8,854 research outputs found
On the geometric dilation of closed curves, graphs, and point sets
The detour between two points u and v (on edges or vertices) of an embedded
planar graph whose edges are curves is the ratio between the shortest path in
in the graph between u and v and their Euclidean distance. The maximum detour
over all pairs of points is called the geometric dilation. Ebbers-Baumann,
Gruene and Klein have shown that every finite point set is contained in a
planar graph whose geometric dilation is at most 1.678, and some point sets
require graphs with dilation at least pi/2 = 1.57... We prove a stronger lower
bound of 1.00000000001*pi/2 by relating graphs with small dilation to a problem
of packing and covering the plane by circular disks.
The proof relies on halving pairs, pairs of points dividing a given closed
curve C in two parts of equal length, and their minimum and maximum distances h
and H. Additionally, we analyze curves of constant halving distance (h=H),
examine the relation of h to other geometric quantities and prove some new
dilation bounds.Comment: 31 pages, 16 figures. The new version is the extended journal
submission; it includes additional material from a conference submission
(ref. [6] in the paper
Lower bounds on the dilation of plane spanners
(I) We exhibit a set of 23 points in the plane that has dilation at least
, improving the previously best lower bound of for the
worst-case dilation of plane spanners.
(II) For every integer , there exists an -element point set
such that the degree 3 dilation of denoted by in the domain of plane geometric spanners. In the
same domain, we show that for every integer , there exists a an
-element point set such that the degree 4 dilation of denoted by
The
previous best lower bound of holds for any degree.
(III) For every integer , there exists an -element point set
such that the stretch factor of the greedy triangulation of is at least
.Comment: Revised definitions in the introduction; 23 pages, 15 figures; 2
table
Computing the Greedy Spanner in Linear Space
The greedy spanner is a high-quality spanner: its total weight, edge count
and maximal degree are asymptotically optimal and in practice significantly
better than for any other spanner with reasonable construction time.
Unfortunately, all known algorithms that compute the greedy spanner of n points
use Omega(n^2) space, which is impractical on large instances. To the best of
our knowledge, the largest instance for which the greedy spanner was computed
so far has about 13,000 vertices.
We present a O(n)-space algorithm that computes the same spanner for points
in R^d running in O(n^2 log^2 n) time for any fixed stretch factor and
dimension. We discuss and evaluate a number of optimizations to its running
time, which allowed us to compute the greedy spanner on a graph with a million
vertices. To our knowledge, this is also the first algorithm for the greedy
spanner with a near-quadratic running time guarantee that has actually been
implemented
Beta-Skeletons have Unbounded Dilation
A fractal construction shows that, for any beta>0, the beta-skeleton of a
point set can have arbitrarily large dilation. In particular this applies to
the Gabriel graph.Comment: 8 pages, 9 figure
Refining Multivariate Value Set Bounds
Over finite fields, if the image of a polynomial map is not the entire field,
then its cardinality can be bounded above by a significantly smaller value.
Earlier results bound the cardinality of the value set using the degree of the
polynomial, but more recent results make use of the powers of all monomials.
In this paper, we explore the geometric properties of the Newton polytope and
show how they allow for tighter upper bounds on the cardinality of the
multivariate value set. We then explore a method which allows for even stronger
upper bounds, regardless of whether one uses the multivariate degree or the
Newton polytope to bound the value set. Effectively, this provides an alternate
proof of Kosters' degree bound, an improved Newton polytope-based bound, and an
improvement of a degree matrix-based result given by Zan and Cao.Comment: 41 pages, 1 figure. Preprint of a dissertation to be published with
Proquest/UMI (in press). Shortened version of manuscript for publication
elsewhere is in preparatio
- …