30 research outputs found
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
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
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
Reachability Preservers: New Extremal Bounds and Approximation Algorithms
We abstract and study \emph{reachability preservers}, a graph-theoretic
primitive that has been implicit in prior work on network design. Given a
directed graph and a set of \emph{demand pairs} , a reachability preserver is a sparse subgraph that preserves
reachability between all demand pairs.
Our first contribution is a series of extremal bounds on the size of
reachability preservers. Our main result states that, for an -node graph and
demand pairs of the form for a small node subset ,
there is always a reachability preserver on edges. We
additionally give a lower bound construction demonstrating that this upper
bound characterizes the settings in which size reachability preservers
are generally possible, in a large range of parameters.
The second contribution of this paper is a new connection between extremal
graph sparsification results and classical Steiner Network Design problems.
Surprisingly, prior to this work, the osmosis of techniques between these two
fields had been superficial. This allows us to improve the state of the art
approximation algorithms for the most basic Steiner-type problem in directed
graphs from the of Chlamatac, Dinitz, Kortsarz, and
Laekhanukit (SODA'17) to .Comment: SODA '1
An Optimal Dual Fault Tolerant Reachability Oracle
Let G=(V,E) be an n-vertices m-edges directed graph. Let s inV be any designated source vertex. We address the problem of reporting the reachability information from s under two vertex failures. We show that it is possible to compute in polynomial time an O(n) size data structure that for any query vertex v, and any pair of failed vertices f_1, f_2, answers in O(1) time whether or not there exists a path from s to v in G{f_1,f_2}.
For the simpler case of single vertex failure such a data structure can be obtained using the dominator-tree from the celebrated work of Lengauer and Tarjan [TOPLAS 1979, Vol. 1]. However, no efficient data structure was known in the past for handling more than one failures. We, in addition, also present a labeling scheme with O(log^3(n))-bit size labels such that for any f_1, f_2, v in Vit is possible to determine in poly-logarithmic time if v is reachable from s in G{f_1,f_2} using only the labels of f1, f_2 and v.
Our data structure can also be seen as an efficient mechanism for verifying double-dominators. For any given x, y, v in V we can determine in O(1) time if the pair (x,y) is a double-dominator of v. Earlier the best known method for this problem was using dominator chain from which verification of double-dominators of only a single vertex was possible