Almost Linear Time Computation of Maximal Repetitions in Run Length Encoded Strings

We consider the problem of computing all maximal repetitions contained in a string that is given in run-length encoding. Given a run-length encoding of a string, we show that the maximum number of maximal repetitions contained in the string is at most m+k-1, where m is the size of the run-length encoding, and k is the number of run-length factors whose exponent is at least 2. We also show an algorithm for computing all maximal repetitions in O(m alpha(m)) time and O(m) space, where alpha denotes the inverse Ackermann function

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 $\tilde{O}(\sqrt{zn})$ time for finding the LZ77 factorization of an input string $T[1..n]$ with $z$ factors. Combined with multiple existing results, this yields an $\tilde{O}(\sqrt{rn})$ time quantum algorithm for finding the RL-BWT encoding with $r$ BWT runs. Note that $r = \tilde{\Theta}(z)$. 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 $\tilde{O}(\sqrt{rn})$ time quantum algorithm for constructing a recently designed $\tilde{O}(r)$ 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 $n$ can be computed in $\tilde{O}(\sqrt{zn})$ time, where $z$ is the number of factors in the LZ77 factorization of their concatenation. This beats the best known $\tilde{O}(n^\frac{2}{3})$ time quantum algorithm when $z$ is sufficiently small

Lyndon Arrays in Sublinear Time

?} with ? ? n. In this case, the string can be stored in O(n log ?) bits (or O(n / log_? n) words) of memory, and reading it takes only O(n / log_? n) time. We show that O(n / log_? n) time and words of space suffice to compute the succinct 2n-bit version of the Lyndon array. The time is optimal for w = O(log n). The algorithm uses precomputed lookup tables to perform significant parts of the computation in constant time. This is possible due to properties of periodic substrings, which we carefully analyze to achieve the desired result. We envision that the algorithm has applications in the computation of runs (maximal periodic substrings), where the Lyndon array plays a central role in both theoretically and practically fast algorithms

Optimal construction of compressed indexes for highly repetitive texts

We propose algorithms that, given the input string of length n over integer alphabet of size σ, construct the Burrows–Wheeler transform (BWT), the permuted longest-common-prefix (PLCP) array, and the LZ77 parsing in O(n/ logσ n + r polylog n) time and working space, where r is the number of runs in the BWT of the input. These are the essential components of many compressed indexes such as compressed suffix tree, FM-index, and grammar and LZ77-based indexes, but also find numerous applications in sequence analysis and data compression. The value of r is a common measure of repetitiveness that is significantly smaller than n if the string is highly repetitive. Since just accessing every symbol of the string requires Ω(n/ logσ n) time, the presented algorithms are time and space optimal for inputs satisfying the assumption n/r ∈ Ω(polylog n) on the repetitiveness. For such inputs our result improves upon the currently fastest general algorithms of Belazzougui (STOC 2014) and Munro et al. (SODA 2017) which run in O(n) time and use O(n/ logσ n) working space. We also show how to use our techniques to obtain optimal solutions on highly repetitive data for other fundamental string processing problems such as: Lyndon factorization, construction of run-length compressed suffix arrays, and some classical “textbook” problems such as computing the longest substring occurring at least some fixed number of times. Copyright © 2019 by SIAMPeer reviewe

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 $used in real-life applications to mark the end of a word. We investigate the impact of this character on words with respect to the BWT. We characterize positions in a word where$ can be inserted in order to turn it into the BWT of a $-terminated word over the same alphabet. We show that, whether and where$ 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

Comparison of LZ77-type Parsings

We investigate the relations between different variants of the LZ77 parsing existing in the literature. All of them are defined as greedily constructed parsings encoding each phrase by reference to a string occurring earlier in the input. They differ by the phrase encodings: encoded by pairs (length + position of an earlier occurrence) or by triples (length + position of an earlier occurrence + the letter following the earlier occurring part); and they differ by allowing or not allowing overlaps between the phrase and its earlier occurrence. For a given string of length $n$ over an alphabet of size $\sigma$, denote the numbers of phrases in the parsings allowing (resp., not allowing) overlaps by $z$ (resp., $\hat{z}$) for "pairs", and by $z_3$ (resp., $\hat{z}_3$) for "triples". We prove the following bounds and provide series of examples showing that these bounds are tight: $\bullet$ $z \le \hat{z} \le z \cdot O(\log\frac{n}{z\log_\sigma z})$ and $z_3 \le \hat{z}_3 \le z_3 \cdot O(\log\frac{n}{z_3\log_\sigma z_3})$; $\bullet$ $\frac{1}2\hat{z} < \hat{z}_3 \le \hat{z}$ and $\frac{1}2 z < z_3 \le z$.Comment: 6 page