59 research outputs found
Sublinear Algorithms for Approximating String Compressibility
We raise the question of approximating the compressibility of a string with respect to a fixed compression scheme, in sublinear time. We study this question in detail for two popular lossless compression schemes: run-length encoding (RLE) and a variant of Lempel-Ziv (LZ77), and present sublinear algorithms for approximating compressibility with respect to both schemes. We also give several lower bounds that show that our algorithms for both schemes cannot be improved significantly.
Our investigation of LZ77 yields results whose interest goes beyond the initial questions we set out to study. In particular, we prove combinatorial structural lemmas that relate the compressibility of a string with respect to LZ77 to the number of distinct short substrings contained in it (its ℓth subword complexity , for small ℓ). In addition, we show that approximating the compressibility with respect to LZ77 is related to approximating the support size of a distribution.National Science Foundation (U.S.) (Award CCF-1065125)National Science Foundation (U.S.) (Award CCF-0728645)Marie Curie International Reintegration Grant PIRG03-GA-2008-231077Israel Science Foundation (Grant 1147/09)Israel Science Foundation (Grant 1675/09
05291 Abstracts Collection -- Sublinear Algorithms
From 17.07.05 to 22.07.05, the Dagstuhl Seminar
05291 ``Sublinear Algorithms\u27\u27 was held
in the International Conference and Research Center (IBFI),
Schloss Dagstuhl.
During the seminar, several participants presented their current
research, and ongoing work and open problems were discussed. Abstracts of
the presentations given during the seminar as well as abstracts of
seminar results and ideas are put together in this paper. The first section
describes the seminar topics and goals in general.
Links to extended abstracts or full papers are provided, if available
Substring Complexity in Sublinear Space
Shannon's entropy is a definitive lower bound for statistical compression.
Unfortunately, no such clear measure exists for the compressibility of
repetitive strings. Thus, ad-hoc measures are employed to estimate the
repetitiveness of strings, e.g., the size of the Lempel-Ziv parse or the
number of equal-letter runs of the Burrows-Wheeler transform. A more recent
one is the size of a smallest string attractor. Unfortunately, Kempa
and Prezza [STOC 2018] showed that computing is NP-hard. Kociumaka et
al. [LATIN 2020] considered a new measure that is based on the function
counting the cardinalities of the sets of substrings of each length of ,
also known as the substring complexity. This new measure is defined as and lower bounds all the measures previously
considered. In particular, always holds and can be
computed in time using working space. Kociumaka et
al. showed that if is given, one can construct an -sized representation of supporting efficient direct
access and efficient pattern matching queries on . Given that for highly
compressible strings, is significantly smaller than , it is natural
to pose the following question: Can we compute efficiently using
sublinear working space?
It is straightforward to show that any algorithm computing using
space requires time through a reduction
from the element distinctness problem [Yao, SIAM J. Comput. 1994]. We present
the following results: an -time and
-space algorithm to compute , for any ; and
an -time and -space algorithm to
compute , for any
Analysis and experimental evaluation of an approximation algorithm for the length of an optimal Lempel-Ziv parsing
We examine a previously known sublinear-time algorithm for approximating the length of a string’s optimal (i.e. shortest) Lempel-Ziv parsing (a.k.a. LZ77 factorization). This length is a measure of compressibility under the LZ77 compression algorithm, so the algorithm also estimates a string’s compressibility. The algorithm’s approximation approach is based on a connection between optimal Lempel-Ziv parsing length and the number of distinct substrings of different lengths in a string. Some aspects of the algorithm are described more explicitly than in earlier work, including the constraints on its input and how to distinguish between strings with short vs. long optimal parsings in sublinear time; several proofs (and pseudocode listings) are also more detailed than in earlier work. An implementation of the algorithm is provided.
We experimentally investigate the algorithm’s practical usefulness for estimating the compressibility of large collections of data. The algorithm is run on real-world data under a wide range of approximation parameter settings. The accuracy of the resulting estimates is evaluated. The estimates turn out to be consistently highly inaccurate, albeit always inside the stated probabilistic error bounds. We conclude that the algorithm is not promising as a practical tool for estimating compressibility. We also examine the empirical connection between optimal parsing length and the number of distinct substrings of different lengths. The latter turns out to be a suprisingly accurate predictor of the former within our test data, which suggests avenues for future work
Compressibility-Aware Quantum Algorithms on Strings
Sublinear time quantum algorithms have been established for many fundamental
problems on strings. This work demonstrates that new, faster quantum algorithms
can be designed when the string is highly compressible. We focus on two popular
and theoretically significant compression algorithms -- the Lempel-Ziv77
algorithm (LZ77) and the Run-length-encoded Burrows-Wheeler Transform (RL-BWT),
and obtain the results below.
We first provide a quantum algorithm running in time
for finding the LZ77 factorization of an input string with
factors. Combined with multiple existing results, this yields an
time quantum algorithm for finding the RL-BWT encoding
with BWT runs. Note that . We complement these
results with lower bounds proving that our algorithms are optimal (up to
polylog factors).
Next, we study the problem of compressed indexing, where we provide a
time quantum algorithm for constructing a recently
designed space structure with equivalent capabilities as the
suffix tree. This data structure is then applied to numerous problems to obtain
sublinear time quantum algorithms when the input is highly compressible. For
example, we show that the longest common substring of two strings of total
length can be computed in time, where is the
number of factors in the LZ77 factorization of their concatenation. This beats
the best known time quantum algorithm when is
sufficiently small
07411 Abstracts Collection -- Algebraic Methods in Computational Complexity
From 07.10. to 12.10., the Dagstuhl Seminar 07411 ``Algebraic Methods in Computational Complexity\u27\u27 was held in the International Conference and Research Center (IBFI),
Schloss Dagstuhl.
During the seminar, several participants presented their current
research, and ongoing work and open problems were discussed. Abstracts of
the presentations given during the seminar as well as abstracts of
seminar results and ideas are put together in this paper. The first section
describes the seminar topics and goals in general.
Links to extended abstracts or full papers are provided, if available
GPU-accelerated k-mer counting
K-mer counting is the process of building a histogram of all substrings of length k for an input string S. The problem itself is quite simple, but counting k-mers efficiently for a very large input string is a difficult task that has been researched extensively. In recent years the performance of k-mer counting algorithms have improved significantly, and there have been efforts to use graphics processing units (GPUs) in k-mer counting. The goal for this thesis was to design, implement and benchmark a GPU accelerated k-mer counting algorithm SNCGPU. The results showed that SNCGPU compares reasonably well to the Gerbil k-mer counting algorithm on a mid-range desktop computer, but does not utilize the resources of a high-end computing platform as efficiently. The implementation of SNCGPU is available as open-source software
Indexing Highly Repetitive String Collections
Two decades ago, a breakthrough in indexing string collections made it
possible to represent them within their compressed space while at the same time
offering indexed search functionalities. As this new technology permeated
through applications like bioinformatics, the string collections experienced a
growth that outperforms Moore's Law and challenges our ability of handling them
even in compressed form. It turns out, fortunately, that many of these rapidly
growing string collections are highly repetitive, so that their information
content is orders of magnitude lower than their plain size. The statistical
compression methods used for classical collections, however, are blind to this
repetitiveness, and therefore a new set of techniques has been developed in
order to properly exploit it. The resulting indexes form a new generation of
data structures able to handle the huge repetitive string collections that we
are facing.
In this survey we cover the algorithmic developments that have led to these
data structures. We describe the distinct compression paradigms that have been
used to exploit repetitiveness, the fundamental algorithmic ideas that form the
base of all the existing indexes, and the various structures that have been
proposed, comparing them both in theoretical and practical aspects. We conclude
with the current challenges in this fascinating field
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