Computation of the suffix array, burrows-wheeler transform and FM-index in V-order

V-order is a total order on strings that determines an instance of Unique Maximal Factorization Families (UMFFs), a generalization of Lyndon words. The fundamental V-comparison of strings can be done in linear time and constant space. V-order has been proposed as an alternative to lexicographic order (lexorder) in the computation of suffix arrays and in the suffix-sorting induced by the Burrows-Wheeler transform (BWT). In line with the recent interest in the connection between suffix arrays and Lyndon factorization, in this paper we obtain similar results for the V-order factorization. Indeed, we show that the results describing the connection between suffix arrays and Lyndon factorization are matched by analogous V-order processing. We also describe a methodology for efficiently computing the FM-Index in V-order, as well as V-order substring pattern matching using backward search

On Bijective Variants of the Burrows-Wheeler Transform

The sort transform (ST) is a modification of the Burrows-Wheeler transform (BWT). Both transformations map an arbitrary word of length n to a pair consisting of a word of length n and an index between 1 and n. The BWT sorts all rotation conjugates of the input word, whereas the ST of order k only uses the first k letters for sorting all such conjugates. If two conjugates start with the same prefix of length k, then the indices of the rotations are used for tie-breaking. Both transforms output the sequence of the last letters of the sorted list and the index of the input within the sorted list. In this paper, we discuss a bijective variant of the BWT (due to Scott), proving its correctness and relations to other results due to Gessel and Reutenauer (1993) and Crochemore, Desarmenien, and Perrin (2005). Further, we present a novel bijective variant of the ST.Comment: 15 pages, presented at the Prague Stringology Conference 2009 (PSC 2009

String Comparison in VV-Order: New Lexicographic Properties & On-line Applications

$V$-order is a global order on strings related to Unique Maximal Factorization Families (UMFFs), which are themselves generalizations of Lyndon words. $V$-order has recently been proposed as an alternative to lexicographical order in the computation of suffix arrays and in the suffix-sorting induced by the Burrows-Wheeler transform. Efficient $V$-ordering of strings thus becomes a matter of considerable interest. In this paper we present new and surprising results on $V$-order in strings, then go on to explore the algorithmic consequences

Universal lossless source coding with the Burrows Wheeler transform

The Burrows Wheeler transform (1994) is a reversible sequence transformation used in a variety of practical lossless source-coding algorithms. In each, the BWT is followed by a lossless source code that attempts to exploit the natural ordering of the BWT coefficients. BWT-based compression schemes are widely touted as low-complexity algorithms giving lossless coding rates better than those of the Ziv-Lempel codes (commonly known as LZ'77 and LZ'78) and almost as good as those achieved by prediction by partial matching (PPM) algorithms. To date, the coding performance claims have been made primarily on the basis of experimental results. This work gives a theoretical evaluation of BWT-based coding. The main results of this theoretical evaluation include: (1) statistical characterizations of the BWT output on both finite strings and sequences of length n → ∞, (2) a variety of very simple new techniques for BWT-based lossless source coding, and (3) proofs of the universality and bounds on the rates of convergence of both new and existing BWT-based codes for finite-memory and stationary ergodic sources. The end result is a theoretical justification and validation of the experimentally derived conclusions: BWT-based lossless source codes achieve universal lossless coding performance that converges to the optimal coding performance more quickly than the rate of convergence observed in Ziv-Lempel style codes and, for some BWT-based codes, within a constant factor of the optimal rate of convergence for finite-memory source

A Grammar Compression Algorithm based on Induced Suffix Sorting

We introduce GCIS, a grammar compression algorithm based on the induced suffix sorting algorithm SAIS, introduced by Nong et al. in 2009. Our solution builds on the factorization performed by SAIS during suffix sorting. We construct a context-free grammar on the input string which can be further reduced into a shorter string by substituting each substring by its correspondent factor. The resulting grammar is encoded by exploring some redundancies, such as common prefixes between suffix rules, which are sorted according to SAIS framework. When compared to well-known compression tools such as Re-Pair and 7-zip, our algorithm is competitive and very effective at handling repetitive string regarding compression ratio, compression and decompression running time

Rust-Bio - a fast and safe bioinformatics library

We present Rust-Bio, the first general purpose bioinformatics library for the innovative Rust programming language. Rust-Bio leverages the unique combination of speed, memory safety and high-level syntax offered by Rust to provide a fast and safe set of bioinformatics algorithms and data structures with a focus on sequence analysis

Low Space External Memory Construction of the Succinct Permuted Longest Common Prefix Array

The longest common prefix (LCP) array is a versatile auxiliary data structure in indexed string matching. It can be used to speed up searching using the suffix array (SA) and provides an implicit representation of the topology of an underlying suffix tree. The LCP array of a string of length $n$ can be represented as an array of length $n$ words, or, in the presence of the SA, as a bit vector of $2n$ bits plus asymptotically negligible support data structures. External memory construction algorithms for the LCP array have been proposed, but those proposed so far have a space requirement of $O(n)$ words (i.e. $O(n \log n)$ bits) in external memory. This space requirement is in some practical cases prohibitively expensive. We present an external memory algorithm for constructing the $2n$ bit version of the LCP array which uses $O(n \log \sigma)$ bits of additional space in external memory when given a (compressed) BWT with alphabet size $\sigma$ and a sampled inverse suffix array at sampling rate $O(\log n)$. This is often a significant space gain in practice where $\sigma$ is usually much smaller than $n$ or even constant. We also consider the case of computing succinct LCP arrays for circular strings

Efficient Construction of the BWT for Repetitive Text Using String Compression

Funding Information: Funding Diego Díaz-Domínguez: Academy of Finland Grant 323233 Gonzalo Navarro: ANID Basal Funds FB0001 and Fondecyt Grant 1-200038, Chile Publisher Copyright: © Diego Daz-Domnguez and Gonzalo Navarro; licensed under Creative Commons License CC-BY 4.0We present a new semi-external algorithm that builds the Burrows-Wheeler transform variant of Bauer et al. (a.k.a., BCR BWT) in linear expected time. Our method uses compression techniques to reduce the computational costs when the input is massive and repetitive. Concretely, we build on induced suffix sorting (ISS) and resort to run-length and grammar compression to maintain our intermediate results in compact form. Our compression format not only saves space, but it also speeds up the required computations. Our experiments show important savings in both space and computation time when the text is repetitive. On average, we are 3.7x faster than the baseline compressed approach, while maintaining a similar memory consumption. These results make our method stand out as the only one (to our knowledge) that can build the BCR BWT of a collection of 25 human genomes (75 GB) in about 7.3 hours, and using only 27 GB of working memory.Peer reviewe