. That is,
it can be colored with colors such that for any (covered) point
there is some disc whose color is distinct from all other colors of discs
containing . They also showed that this bound is asymptotically tight. In
this paper we prove the following stronger results:
\begin{enumerate} \item [(i)] Any set of discs in the plane can be
colored with a total of at most colors such that (a) for any
point that is covered by at least discs, there are at least
distinct discs each of which is colored by a color distinct from all other
discs containing and (b) for any point covered by at most discs,
all discs covering are colored distinctively. We call such a coloring a
{\em -Strong Conflict-Free} coloring. We extend this result to pseudo-discs
and arbitrary regions with linear union-complexity.
\item [(ii)] More generally, for families of simple closed Jordan regions
with union-complexity bounded by , we prove that there exists
a -Strong Conflict-Free coloring with at most colors.
\item [(iii)] We prove that any set of axis-parallel rectangles can be
-Strong Conflict-Free colored with at most colors.
\item [(iv)] We provide a general framework for -Strong Conflict-Free
coloring arbitrary hypergraphs. This framework relates the notion of -Strong
Conflict-Free coloring and the recently studied notion of -colorful
coloring. \end{enumerate}
All of our proofs are constructive. That is, there exist polynomial time
algorithms for computing such colorings
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