16,309 research outputs found
Noise-Resilient Group Testing: Limitations and Constructions
We study combinatorial group testing schemes for learning -sparse Boolean
vectors using highly unreliable disjunctive measurements. We consider an
adversarial noise model that only limits the number of false observations, and
show that any noise-resilient scheme in this model can only approximately
reconstruct the sparse vector. On the positive side, we take this barrier to
our advantage and show that approximate reconstruction (within a satisfactory
degree of approximation) allows us to break the information theoretic lower
bound of that is known for exact reconstruction of
-sparse vectors of length via non-adaptive measurements, by a
multiplicative factor .
Specifically, we give simple randomized constructions of non-adaptive
measurement schemes, with measurements, that allow efficient
reconstruction of -sparse vectors up to false positives even in the
presence of false positives and false negatives within the
measurement outcomes, for any constant . We show that, information
theoretically, none of these parameters can be substantially improved without
dramatically affecting the others. Furthermore, we obtain several explicit
constructions, in particular one matching the randomized trade-off but using measurements. We also obtain explicit constructions
that allow fast reconstruction in time \poly(m), which would be sublinear in
for sufficiently sparse vectors. The main tool used in our construction is
the list-decoding view of randomness condensers and extractors.Comment: Full version. A preliminary summary of this work appears (under the
same title) in proceedings of the 17th International Symposium on
Fundamentals of Computation Theory (FCT 2009
Near-Linear Time Insertion-Deletion Codes and (1+)-Approximating Edit Distance via Indexing
We introduce fast-decodable indexing schemes for edit distance which can be
used to speed up edit distance computations to near-linear time if one of the
strings is indexed by an indexing string . In particular, for every length
and every , one can in near linear time construct a string
with , such that, indexing
any string , symbol-by-symbol, with results in a string where for which edit
distance computations are easy, i.e., one can compute a
-approximation of the edit distance between and any other
string in time.
Our indexing schemes can be used to improve the decoding complexity of
state-of-the-art error correcting codes for insertions and deletions. In
particular, they lead to near-linear time decoding algorithms for the
insertion-deletion codes of [Haeupler, Shahrasbi; STOC `17] and faster decoding
algorithms for list-decodable insertion-deletion codes of [Haeupler, Shahrasbi,
Sudan; ICALP `18]. Interestingly, the latter codes are a crucial ingredient in
the construction of fast-decodable indexing schemes
Multi-path Summation for Decoding 2D Topological Codes
Fault tolerance is a prerequisite for scalable quantum computing.
Architectures based on 2D topological codes are effective for near-term
implementations of fault tolerance. To obtain high performance with these
architectures, we require a decoder which can adapt to the wide variety of
error models present in experiments. The typical approach to the problem of
decoding the surface code is to reduce it to minimum-weight perfect matching in
a way that provides a suboptimal threshold error rate, and is specialized to
correct a specific error model. Recently, optimal threshold error rates for a
variety of error models have been obtained by methods which do not use
minimum-weight perfect matching, showing that such thresholds can be achieved
in polynomial time. It is an open question whether these results can also be
achieved by minimum-weight perfect matching. In this work, we use belief
propagation and a novel algorithm for producing edge weights to increase the
utility of minimum-weight perfect matching for decoding surface codes. This
allows us to correct depolarizing errors using the rotated surface code,
obtaining a threshold of . This is larger than the threshold
achieved by previous matching-based decoders (), though
still below the known upper bound of .Comment: 19 pages, 13 figures, published in Quantum, available at
https://quantum-journal.org/papers/q-2018-10-19-102
Error-tolerant Finite State Recognition with Applications to Morphological Analysis and Spelling Correction
Error-tolerant recognition enables the recognition of strings that deviate
mildly from any string in the regular set recognized by the underlying finite
state recognizer. Such recognition has applications in error-tolerant
morphological processing, spelling correction, and approximate string matching
in information retrieval. After a description of the concepts and algorithms
involved, we give examples from two applications: In the context of
morphological analysis, error-tolerant recognition allows misspelled input word
forms to be corrected, and morphologically analyzed concurrently. We present an
application of this to error-tolerant analysis of agglutinative morphology of
Turkish words. The algorithm can be applied to morphological analysis of any
language whose morphology is fully captured by a single (and possibly very
large) finite state transducer, regardless of the word formation processes and
morphographemic phenomena involved. In the context of spelling correction,
error-tolerant recognition can be used to enumerate correct candidate forms
from a given misspelled string within a certain edit distance. Again, it can be
applied to any language with a word list comprising all inflected forms, or
whose morphology is fully described by a finite state transducer. We present
experimental results for spelling correction for a number of languages. These
results indicate that such recognition works very efficiently for candidate
generation in spelling correction for many European languages such as English,
Dutch, French, German, Italian (and others) with very large word lists of root
and inflected forms (some containing well over 200,000 forms), generating all
candidate solutions within 10 to 45 milliseconds (with edit distance 1) on a
SparcStation 10/41. For spelling correction in Turkish, error-tolerantComment: Replaces 9504031. gzipped, uuencoded postscript file. To appear in
Computational Linguistics Volume 22 No:1, 1996, Also available as
ftp://ftp.cs.bilkent.edu.tr/pub/ko/clpaper9512.ps.
Distributed PCP Theorems for Hardness of Approximation in P
We present a new distributed model of probabilistically checkable proofs
(PCP). A satisfying assignment to a CNF formula is
shared between two parties, where Alice knows , Bob knows
, and both parties know . The goal is to have
Alice and Bob jointly write a PCP that satisfies , while
exchanging little or no information. Unfortunately, this model as-is does not
allow for nontrivial query complexity. Instead, we focus on a non-deterministic
variant, where the players are helped by Merlin, a third party who knows all of
.
Using our framework, we obtain, for the first time, PCP-like reductions from
the Strong Exponential Time Hypothesis (SETH) to approximation problems in P.
In particular, under SETH we show that there are no truly-subquadratic
approximation algorithms for Bichromatic Maximum Inner Product over
{0,1}-vectors, Bichromatic LCS Closest Pair over permutations, Approximate
Regular Expression Matching, and Diameter in Product Metric. All our
inapproximability factors are nearly-tight. In particular, for the first two
problems we obtain nearly-polynomial factors of ; only
-factor lower bounds (under SETH) were known before
Edit Distance: Sketching, Streaming and Document Exchange
We show that in the document exchange problem, where Alice holds and Bob holds , Alice can send Bob a message of
size bits such that Bob can recover using the
message and his input if the edit distance between and is no more
than , and output "error" otherwise. Both the encoding and decoding can be
done in time . This result significantly
improves the previous communication bounds under polynomial encoding/decoding
time. We also show that in the referee model, where Alice and Bob hold and
respectively, they can compute sketches of and of sizes
bits (the encoding), and send to the referee, who can
then compute the edit distance between and together with all the edit
operations if the edit distance is no more than , and output "error"
otherwise (the decoding). To the best of our knowledge, this is the first
result for sketching edit distance using bits.
Moreover, the encoding phase of our sketching algorithm can be performed by
scanning the input string in one pass. Thus our sketching algorithm also
implies the first streaming algorithm for computing edit distance and all the
edits exactly using bits of space.Comment: Full version of an article to be presented at the 57th Annual IEEE
Symposium on Foundations of Computer Science (FOCS 2016
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