577 research outputs found
Distributed Constructions of Dual-Failure Fault-Tolerant Distance Preservers
Fault tolerant distance preservers (spanners) are sparse subgraphs that preserve (approximate) distances between given pairs of vertices under edge or vertex failures. So-far, these structures have been studied thoroughly mainly from a centralized viewpoint. Despite the fact fault tolerant preservers are mainly motivated by the error-prone nature of distributed networks, not much is known on the distributed computational aspects of these structures.
In this paper, we present distributed algorithms for constructing fault tolerant distance preservers and +2 additive spanners that are resilient to at most two edge faults. Prior to our work, the only non-trivial constructions known were for the single fault and single source setting by [Ghaffari and Parter SPAA\u2716].
Our key technical contribution is a distributed algorithm for computing distance preservers w.r.t. a subset S of source vertices, resilient to two edge faults. The output structure contains a BFS tree BFS(s,G ? {e?,e?}) for every s ? S and every e?,e? ? G. The distributed construction of this structure is based on a delicate balance between the edge congestion (formed by running multiple BFS trees simultaneously) and the sparsity of the output subgraph. No sublinear-round algorithms for constructing these structures have been known before
Sparse Fault-Tolerant BFS Trees
This paper addresses the problem of designing a sparse {\em fault-tolerant}
BFS tree, or {\em FT-BFS tree} for short, namely, a sparse subgraph of the
given network such that subsequent to the failure of a single edge or
vertex, the surviving part of still contains a BFS spanning tree for
(the surviving part of) . Our main results are as follows. We present an
algorithm that for every -vertex graph and source node constructs a
(single edge failure) FT-BFS tree rooted at with O(n \cdot
\min\{\Depth(s), \sqrt{n}\}) edges, where \Depth(s) is the depth of the BFS
tree rooted at . This result is complemented by a matching lower bound,
showing that there exist -vertex graphs with a source node for which any
edge (or vertex) FT-BFS tree rooted at has edges. We then
consider {\em fault-tolerant multi-source BFS trees}, or {\em FT-MBFS trees}
for short, aiming to provide (following a failure) a BFS tree rooted at each
source for some subset of sources . Again, tight bounds
are provided, showing that there exists a poly-time algorithm that for every
-vertex graph and source set of size constructs a
(single failure) FT-MBFS tree from each source , with
edges, and on the other hand there exist
-vertex graphs with source sets of cardinality , on
which any FT-MBFS tree from has edges.
Finally, we propose an approximation algorithm for constructing
FT-BFS and FT-MBFS structures. The latter is complemented by a hardness result
stating that there exists no approximation algorithm for these
problems under standard complexity assumptions
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 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)
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
Dual Failure Resilient BFS Structure
We study {\em breadth-first search (BFS)} spanning trees, and address the
problem of designing a sparse {\em fault-tolerant} BFS structure, or {\em
FT-BFS } for short, resilient to the failure of up to two edges in the given
undirected unweighted graph , i.e., a sparse subgraph of such that
subsequent to the failure of up to two edges, the surviving part of
still contains a BFS spanning tree for (the surviving part of) . FT-BFS
structures, as well as the related notion of replacement paths, have been
studied so far for the restricted case of a single failure. It has been noted
widely that when concerning shortest-paths in a variety of contexts, there is a
sharp qualitative difference between a single failure and two or more failures.
Our main results are as follows. We present an algorithm that for every
-vertex unweighted undirected graph and source node constructs a
(two edge failure) FT-BFS structure rooted at with edges. To
provide a useful theory of shortest paths avoiding 2 edges failures, we take a
principled approach to classifying the arrangement these paths. We believe that
the structural analysis provided in this paper may decrease the barrier for
understanding the general case of faults and pave the way to the
future design of -fault resilient structures for . We also provide
a matching lower bound, which in fact holds for the general case of
and multiple sources . It shows that for every , and
integer , there exist -vertex graphs with a source set
of cardinality for which any FT-BFS structure rooted
at each , resilient to up to -edge faults has
edges
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