390 research outputs found
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
Distance Oracles for Time-Dependent Networks
We present the first approximate distance oracle for sparse directed networks
with time-dependent arc-travel-times determined by continuous, piecewise
linear, positive functions possessing the FIFO property.
Our approach precomputes approximate distance summaries from
selected landmark vertices to all other vertices in the network. Our oracle
uses subquadratic space and time preprocessing, and provides two sublinear-time
query algorithms that deliver constant and approximate
shortest-travel-times, respectively, for arbitrary origin-destination pairs in
the network, for any constant . Our oracle is based only on
the sparsity of the network, along with two quite natural assumptions about
travel-time functions which allow the smooth transition towards asymmetric and
time-dependent distance metrics.Comment: A preliminary version appeared as Technical Report ECOMPASS-TR-025 of
EU funded research project eCOMPASS (http://www.ecompass-project.eu/). An
extended abstract also appeared in the 41st International Colloquium on
Automata, Languages, and Programming (ICALP 2014, track-A
Near-Optimal Deterministic Single-Source Distance Sensitivity Oracles
Given a graph with a source vertex , the Single Source Replacement Paths
(SSRP) problem is to compute, for every vertex and edge , the length
of a shortest path from to that avoids . A Single-Source
Distance Sensitivity Oracle (Single-Source DSO) is a data structure that
answers queries of the form by returning the distance . We
show how to deterministically compress the output of the SSRP problem on
-vertex, -edge graphs with integer edge weights in the range into
a Single-Source DSO of size with query time
. The space requirement is optimal (up to the word size) and
our techniques can also handle vertex failures.
Chechik and Cohen [SODA 2019] presented a combinatorial, randomized
time SSRP algorithm for undirected and
unweighted graphs. Grandoni and Vassilevska Williams [FOCS 2012, TALG 2020]
gave an algebraic, randomized time SSRP algorithm
for graphs with integer edge weights in the range , where
is the matrix multiplication exponent. We derandomize both algorithms for
undirected graphs in the same asymptotic running time and apply our compression
to obtain deterministic Single-Source DSOs. The
and preprocessing times are polynomial improvements
over previous -space oracles.
On sparse graphs with edges, for any
constant , we reduce the preprocessing to randomized
time. This is
the first truly subquadratic time algorithm for building Single-Source DSOs on
sparse graphs.Comment: Full version of a paper to appear at ESA 2021. Abstract shortened to
meet ArXiv requirement
Space-Efficient Fault-Tolerant Diameter Oracles
We design -edge fault-tolerant diameter oracles (-FDOs). We preprocess
a given graph on vertices and edges, and a positive integer , to
construct a data structure that, when queried with a set of
edges, returns the diameter of .
For a single failure () in an unweighted directed graph of diameter ,
there exists an approximate FDO by Henzinger et al. [ITCS 2017] with stretch
, constant query time, space , and a combinatorial
preprocessing time of .We
present an FDO for directed graphs with the same stretch, query time, and
space. It has a preprocessing time of .
The preprocessing time nearly matches a conditional lower bound for
combinatorial algorithms, also by Henzinger et al. With fast matrix
multiplication, we achieve a preprocessing time of . We further prove an information-theoretic lower bound
showing that any FDO with stretch better than requires bits
of space.
For multiple failures () in undirected graphs with non-negative edge
weights, we give an -FDO with stretch , query time ,
space, and preprocessing time . We
complement this with a lower bound excluding any finite stretch in
space. We show that for unweighted graphs with polylogarithmic diameter and up
to failures, one can swap approximation for query
time and space. We present an exact combinatorial -FDO with preprocessing
time , query time , and space . When using
fast matrix multiplication instead, the preprocessing time can be improved to
, where is the matrix multiplication
exponent.Comment: Full version of a paper to appear at MFCS'21. Abstract shortened to
meet ArXiv requirement
Connectivity Oracles for Graphs Subject to Vertex Failures
We introduce new data structures for answering connectivity queries in graphs
subject to batched vertex failures. A deterministic structure processes a batch
of failed vertices in time and thereafter
answers connectivity queries in time. It occupies space . We develop a randomized Monte Carlo version of our data structure
with update time , query time , and space
for any failure bound . This is the first connectivity oracle for
general graphs that can efficiently deal with an unbounded number of vertex
failures.
We also develop a more efficient Monte Carlo edge-failure connectivity
oracle. Using space , edge failures are processed in time and thereafter, connectivity queries are answered in
time, which are correct w.h.p.
Our data structures are based on a new decomposition theorem for an
undirected graph , which is of independent interest. It states that
for any terminal set we can remove a set of
vertices such that the remaining graph contains a Steiner forest for with
maximum degree
Deep Distance Sensitivity Oracles
One of the most fundamental graph problems is finding a shortest path from a
source to a target node. While in its basic forms the problem has been studied
extensively and efficient algorithms are known, it becomes significantly harder
as soon as parts of the graph are susceptible to failure. Although one can
recompute a shortest replacement path after every outage, this is rather
inefficient both in time and/or storage. One way to overcome this problem is to
shift computational burden from the queries into a pre-processing step, where a
data structure is computed that allows for fast querying of replacement paths,
typically referred to as a Distance Sensitivity Oracle (DSO). While DSOs have
been extensively studied in the theoretical computer science community, to the
best of our knowledge this is the first work to construct DSOs using deep
learning techniques. We show how to use deep learning to utilize a
combinatorial structure of replacement paths. More specifically, we utilize the
combinatorial structure of replacement paths as a concatenation of shortest
paths and use deep learning to find the pivot nodes for stitching shortest
paths into replacement paths.Comment: arXiv admin note: text overlap with arXiv:2007.11495 by other author
Near-Optimal Distributed Computation of Small Vertex Cuts
We present near-optimal algorithms for detecting small vertex cuts in the {CONGEST} model of distributed computing. Despite extensive research in this area, our understanding of the vertex connectivity of a graph is still incomplete, especially in the distributed setting. To this date, all distributed algorithms for detecting cut vertices suffer from an inherent dependency in the maximum degree of the graph, ?. Hence, in particular, there is no truly sub-linear time algorithm for this problem, not even for detecting a single cut vertex. We take a new algorithmic approach for vertex connectivity which allows us to bypass the existing ? barrier.
- As a warm-up to our approach, we show a simple O?(D)-round randomized algorithm for computing all cut vertices in a D-diameter n-vertex graph. This improves upon the O(D+?/log n)-round algorithm of [Pritchard and Thurimella, ICALP 2008].
- Our key technical contribution is an O?(D)-round randomized algorithm for computing all cut pairs in the graph, improving upon the state-of-the-art O(? ? D)?-round algorithm by [Parter, DISC \u2719]. Note that even for the considerably simpler setting of edge cuts, currently O?(D)-round algorithms are currently known only for detecting pairs of cut edges.
Our approach is based on employing the well-known linear graph sketching technique [Ahn, Guha and McGregor, SODA 2012] along with the heavy-light tree decomposition of [Sleator and Tarjan, STOC 1981] . Combining this with a careful characterization of the survivable subgraphs, allows us to determine the connectivity of G ? {x,y} for every pair x,y ? V, using O?(D)-rounds. We believe that the tools provided in this paper are useful for omitting the ?-dependency even for larger cut values
Automatic Quality Estimation for ASR System Combination
Recognizer Output Voting Error Reduction (ROVER) has been widely used for
system combination in automatic speech recognition (ASR). In order to select
the most appropriate words to insert at each position in the output
transcriptions, some ROVER extensions rely on critical information such as
confidence scores and other ASR decoder features. This information, which is
not always available, highly depends on the decoding process and sometimes
tends to over estimate the real quality of the recognized words. In this paper
we propose a novel variant of ROVER that takes advantage of ASR quality
estimation (QE) for ranking the transcriptions at "segment level" instead of:
i) relying on confidence scores, or ii) feeding ROVER with randomly ordered
hypotheses. We first introduce an effective set of features to compensate for
the absence of ASR decoder information. Then, we apply QE techniques to perform
accurate hypothesis ranking at segment-level before starting the fusion
process. The evaluation is carried out on two different tasks, in which we
respectively combine hypotheses coming from independent ASR systems and
multi-microphone recordings. In both tasks, it is assumed that the ASR decoder
information is not available. The proposed approach significantly outperforms
standard ROVER and it is competitive with two strong oracles that e xploit
prior knowledge about the real quality of the hypotheses to be combined.
Compared to standard ROVER, the abs olute WER improvements in the two
evaluation scenarios range from 0.5% to 7.3%
- …