121 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
Vertex Fault Tolerant Additive Spanners
A {\em fault-tolerant} structure for a network is required to continue
functioning following the failure of some of the network's edges or vertices.
In this paper, we address the problem of designing a {\em fault-tolerant}
additive spanner, namely, a subgraph of the network such that
subsequent to the failure of a single vertex, the surviving part of still
contains an \emph{additive} spanner for (the surviving part of) , satisfying
for every
. Recently, the problem of constructing fault-tolerant additive
spanners resilient to the failure of up to \emph{edges} has been considered
by Braunschvig et. al. The problem of handling \emph{vertex} failures was left
open therein. In this paper we develop new techniques for constructing additive
FT-spanners overcoming the failure of a single vertex in the graph. Our first
result is an FT-spanner with additive stretch and
edges. Our second result is an FT-spanner with additive stretch and
edges. The construction algorithm consists of two main
components: (a) constructing an FT-clustering graph and (b) applying a modified
path-buying procedure suitably adopted to failure prone settings. Finally, we
also describe two constructions for {\em fault-tolerant multi-source additive
spanners}, aiming to guarantee a bounded additive stretch following a vertex
failure, for every pair of vertices in for a given subset of
sources . The additive stretch bounds of our constructions are 4
and 8 (using a different number of edges)
Fault-tolerant additive weighted geometric spanners
Let S be a set of n points and let w be a function that assigns non-negative
weights to points in S. The additive weighted distance d_w(p, q) between two
points p,q belonging to S is defined as w(p) + d(p, q) + w(q) if p \ne q and it
is zero if p = q. Here, d(p, q) denotes the (geodesic) Euclidean distance
between p and q. A graph G(S, E) is called a t-spanner for the additive
weighted set S of points if for any two points p and q in S the distance
between p and q in graph G is at most t.d_w(p, q) for a real number t > 1.
Here, d_w(p,q) is the additive weighted distance between p and q. For some
integer k \geq 1, a t-spanner G for the set S is a (k, t)-vertex fault-tolerant
additive weighted spanner, denoted with (k, t)-VFTAWS, if for any set S'
\subset S with cardinality at most k, the graph G \ S' is a t-spanner for the
points in S \ S'. For any given real number \epsilon > 0, we obtain the
following results:
- When the points in S belong to Euclidean space R^d, an algorithm to compute
a (k,(2 + \epsilon))-VFTAWS with O(kn) edges for the metric space (S, d_w).
Here, for any two points p, q \in S, d(p, q) is the Euclidean distance between
p and q in R^d.
- When the points in S belong to a simple polygon P, for the metric space (S,
d_w), one algorithm to compute a geodesic (k, (2 + \epsilon))-VFTAWS with
O(\frac{k n}{\epsilon^{2}}\lg{n}) edges and another algorithm to compute a
geodesic (k, (\sqrt{10} + \epsilon))-VFTAWS with O(kn(\lg{n})^2) edges. Here,
for any two points p, q \in S, d(p, q) is the geodesic Euclidean distance along
the shortest path between p and q in P.
- When the points in lie on a terrain T, an algorithm to compute a
geodesic (k, (2 + \epsilon))-VFTAWS with O(\frac{k n}{\epsilon^{2}}\lg{n})
edges.Comment: a few update
Multiple-Edge-Fault-Tolerant Approximate Shortest-Path Trees
Let be an -node and -edge positively real-weighted undirected
graph. For any given integer , we study the problem of designing a
sparse \emph{f-edge-fault-tolerant} (-EFT) {\em -approximate
single-source shortest-path tree} (-ASPT), namely a subgraph of
having as few edges as possible and which, following the failure of a set
of at most edges in , contains paths from a fixed source that are
stretched at most by a factor of . To this respect, we provide an
algorithm that efficiently computes an -EFT -ASPT of size . Our structure improves on a previous related construction designed for
\emph{unweighted} graphs, having the same size but guaranteeing a larger
stretch factor of , plus an additive term of .
Then, we show how to convert our structure into an efficient -EFT
\emph{single-source distance oracle} (SSDO), that can be built in
time, has size , and is able to report,
after the failure of the edge set , in time a
-approximate distance from the source to any node, and a
corresponding approximate path in the same amount of time plus the path's size.
Such an oracle is obtained by handling another fundamental problem, namely that
of updating a \emph{minimum spanning forest} (MSF) of after that a
\emph{batch} of simultaneous edge modifications (i.e., edge insertions,
deletions and weight changes) is performed. For this problem, we build in time a \emph{sensitivity} oracle of size , that
reports in time the (at most ) edges either exiting from
or entering into the MSF. [...]Comment: 16 pages, 4 figure
Vertex Fault-Tolerant Emulators
A -spanner of a graph is a sparse subgraph that preserves its shortest
path distances up to a multiplicative stretch factor of , and a -emulator
is similar but not required to be a subgraph of . A classic theorem by
Thorup and Zwick [JACM '05] shows that, despite the extra flexibility available
to emulators, the size/stretch tradeoffs for spanners and emulators are
equivalent. Our main result is that this equivalence in tradeoffs no longer
holds in the commonly-studied setting of graphs with vertex failures. That is:
we introduce a natural definition of vertex fault-tolerant emulators, and then
we show a three-way tradeoff between size, stretch, and fault-tolerance for
these emulators that polynomially surpasses the tradeoff known to be optimal
for spanners.
We complement our emulator upper bound with a lower bound construction that
is essentially tight (within factors of the upper bound) when the
stretch is and is either a fixed odd integer or . We also show
constructions of fault-tolerant emulators with additive error, demonstrating
that these also enjoy significantly improved tradeoffs over those available for
fault-tolerant additive spanners.Comment: To appear in ITCS 202
Optimal Vertex Fault Tolerant Spanners (for fixed stretch)
A -spanner of a graph is a sparse subgraph whose shortest path
distances match those of up to a multiplicative error . In this paper we
study spanners that are resistant to faults. A subgraph is an
vertex fault tolerant (VFT) -spanner if is a -spanner
of for any small set of vertices that might "fail." One
of the main questions in the area is: what is the minimum size of an fault
tolerant -spanner that holds for all node graphs (as a function of ,
and )? This question was first studied in the context of geometric
graphs [Levcopoulos et al. STOC '98, Czumaj and Zhao SoCG '03] and has more
recently been considered in general undirected graphs [Chechik et al. STOC '09,
Dinitz and Krauthgamer PODC '11].
In this paper, we settle the question of the optimal size of a VFT spanner,
in the setting where the stretch factor is fixed. Specifically, we prove
that every (undirected, possibly weighted) -node graph has a
-spanner resilient to vertex faults with edges, and this is fully optimal (unless the famous Erdos Girth
Conjecture is false). Our lower bound even generalizes to imply that no data
structure capable of approximating similarly can
beat the space usage of our spanner in the worst case. We also consider the
edge fault tolerant (EFT) model, defined analogously with edge failures rather
than vertex failures. We show that the same spanner upper bound applies in this
setting. Our data structure lower bound extends to the case (and hence we
close the EFT problem for -approximations), but it falls to for . We leave it as an open problem to
close this gap.Comment: To appear in SODA 201
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