827 research outputs found
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
Erdos-Szekeres-type theorems for monotone paths and convex bodies
For any sequence of positive integers j_1 < j_2 < ... < j_n, the k-tuples
(j_i,j_{i + 1},...,j_{i + k-1}), i=1, 2,..., n - k+1, are said to form a
monotone path of length n. Given any integers n\ge k\ge 2 and q\ge 2, what is
the smallest integer N with the property that no matter how we color all
k-element subsets of [N]=\{1,2,..., N\} with q colors, we can always find a
monochromatic monotone path of length n? Denoting this minimum by N_k(q,n), it
follows from the seminal 1935 paper of Erd\H os and Szekeres that
N_2(q,n)=(n-1)^q+1 and N_3(2,n) = {2n -4\choose n-2} + 1. Determining the other
values of these functions appears to be a difficult task. Here we show that
2^{(n/q)^{q-1}} \leq N_3(q,n) \leq 2^{n^{q-1}\log n}, for q \geq 2 and n \geq
q+2. Using a stepping-up approach that goes back to Erdos and Hajnal, we prove
analogous bounds on N_k(q,n) for larger values of k, which are towers of height
k-1 in n^{q-1}. As a geometric application, we prove the following extension of
the Happy Ending Theorem. Every family of at least M(n)=2^{n^2 \log n} plane
convex bodies in general position, any pair of which share at most two boundary
points, has n members in convex position, that is, it has n members such that
each of them contributes a point to the boundary of the convex hull of their
union.Comment: 32 page
More on Decomposing Coverings by Octants
In this note we improve our upper bound given earlier by showing that every
9-fold covering of a point set in the space by finitely many translates of an
octant decomposes into two coverings, and our lower bound by a construction for
a 4-fold covering that does not decompose into two coverings. The same bounds
also hold for coverings of points in by finitely many homothets or
translates of a triangle. We also prove that certain dynamic interval coloring
problems are equivalent to the above question
Online and quasi-online colorings of wedges and intervals
We consider proper online colorings of hypergraphs defined by geometric
regions. We prove that there is an online coloring algorithm that colors
intervals of the real line using colors such that for every
point , contained in at least intervals, not all the intervals
containing have the same color. We also prove the corresponding result
about online coloring a family of wedges (quadrants) in the plane that are the
translates of a given fixed wedge. These results contrast the results of the
first and third author showing that in the quasi-online setting 12 colors are
enough to color wedges (independent of and ). We also consider
quasi-online coloring of intervals. In all cases we present efficient coloring
algorithms
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