660 research outputs found
Stabbing line segments with disks: complexity and approximation algorithms
Computational complexity and approximation algorithms are reported for a
problem of stabbing a set of straight line segments with the least cardinality
set of disks of fixed radii where the set of segments forms a straight
line drawing of a planar graph without edge crossings. Close
geometric problems arise in network security applications. We give strong
NP-hardness of the problem for edge sets of Delaunay triangulations, Gabriel
graphs and other subgraphs (which are often used in network design) for and some constant where and
are Euclidean lengths of the longest and shortest graph edges
respectively. Fast -time -approximation algorithm is
proposed within the class of straight line drawings of planar graphs for which
the inequality holds uniformly for some constant
i.e. when lengths of edges of are uniformly bounded from above by
some linear function of Comment: 12 pages, 1 appendix, 15 bibliography items, 6th International
Conference on Analysis of Images, Social Networks and Texts (AIST-2017
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
Double Bubbles Minimize
The classical isoperimetric inequality in R^3 states that the surface of
smallest area enclosing a given volume is a sphere. We show that the least area
surface enclosing two equal volumes is a double bubble, a surface made of two
pieces of round spheres separated by a flat disk, meeting along a single circle
at an angle of 120 degrees.Comment: 57 pages, 32 figures. Includes the complete code for a C++ program as
described in the article. You can obtain this code by viewing the source of
this articl
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