5,442 research outputs found
On the stretch factor of the Theta-4 graph
In this paper we show that the \theta-graph with 4 cones has constant stretch
factor, i.e., there is a path between any pair of vertices in this graph whose
length is at most a constant times the Euclidean distance between that pair of
vertices. This is the last \theta-graph for which it was not known whether its
stretch factor was bounded
Spanning Properties of Theta-Theta Graphs
We study the spanning properties of Theta-Theta graphs. Similar in spirit
with the Yao-Yao graphs, Theta-Theta graphs partition the space around each
vertex into a set of k cones, for some fixed integer k > 1, and select at most
one edge per cone. The difference is in the way edges are selected. Yao-Yao
graphs select an edge of minimum length, whereas Theta-Theta graphs select an
edge of minimum orthogonal projection onto the cone bisector. It has been
established that the Yao-Yao graphs with parameter k = 6k' have spanning ratio
11.67, for k' >= 6. In this paper we establish a first spanning ratio of
for Theta-Theta graphs, for the same values of . We also extend the class of
Theta-Theta spanners with parameter 6k', and establish a spanning ratio of
for k' >= 5. We surmise that these stronger results are mainly due to a
tighter analysis in this paper, rather than Theta-Theta being superior to
Yao-Yao as a spanner. We also show that the spanning ratio of Theta-Theta
graphs decreases to 4.64 as k' increases to 8. These are the first results on
the spanning properties of Theta-Theta graphs.Comment: 20 pages, 6 figures, 3 table
An Infinite Class of Sparse-Yao Spanners
We show that, for any integer k > 5, the Sparse-Yao graph YY_{6k} (also known
as Yao-Yao) is a spanner with stretch factor 11.67. The stretch factor drops
down to 4.75 for k > 7.Comment: 17 pages, 12 figure
Exponential Time Complexity of the Permanent and the Tutte Polynomial
We show conditional lower bounds for well-studied #P-hard problems:
(a) The number of satisfying assignments of a 2-CNF formula with n variables
cannot be counted in time exp(o(n)), and the same is true for computing the
number of all independent sets in an n-vertex graph.
(b) The permanent of an n x n matrix with entries 0 and 1 cannot be computed
in time exp(o(n)).
(c) The Tutte polynomial of an n-vertex multigraph cannot be computed in time
exp(o(n)) at most evaluation points (x,y) in the case of multigraphs, and it
cannot be computed in time exp(o(n/polylog n)) in the case of simple graphs.
Our lower bounds are relative to (variants of) the Exponential Time
Hypothesis (ETH), which says that the satisfiability of n-variable 3-CNF
formulas cannot be decided in time exp(o(n)). We relax this hypothesis by
introducing its counting version #ETH, namely that the satisfying assignments
cannot be counted in time exp(o(n)). In order to use #ETH for our lower bounds,
we transfer the sparsification lemma for d-CNF formulas to the counting
setting
The Stretch - Length Tradeoff in Geometric Networks: Average Case and Worst Case Study
Consider a network linking the points of a rate- Poisson point process on
the plane. Write \Psi^{\mbox{ave}}(s) for the minimum possible mean length
per unit area of such a network, subject to the constraint that the
route-length between every pair of points is at most times the Euclidean
distance. We give upper and lower bounds on the function
\Psi^{\mbox{ave}}(s), and on the analogous "worst-case" function
\Psi^{\mbox{worst}}(s) where the point configuration is arbitrary subject to
average density one per unit area. Our bounds are numerically crude, but raise
the question of whether there is an exponent such that each function
has as .Comment: 33 page
The complexity of approximating the complex-valued Potts model
We study the complexity of approximating the partition function of the
-state Potts model and the closely related Tutte polynomial for complex
values of the underlying parameters. Apart from the classical connections with
quantum computing and phase transitions in statistical physics, recent work in
approximate counting has shown that the behaviour in the complex plane, and
more precisely the location of zeros, is strongly connected with the complexity
of the approximation problem, even for positive real-valued parameters.
Previous work in the complex plane by Goldberg and Guo focused on , which
corresponds to the case of the Ising model; for , the behaviour in the
complex plane is not as well understood and most work applies only to the
real-valued Tutte plane.
Our main result is a complete classification of the complexity of the
approximation problems for all non-real values of the parameters, by
establishing \#P-hardness results that apply even when restricted to planar
graphs. Our techniques apply to all and further complement/refine
previous results both for the Ising model and the Tutte plane, answering in
particular a question raised by Bordewich, Freedman, Lov\'{a}sz and Welsh in
the context of quantum computations.Comment: 58 pages. Changes on version 2: minor change
Upper and Lower Bounds for Competitive Online Routing on Delaunay Triangulations
Consider a weighted graph G where vertices are points in the plane and edges
are line segments. The weight of each edge is the Euclidean distance between
its two endpoints. A routing algorithm on G has a competitive ratio of c if the
length of the path produced by the algorithm from any vertex s to any vertex t
is at most c times the length of the shortest path from s to t in G. If the
length of the path is at most c times the Euclidean distance from s to t, we
say that the routing algorithm on G has a routing ratio of c.We present an
online routing algorithm on the Delaunay triangulation with competitive and
routing ratios of 5.90. This improves upon the best known algorithm that has
competitive and routing ratio 15.48. The algorithm is a generalization of the
deterministic 1-local routing algorithm by Chew on the L1-Delaunay
triangulation. When a message follows the routing path produced by our
algorithm, its header need only contain the coordinates of s and t. This is an
improvement over the currently known competitive routing algorithms on the
Delaunay triangulation, for which the header of a message must additionally
contain partial sums of distances along the routing path.We also show that the
routing ratio of any deterministic k-local algorithm is at least 1.70 for the
Delaunay triangulation and 2.70 for the L1-Delaunay triangulation. In the case
of the L1-Delaunay triangulation, this implies that even though there exists a
path between two points x and y whose length is at most 2.61|[xy]| (where
|[xy]| denotes the length of the line segment [xy]), it is not always possible
to route a message along a path of length less than 2.70|[xy]|. From these
bounds on the routing ratio, we derive lower bounds on the competitive ratio of
1.23 for Delaunay triangulations and 1.12 for L1-Delaunay triangulations
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