14 research outputs found
Fully dynamic data structure for LCE queries in compressed space
A Longest Common Extension (LCE) query on a text of length asks for
the length of the longest common prefix of suffixes starting at given two
positions. We show that the signature encoding of size [Mehlhorn et al., Algorithmica 17(2):183-198,
1997] of , which can be seen as a compressed representation of , has a
capability to support LCE queries in time,
where is the answer to the query, is the size of the Lempel-Ziv77
(LZ77) factorization of , and is an integer that can be handled
in constant time under word RAM model. In compressed space, this is the fastest
deterministic LCE data structure in many cases. Moreover, can be
enhanced to support efficient update operations: After processing
in time, we can insert/delete any (sub)string of length
into/from an arbitrary position of in time, where . This yields
the first fully dynamic LCE data structure. We also present efficient
construction algorithms from various types of inputs: We can construct
in time from uncompressed string ; in
time from grammar-compressed string
represented by a straight-line program of size ; and in time from LZ77-compressed string with factors. On top
of the above contributions, we show several applications of our data structures
which improve previous best known results on grammar-compressed string
processing.Comment: arXiv admin note: text overlap with arXiv:1504.0695
Minimal Suffix and Rotation of a Substring in Optimal Time
For a text given in advance, the substring minimal suffix queries ask to
determine the lexicographically minimal non-empty suffix of a substring
specified by the location of its occurrence in the text. We develop a data
structure answering such queries optimally: in constant time after linear-time
preprocessing. This improves upon the results of Babenko et al. (CPM 2014),
whose trade-off solution is characterized by product of these
time complexities. Next, we extend our queries to support concatenations of
substrings, for which the construction and query time is preserved. We
apply these generalized queries to compute lexicographically minimal and
maximal rotations of a given substring in constant time after linear-time
preprocessing.
Our data structures mainly rely on properties of Lyndon words and Lyndon
factorizations. We combine them with further algorithmic and combinatorial
tools, such as fusion trees and the notion of order isomorphism of strings
Indexing the Bijective BWT
The Burrows-Wheeler transform (BWT) is a permutation whose applications are prevalent in data compression and text indexing. The bijective BWT is a bijective variant of it that has not yet been studied for text indexing applications. We fill this gap by proposing a self-index built on the bijective BWT . The self-index applies the backward search technique of the FM-index to find a pattern P with O(|P| lg|P|) backward search steps
Rank, select and access in grammar-compressed strings
Given a string of length on a fixed alphabet of symbols, a
grammar compressor produces a context-free grammar of size that
generates and only . In this paper we describe data structures to
support the following operations on a grammar-compressed string:
\mbox{rank}_c(S,i) (return the number of occurrences of symbol before
position in ); \mbox{select}_c(S,i) (return the position of the th
occurrence of in ); and \mbox{access}(S,i,j) (return substring
). For rank and select we describe data structures of size
bits that support the two operations in time. We
propose another structure that uses
bits and that supports the two queries in , where
is an arbitrary constant. To our knowledge, we are the first to
study the asymptotic complexity of rank and select in the grammar-compressed
setting, and we provide a hardness result showing that significantly improving
the bounds we achieve would imply a major breakthrough on a hard
graph-theoretical problem. Our main result for access is a method that requires
bits of space and time to extract
consecutive symbols from . Alternatively, we can achieve query time using bits of space. This matches a lower bound stated by Verbin
and Yu for strings where is polynomially related to .Comment: 16 page
Sensitivity of the Burrows-Wheeler Transform to small modifications, and other problems on string compressors in Bioinformatics
Extensive amount of data is produced in textual form nowadays, especially in bioinformatics. Several algorithms exist to store and process this data efficiently in compressed space. In this thesis, we focus on both combinatorial and practical aspects of two of the most widely used algorithms for compressing text in bioinformatics: the Burrows-Wheeler Transform (BWT) and Lempel-Ziv compression (LZ77). In the first part, we focus on combinatorial aspects of the BWT. Given a word v, r = r(v) denotes the number of maximal equal-letter runs in BWT(v). First, we investigate the relationship between r of a word and r of its reverse. We prove that there exist words for which these two values differ by a logarithmic factor in the length of the word. In other words, although the repetitiveness in the two words is preserved, the number of runs can change by a non-constant factor. This suggests that the number of runs may not be an ideal repetitiveness measure. The second combinatorial aspect we are interested in is how small alterations in a word may affect its BWT in a relevant way. We prove that the number of runs of the BWT of a word can change (increase or decrease) by up to a logarithmic factor in the length of the word by just adding, removing, or substituting a single character. We then consider the special character can be inserted in order to turn it into the BWT of a is allowed, depends entirely on the structure of a specific permutation of the indices of the word, which is called the standard permutation of the word. The final part of this thesis treats more applied aspects of text compressors. In bioinformatics, BWT-based compressed data structures are widely used for pattern matching. We give an algorithm based on the BWT to find Maximal Unique Matches (MUMs) of a pattern with respect to a reference text in compressed space, extending an existing tool called PHONI [Boucher et. al, DCC 2021]. Finally, we study some aspects of the Lempel-Ziv 77 (LZ77) factorization of a word. Modeling DNA short reads, we provide a bound on the compression size of the concatenation of regular samples of a word