16 research outputs found
Decomposition of Multiple Coverings into More Parts
We prove that for every centrally symmetric convex polygon Q, there exists a
constant alpha such that any alpha*k-fold covering of the plane by translates
of Q can be decomposed into k coverings. This improves on a quadratic upper
bound proved by Pach and Toth (SoCG'07). The question is motivated by a sensor
network problem, in which a region has to be monitored by sensors with limited
battery lifetime
Making Triangles Colorful
We prove that for any point set P in the plane, a triangle T, and a positive
integer k, there exists a coloring of P with k colors such that any homothetic
copy of T containing at least ck^8 points of P, for some constant c, contains
at least one of each color. This is the first polynomial bound for range spaces
induced by homothetic polygons. The only previously known bound for this
problem applies to the more general case of octants in R^3, but is doubly
exponential.Comment: 6 page
Coloring Hypergraphs Induced by Dynamic Point Sets and Bottomless Rectangles
We consider a coloring problem on dynamic, one-dimensional point sets: points
appearing and disappearing on a line at given times. We wish to color them with
k colors so that at any time, any sequence of p(k) consecutive points, for some
function p, contains at least one point of each color.
We prove that no such function p(k) exists in general. However, in the
restricted case in which points appear gradually, but never disappear, we give
a coloring algorithm guaranteeing the property at any time with p(k)=3k-2. This
can be interpreted as coloring point sets in R^2 with k colors such that any
bottomless rectangle containing at least 3k-2 points contains at least one
point of each color. Here a bottomless rectangle is an axis-aligned rectangle
whose bottom edge is below the lowest point of the set. For this problem, we
also prove a lower bound p(k)>ck, where c>1.67. Hence for every k there exists
a point set, every k-coloring of which is such that there exists a bottomless
rectangle containing ck points and missing at least one of the k colors.
Chen et al. (2009) proved that no such function exists in the case of
general axis-aligned rectangles. Our result also complements recent results
from Keszegh and Palvolgyi on cover-decomposability of octants (2011, 2012).Comment: A preliminary version was presented by a subset of the authors to the
European Workshop on Computational Geometry, held in Assisi (Italy) on March
19-21, 201
Colorful Strips
Given a planar point set and an integer , we wish to color the points with
colors so that any axis-aligned strip containing enough points contains all
colors. The goal is to bound the necessary size of such a strip, as a function
of . We show that if the strip size is at least , such a coloring
can always be found. We prove that the size of the strip is also bounded in any
fixed number of dimensions. In contrast to the planar case, we show that
deciding whether a 3D point set can be 2-colored so that any strip containing
at least three points contains both colors is NP-complete.
We also consider the problem of coloring a given set of axis-aligned strips,
so that any sufficiently covered point in the plane is covered by colors.
We show that in dimensions the required coverage is at most .
Lower bounds are given for the two problems. This complements recent
impossibility results on decomposition of strip coverings with arbitrary
orientations. Finally, we study a variant where strips are replaced by wedges
Making Octants Colorful and Related Covering Decomposition Problems
We give new positive results on the long-standing open problem of geometric
covering decomposition for homothetic polygons. In particular, we prove that
for any positive integer k, every finite set of points in R^3 can be colored
with k colors so that every translate of the negative octant containing at
least k^6 points contains at least one of each color. The best previously known
bound was doubly exponential in k. This yields, among other corollaries, the
first polynomial bound for the decomposability of multiple coverings by
homothetic triangles. We also investigate related decomposition problems
involving intervals appearing on a line. We prove that no algorithm can
dynamically maintain a decomposition of a multiple covering by intervals under
insertion of new intervals, even in a semi-online model, in which some coloring
decisions can be delayed. This implies that a wide range of sweeping plane
algorithms cannot guarantee any bound even for special cases of the octant
problem.Comment: version after revision process; minor changes in the expositio
Set It and Forget It: Approximating the Set Once Strip Cover Problem
We consider the Set Once Strip Cover problem, in which n wireless sensors are
deployed over a one-dimensional region. Each sensor has a fixed battery that
drains in inverse proportion to a radius that can be set just once, but
activated at any time. The problem is to find an assignment of radii and
activation times that maximizes the length of time during which the entire
region is covered. We show that this problem is NP-hard. Second, we show that
RoundRobin, the algorithm in which the sensors simply take turns covering the
entire region, has a tight approximation guarantee of 3/2 in both Set Once
Strip Cover and the more general Strip Cover problem, in which each radius may
be set finitely-many times. Moreover, we show that the more general class of
duty cycle algorithms, in which groups of sensors take turns covering the
entire region, can do no better. Finally, we give an optimal O(n^2 log n)-time
algorithm for the related Set Radius Strip Cover problem, in which all sensors
must be activated immediately.Comment: briefly announced at SPAA 201
Octants are cover-decomposable into many coverings
We prove that octants are cover-decomposable into multiple coverings, i.e., for any k there is an m(k)m(k) such that any m(k)m(k)-fold covering of any subset of the space with a finite number of translates of a given octant can be decomposed into k coverings. As a corollary, we obtain that any m(k)m(k)-fold covering of any subset of the plane with a finite number of homothetic copies of a given triangle can be decomposed into k coverings. Previously only some weaker bounds were known for related problems [20]
A Characterization of Visibility Graphs for Pseudo-Polygons
In this paper, we give a characterization of the visibility graphs of
pseudo-polygons. We first identify some key combinatorial properties of
pseudo-polygons, and we then give a set of five necessary conditions based off
our identified properties. We then prove that these necessary conditions are
also sufficient via a reduction to a characterization of vertex-edge visibility
graphs given by O'Rourke and Streinu