42,371 research outputs found
Unsplittable coverings in the plane
A system of sets forms an {\em -fold covering} of a set if every point
of belongs to at least of its members. A -fold covering is called a
{\em covering}. The problem of splitting multiple coverings into several
coverings was motivated by classical density estimates for {\em sphere
packings} as well as by the {\em planar sensor cover problem}. It has been the
prevailing conjecture for 35 years (settled in many special cases) that for
every plane convex body , there exists a constant such that every
-fold covering of the plane with translates of splits into
coverings. In the present paper, it is proved that this conjecture is false for
the unit disk. The proof can be generalized to construct, for every , an
unsplittable -fold covering of the plane with translates of any open convex
body which has a smooth boundary with everywhere {\em positive curvature}.
Somewhat surprisingly, {\em unbounded} open convex sets do not misbehave,
they satisfy the conjecture: every -fold covering of any region of the plane
by translates of such a set splits into two coverings. To establish this
result, we prove a general coloring theorem for hypergraphs of a special type:
{\em shift-chains}. We also show that there is a constant such that, for
any positive integer , every -fold covering of a region with unit disks
splits into two coverings, provided that every point is covered by {\em at
most} sets
A Bicriteria Approximation for the Reordering Buffer Problem
In the reordering buffer problem (RBP), a server is asked to process a
sequence of requests lying in a metric space. To process a request the server
must move to the corresponding point in the metric. The requests can be
processed slightly out of order; in particular, the server has a buffer of
capacity k which can store up to k requests as it reads in the sequence. The
goal is to reorder the requests in such a manner that the buffer constraint is
satisfied and the total travel cost of the server is minimized. The RBP arises
in many applications that require scheduling with a limited buffer capacity,
such as scheduling a disk arm in storage systems, switching colors in paint
shops of a car manufacturing plant, and rendering 3D images in computer
graphics.
We study the offline version of RBP and develop bicriteria approximations.
When the underlying metric is a tree, we obtain a solution of cost no more than
9OPT using a buffer of capacity 4k + 1 where OPT is the cost of an optimal
solution with buffer capacity k. Constant factor approximations were known
previously only for the uniform metric (Avigdor-Elgrabli et al., 2012). Via
randomized tree embeddings, this implies an O(log n) approximation to cost and
O(1) approximation to buffer size for general metrics. Previously the best
known algorithm for arbitrary metrics by Englert et al. (2007) provided an
O(log^2 k log n) approximation without violating the buffer constraint.Comment: 13 page
Recursive tilings and space-filling curves with little fragmentation
This paper defines the Arrwwid number of a recursive tiling (or space-filling
curve) as the smallest number w such that any ball Q can be covered by w tiles
(or curve sections) with total volume O(vol(Q)). Recursive tilings and
space-filling curves with low Arrwwid numbers can be applied to optimise disk,
memory or server access patterns when processing sets of points in
d-dimensional space. This paper presents recursive tilings and space-filling
curves with optimal Arrwwid numbers. For d >= 3, we see that regular cube
tilings and space-filling curves cannot have optimal Arrwwid number, and we see
how to construct alternatives with better Arrwwid numbers.Comment: Manuscript accompanying abstract in EuroCG 2010, including full
proofs, 20 figures, references, discussion et
Minimum-Cost Coverage of Point Sets by Disks
We consider a class of geometric facility location problems in which the goal
is to determine a set X of disks given by their centers (t_j) and radii (r_j)
that cover a given set of demand points Y in the plane at the smallest possible
cost. We consider cost functions of the form sum_j f(r_j), where f(r)=r^alpha
is the cost of transmission to radius r. Special cases arise for alpha=1 (sum
of radii) and alpha=2 (total area); power consumption models in wireless
network design often use an exponent alpha>2. Different scenarios arise
according to possible restrictions on the transmission centers t_j, which may
be constrained to belong to a given discrete set or to lie on a line, etc. We
obtain several new results, including (a) exact and approximation algorithms
for selecting transmission points t_j on a given line in order to cover demand
points Y in the plane; (b) approximation algorithms (and an algebraic
intractability result) for selecting an optimal line on which to place
transmission points to cover Y; (c) a proof of NP-hardness for a discrete set
of transmission points in the plane and any fixed alpha>1; and (d) a
polynomial-time approximation scheme for the problem of computing a minimum
cost covering tour (MCCT), in which the total cost is a linear combination of
the transmission cost for the set of disks and the length of a tour/path that
connects the centers of the disks.Comment: 10 pages, 4 figures, Latex, to appear in ACM Symposium on
Computational Geometry 200
Topological Stability of Kinetic -Centers
We study the -center problem in a kinetic setting: given a set of
continuously moving points in the plane, determine a set of (moving)
disks that cover at every time step, such that the disks are as small as
possible at any point in time. Whereas the optimal solution over time may
exhibit discontinuous changes, many practical applications require the solution
to be stable: the disks must move smoothly over time. Existing results on this
problem require the disks to move with a bounded speed, but this model is very
hard to work with. Hence, the results are limited and offer little theoretical
insight. Instead, we study the topological stability of -centers.
Topological stability was recently introduced and simply requires the solution
to change continuously, but may do so arbitrarily fast. We prove upper and
lower bounds on the ratio between the radii of an optimal but unstable solution
and the radii of a topologically stable solution---the topological stability
ratio---considering various metrics and various optimization criteria. For we provide tight bounds, and for small we can obtain nontrivial
lower and upper bounds. Finally, we provide an algorithm to compute the
topological stability ratio in polynomial time for constant
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