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

Space-efficient merging of succinct de Bruijn graphs

We propose a new algorithm for merging succinct representations of de Bruijn graphs introduced in [Bowe et al. WABI 2012]. Our algorithm is based on the lightweight BWT merging approach by Holt and McMillan [Bionformatics 2014, ACM-BCB 2014]. Our algorithm has the same asymptotic cost of the state of the art tool for the same problem presented by Muggli et al. [bioRxiv 2017, Bioinformatics 2019], but it uses less than half of its working space. A novel important feature of our algorithm, not found in any of the existing tools, is that it can compute the Variable Order succinct representation of the union graph within the same asymptotic time/space bounds.Comment: Accepted to SPIRE'1

Space efficient merging of de Bruijn graphs and Wheeler graphs

The merging of succinct data structures is a well established technique for the space efficient construction of large succinct indexes. In the first part of the paper we propose a new algorithm for merging succinct representations of de Bruijn graphs. Our algorithm has the same asymptotic cost of the state of the art algorithm for the same problem but it uses less than half of its working space. A novel important feature of our algorithm, not found in any of the existing tools, is that it can compute the Variable Order succinct representation of the union graph within the same asymptotic time/space bounds. In the second part of the paper we consider the more general problem of merging succinct representations of Wheeler graphs, a recently introduced graph family which includes as special cases de Bruijn graphs and many other known succinct indexes based on the BWT or one of its variants. We show that Wheeler graphs merging is in general a much more difficult problem, and we provide a space efficient algorithm for the slightly simplified problem of determining whether the union graph has an ordering that satisfies the Wheeler conditions.Comment: 24 pages, 10 figures. arXiv admin note: text overlap with arXiv:1902.0288

A new class of string transformations for compressed text indexing

Introduced about thirty years ago in the field of data compression, the Burrows-Wheeler Transform (BWT) is a string transformation that, besides being a booster of the performance of memoryless compressors, plays a fundamental role in the design of efficient self-indexing compressed data structures. Finding other string transformations with the same remarkable properties of BWT has been a challenge for many researchers for a long time. In this paper, we introduce a whole class of new string transformations, called local orderings-based transformations, which have all the “myriad virtues” of BWT. As a further result, we show that such new string transformations can be used for the construction of the recently introduced r-index, which makes them suitable also for highly repetitive collections. In this context, we consider the problem of finding, for a given string, the BWT variant that minimizes the number of runs in the transformed string

A representation of a compressed de Bruijn graph for pan-genome analysis that enables search

Recently, Marcus et al. (Bioinformatics 2014) proposed to use a compressed de Bruijn graph to describe the relationship between the genomes of many individuals/strains of the same or closely related species. They devised an $O(n \log g)$ time algorithm called splitMEM that constructs this graph directly (i.e., without using the uncompressed de Bruijn graph) based on a suffix tree, where $n$ is the total length of the genomes and $g$ is the length of the longest genome. In this paper, we present a construction algorithm that outperforms their algorithm in theory and in practice. Moreover, we propose a new space-efficient representation of the compressed de Bruijn graph that adds the possibility to search for a pattern (e.g. an allele - a variant form of a gene) within the pan-genome.Comment: Submitted to Algorithmica special issue of CPM201

Wheeler graphs: A framework for BWT-based data structures

The famous Burrows\u2013Wheeler Transform (BWT) was originally defined for a single string but variations have been developed for sets of strings, labeled trees, de Bruijn graphs, etc. In this paper we propose a framework that includes many of these variations and that we hope will simplify the search for more. We first define Wheeler graphs and show they have a property we call path coherence. We show that if the state diagram of a finite-state automaton is a Wheeler graph then, by its path coherence, we can order the nodes such that, for any string, the nodes reachable from the initial state or states by processing that string are consecutive. This means that even if the automaton is non-deterministic, we can still store it compactly and process strings with it quickly. We then rederive several variations of the BWT by designing straightforward finite-state automata for the relevant problems and showing that their state diagrams are Wheeler graphs

Detecting Mutations by eBWT

In this paper we develop a theory describing how the extended Burrows-Wheeler Transform (EBWT) of a collection of DNA fragments tends to cluster together the copies of nucleotides sequenced from a genome G. Our theory accurately predicts how many copies of any nucleotide are expected inside each such cluster, and how an elegant and precise LCP array based procedure can locate these clusters in the EBWT. Our findings are very general and can be applied to a wide range of different problems. In this paper, we consider the case of alignment-free and reference-free SNPs discovery in multiple collections of reads. We note that, in accordance with our theoretical results, SNPs are clustered in the EBWT of the reads collection, and we develop a tool finding SNPs with a simple scan of the EBWT and LCP arrays. Preliminary results show that our method requires much less coverage than state-of-the-art tools while drastically improving precision and sensitivity

Compressing and indexing aligned readsets

Compressed full-text indexes are one of the main success stories of bioinformatics data structures but even they struggle to handle some DNA readsets. This may seem surprising since, at least when dealing with short reads from the same individual, the readset will be highly repetitive and, thus, highly compressible. If we are not careful, however, this advantage can be more than offset by two disadvantages: first, since most base pairs are included in at least tens reads each, the uncompressed readset is likely to be at least an order of magnitude larger than the individual's uncompressed genome; second, these indexes usually pay some space overhead for each string they store, and the total overhead can be substantial when dealing with millions of reads. The most successful compressed full-text indexes for readsets so far are based on the Extended Burrows-Wheeler Transform (EBWT) and use a sorting heuristic to try to reduce the space overhead per read, but they still treat the reads as separate strings and thus may not take full advantage of the readset's structure. For example, if we have already assembled an individual's genome from the readset, then we can usually use it to compress the readset well: e.g., we store the gap-coded list of reads' starting positions; we store the list of their lengths, which is often highly compressible; and we store information about the sequencing errors, which are rare with short reads. There is nowhere, however, where we can plug an assembled genome into the EBWT. In this paper we show how to use one or more assembled or partially assembled genome as the basis for a compressed full-text index of its readset. Specifically, we build a labelled tree by taking the assembled genome as a trunk and grafting onto it the reads that align to it, at the starting positions of their alignments. Next, we compute the eXtended Burrows-Wheeler Transform (XBWT) of the resulting labelled tree and build a compressed full-text index on that. Although this index can occasionally return false positives, it is usually much more compact than the alternatives. Following the established practice for datasets with many repetitions, we compare different full-text indices by looking at the number of runs in the transformed strings. For a human Chr19 readset our preliminary experiments show that eliminating separators characters from the EBWT reduces the number of runs by 19%, from 220 million to 178 million, and using the XBWT reduces it by a further 15%, to 150 million