16 research outputs found
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
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]
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
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
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
Hitting sets and colorings of hypergraphs
In this paper we study the minimal size of edges in hypergraph families which
guarantees the existence of a polychromatic coloring, that is, a -coloring
of a vertex set such that every hyperedge contains a vertex of all color
classes. We also investigate the connection of this problem with -shallow
hitting sets: sets of vertices that intersect each hyperedge in at least one
and at most vertices.
We determine in some hypergraph families the minimal for which a
-shallow hitting set exists.
We also study this problem for a special hypergraph family, which is induced
by arithmetic progressions with a difference from a given set. We show
connections between some geometric hypergraph families and the latter, and
prove relations between the set of differences and polychromatic colorability
Proper Coloring of Geometric Hypergraphs
We study whether for a given planar family F there is an m such that any finite set of points can be 3-colored so that anymember ofF that contains at leastm points contains two points with different colors. We conjecture that if F is a family of pseudo-disks, then such an m exists. We prove this in the special case when F is the family of all homothetic copies of a given convex polygon. We also study the problem in higher dimensions
Indecomposable Coverings with Concave Polygons
We show that for any concave polygon that has no parallel sides and for any k, there is a k-fold covering of some point set by the translates of this polygon that cannot be decomposed into two coverings. Moreover, we give a complete classification of open polygons with this property. We also construct for any polytope (having dimension at least three) and for any k, a k-fold covering of the space by its translates that cannot be decomposed into two coverings
Maximizing Barrier Coverage Lifetime with Mobile Sensors
Sensor networks are ubiquitously used for detection and tracking and as a
result covering is one of the main tasks of such networks. We study the problem
of maximizing the coverage lifetime of a barrier by mobile sensors with limited
battery powers, where the coverage lifetime is the time until there is a
breakdown in coverage due to the death of a sensor. Sensors are first deployed
and then coverage commences. Energy is consumed in proportion to the distance
traveled for mobility, while for coverage, energy is consumed in direct
proportion to the radius of the sensor raised to a constant exponent. We study
two variants which are distinguished by whether the sensing radii are given as
part of the input or can be optimized, the fixed radii problem and the variable
radii problem. We design parametric search algorithms for both problems for the
case where the final order of the sensors is predetermined and for the case
where sensors are initially located at barrier endpoints. In contrast, we show
that the variable radii problem is strongly NP-hard and provide hardness of
approximation results for fixed radii for the case where all the sensors are
initially co-located at an internal point of the barrier