When a Dollar Makes a BWT

TheBurrows-Wheeler-Transform(BWT)isareversiblestring transformation which plays a central role in text compression and is fun- damental in many modern bioinformatics applications. The BWT is a permutation of the characters, which is in general better compressible and allows to answer several different query types more efficiently than the original string. It is easy to see that not every string is a BWT image, and exact charac- terizations of BWT images are known. We investigate a related combi- natorial question. In many applications, a sentinel character $is added to mark the end of the string, and thus the BWT of a string ending with$ contains exactly one $character. We ask, given a string w, in which positions, if any, can the$-character be inserted to turn w into the BWT image of a word ending with the sentinel character. We show that this depends only on the standard permutation of w and give a combinatorial characterization of such positions via this permutation. We then develop an O(n log n)-time algorithm for identifying all such positions, improving on the naive quadratic time algorithm

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

Variable-order reference-free variant discovery with the Burrows-Wheeler Transform

International audienceBackground: In [Prezza et al., AMB 2019], a new reference-free and alignment-free framework for the detection of SNPs was suggested and tested. The framework, based on the Burrows-Wheeler Transform (BWT), significantly improves sensitivity and precision of previous de Bruijn graphs based tools by overcoming several of their limitations, namely: (i) the need to establish a fixed value, usually small, for the order k, (ii) the loss of important information such as k-mer coverage and adjacency of k-mers within the same read, and (iii) bad performance in repeated regions longer than k bases. The preliminary tool, however, was able to identify only SNPs and it was too slow and memory consuming due to the use of additional heavy data structures (namely, the Suffix and LCP arrays), besides the BWT. Results: In this paper, we introduce a new algorithm and the corresponding tool ebwt2InDel that (i) extend the framework of [Prezza et al., AMB 2019] to detect also INDELs, and (ii) implements recent algorithmic findings that allow to perform the whole analysis using just the BWT, thus reducing the working space by one order of magnitude and allowing the analysis of full genomes. Finally, we describe a simple strategy for effectively parallelizing our tool for SNP detection only. On a 24-cores machine, the parallel version of our tool is one order of magnitude faster than the sequential one. The tool ebwt2InDel is available at github.com/nicolaprezza/ebwt2InDel. Conclusions: Results on a synthetic dataset covered at 30x (Human chromosome 1) show that our tool is indeed able to find up to 83% of the SNPs and 72% of the existing INDELs. These percentages considerably improve the 71% of SNPs and 51% of INDELs found by the state-of-the art tool based on de Bruijn graphs. We furthermore repor

When a dollar makes a BWT

The Burrows-Wheeler-Transform (BWT) is a reversible string transformation which plays a central role in text compression and is fundamental in many modern bioinformatics applications. The BWT is a permutation of the characters, which is in general better compressible and allows to answer several different query types more efficiently than the original string. It is easy to see that not every string is a BWT image, and exact characterizations of BWT images are known. We investigate a related combinatorial question. In many applications, a sentinel character $is added to mark the end of the string, and thus the BWT of a string ending with$ contains exactly one $-character. Given a string w, we ask in which positions, if any, the$-character can be inserted to turn w into the BWT image of a word ending with $. We show that this depends only on the standard permutation of w and present a O(nlogn)-time algorithm for identifying all such positions, improving on the naive quadratic time algorithm. We also give a combinatorial characterization of such positions and develop bounds on their number and value. This is an extended version of [Giuliani et al. ICTCS 2019]