11,711 research outputs found
Improved Purely Additive Fault-Tolerant Spanners
Let be an unweighted -node undirected graph. A \emph{-additive
spanner} of is a spanning subgraph of such that distances in
are stretched at most by an additive term w.r.t. the corresponding
distances in . A natural research goal related with spanners is that of
designing \emph{sparse} spanners with \emph{low} stretch.
In this paper, we focus on \emph{fault-tolerant} additive spanners, namely
additive spanners which are able to preserve their additive stretch even when
one edge fails. We are able to improve all known such spanners, in terms of
either sparsity or stretch. In particular, we consider the sparsest known
spanners with stretch , , and , and reduce the stretch to , ,
and , respectively (while keeping the same sparsity).
Our results are based on two different constructions. On one hand, we show
how to augment (by adding a \emph{small} number of edges) a fault-tolerant
additive \emph{sourcewise spanner} (that approximately preserves distances only
from a given set of source nodes) into one such spanner that preserves all
pairwise distances. On the other hand, we show how to augment some known
fault-tolerant additive spanners, based on clustering techniques. This way we
decrease the additive stretch without any asymptotic increase in their size. We
also obtain improved fault-tolerant additive spanners for the case of one
vertex failure, and for the case of edge failures.Comment: 17 pages, 4 figures, ESA 201
Optimal Euclidean spanners: really short, thin and lanky
In a seminal STOC'95 paper, titled "Euclidean spanners: short, thin and
lanky", Arya et al. devised a construction of Euclidean (1+\eps)-spanners
that achieves constant degree, diameter , and weight , and has running time . This construction
applies to -point constant-dimensional Euclidean spaces. Moreover, Arya et
al. conjectured that the weight bound can be improved by a logarithmic factor,
without increasing the degree and the diameter of the spanner, and within the
same running time.
This conjecture of Arya et al. became a central open problem in the area of
Euclidean spanners.
In this paper we resolve the long-standing conjecture of Arya et al. in the
affirmative. Specifically, we present a construction of spanners with the same
stretch, degree, diameter, and running time, as in Arya et al.'s result, but
with optimal weight .
Moreover, our result is more general in three ways. First, we demonstrate
that the conjecture holds true not only in constant-dimensional Euclidean
spaces, but also in doubling metrics. Second, we provide a general tradeoff
between the three involved parameters, which is tight in the entire range.
Third, we devise a transformation that decreases the lightness of spanners in
general metrics, while keeping all their other parameters in check. Our main
result is obtained as a corollary of this transformation.Comment: A technical report of this paper was available online from April 4,
201
Additive Spanners: A Simple Construction
We consider additive spanners of unweighted undirected graphs. Let be a
graph and a subgraph of . The most na\"ive way to construct an additive
-spanner of is the following: As long as is not an additive
-spanner repeat: Find a pair that violates the
spanner-condition and a shortest path from to in . Add the edges of
this path to .
We show that, with a very simple initial graph , this na\"ive method gives
additive - and -spanners of sizes matching the best known upper bounds.
For additive -spanners we start with and end with
edges in the spanner. For additive -spanners we start with containing
arbitrary edges incident to each node and end with a
spanner of size .Comment: To appear at proceedings of the 14th Scandinavian Symposium and
Workshop on Algorithm Theory (SWAT 2014
Light Spanners
A -spanner of a weighted undirected graph , is a subgraph
such that for all . The sparseness of
the spanner can be measured by its size (the number of edges) and weight (the
sum of all edge weights), both being important measures of the spanner's
quality -- in this work we focus on the latter.
Specifically, it is shown that for any parameters and ,
any weighted graph on vertices admits a
-stretch spanner of weight at most , where is the weight of a minimum
spanning tree of . Our result is obtained via a novel analysis of the
classic greedy algorithm, and improves previous work by a factor of .Comment: 10 pages, 1 figure, to appear in ICALP 201
Volumetric Spanners: an Efficient Exploration Basis for Learning
Numerous machine learning problems require an exploration basis - a mechanism
to explore the action space. We define a novel geometric notion of exploration
basis with low variance, called volumetric spanners, and give efficient
algorithms to construct such a basis.
We show how efficient volumetric spanners give rise to the first efficient
and optimal regret algorithm for bandit linear optimization over general convex
sets. Previously such results were known only for specific convex sets, or
under special conditions such as the existence of an efficient self-concordant
barrier for the underlying set
- …