885 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
New Pairwise Spanners
Let G = (V,E) be an undirected unweighted graph on n vertices. A subgraph H of G is called an (all-pairs) purely additive spanner with stretch beta if for every (u,v) in V times V, mathsf{dist}_H(u,v) le mathsf{dist}_G(u,v) + beta. The problem of computing sparse spanners with small stretch beta is well-studied. Here we consider the following relaxation: we are given psubseteq V times V and we seek a sparse subgraph H where mathsf{dist}_H(u,v)le mathsf{dist}_G(u,v) + beta for each (u,v) in p. Such a subgraph is called a pairwise spanner with additive stretch beta and our goal is to construct
such subgraphs that are sparser than all-pairs spanners with the same stretch. We show sparse pairwise spanners with additive stretch 4 and with additive stretch 6. We also consider the following special cases: p = S times V and p = S times T, where Ssubseteq V and Tsubseteq V, and show sparser pairwise spanners for these cases
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