SUMMARY: In this article, we introduce the Conway-Bromage-Lyndon (CBL) structure, a compressed, dynamic and exact method for representing k-mer sets. Originating from Conway and Bromage's concept, CBL innovatively employs the smallest cyclic rotations of k-mers, akin to Lyndon words, to leverage lexicographic redundancies. In order to support dynamic operations and set operations, we propose a dynamic bit vector structure that draws a parallel with Elias-Fano's scheme. This structure is encapsulated in a Rust library, demonstrating a balanced blend of construction efficiency, cache locality, and compression. Our findings suggest that CBL outperforms existing dynamic k-mer set methods. Unique to this work, CBL stands out as the only known exact k-mer structure offering in-place set operations. Its different combined abilities position it as a flexible Swiss knife structure for k-mer set management. AVAILABILITY AND IMPLEMENTATION: https://github.com/imartayan/CBL.

Abstract We study the problem of matching a string in a labeled graph. Previous research has shown that unless theOrthogonal Vectors Hypothesis(OVH) is false, one cannot solve this problem in strongly sub-quadratic time, nor index the graph in polynomial time to answer queries efficiently (Equi et al. ICALP 2019, SOFSEM 2021). These conditional lower-bounds cover even deterministic graphs with binary alphabet, but there naturally exist also graph classes that are easy to index: For example,Wheeler graphs(Gagie et al. Theor. Comp. Sci.2017) cover graphs admitting a Burrows-Wheeler transform -based indexing scheme. However, it is NP-complete to recognize if a graph is a Wheeler graph (Gibney, Thankachan, ESA 2019). We propose an approach to alleviate the construction bottleneck of Wheeler graphs. Rather than starting from an arbitrary graph, we study graphs induced frommultiple sequence alignments().Elastic degenerate strings(Bernadini et al. SPIRE 2017, ICALP 2019) can be seen as such graphs, and we introduce here their generalization:elastic founder graphs. We first prove that even such induced graphs are hard to index under OVH. Then we introduce two subclasses, repeat-free and semi-repeat-free graphs, that are easy to index. We give a linear time algorithm to construct a repeat-free (non-elastic) founder graph from a gapless , and (parameterized) near-linear time algorithms to construct a semi-repeat-free (repeat-free, respectively) elastic founder graph from general . Finally, we show that repeat-free founder graphs admit a reduction to Wheeler graphs in polynomial time.

MOTIVATION: Seeking probabilistic motifs in a sequence is a common task to annotate putative transcription factor binding sites or other RNA/DNA binding sites. Useful motif representations include position weight matrices (PWMs), dinucleotide PWMs (di-PWMs), and hidden Markov models (HMMs). Dinucleotide PWMs not only combine the simplicity of PWMs-a matrix form and a cumulative scoring function-but also incorporate dependency between adjacent positions in the motif (unlike PWMs which disregard any dependency). For instance to represent binding sites, the HOCOMOCO database provides di-PWM motifs derived from experimental data. Currently, two programs, SPRy-SARUS and MOODS, can search for occurrences of di-PWMs in sequences. RESULTS: We propose a Python package called dipwmsearch, which provides an original and efficient algorithm for this task (it first enumerates matching words for the di-PWM, and then searches these all at once in the sequence, even if the latter contains IUPAC codes). The user benefits from an easy installation via Pypi or conda, a comprehensive documentation, and executable scripts that facilitate the use of di-PWMs. AVAILABILITY AND IMPLEMENTATION: dipwmsearch is available at https://pypi.org/project/dipwmsearch/ and https://gite.lirmm.fr/rivals/dipwmsearch/ under Cecill license.

International audience

The hierarchical overlap graph (HOG) is a graph that encodes overlaps from a given set P of n strings, as the overlap graph does. A best known algorithm constructs HOG in O(||P|| log n) time and O(||P||) space, where ||P|| is the sum of lengths of the strings in P. In this paper we present a new algorithm to construct HOG in O(||P||) time and space. Hence, the construction time and space of HOG are better than those of the overlap graph, which are O(||P|| + n^2).

MOTIVATION: Variant calling workflows that utilize a single reference sequence are the de facto standard elementary genomic analysis routine for resequencing projects. Various ways to enhance the reference with pangenomic information have been proposed, but scalability combined with seamless integration to existing workflows remains a challenge. RESULTS: We present PanVC with founder sequences, a scalable and accurate variant calling workflow based on a multiple alignment of reference sequences. Scalability is achieved by removing duplicate parts up to a limit into a founder multiple alignment, that is then indexed using a hybrid scheme that exploits general purpose read aligners. Our implemented workflow uses GATK or BCFtools for variant calling, but the various steps of our workflow (e.g. vcf2multialign tool, founder reconstruction) can be of independent interest as a basis for creating novel pangenome analysis workflows beyond variant calling. AVAILABILITY AND IMPLEMENTATION: Our open access tools and instructions how to reproduce our experiments are available at the following address: https://github.com/algbio/panvc-founders. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

We introduce a compact pangenome representation based on an optimal segmentation concept that aims to reconstruct founder sequences from a multiple sequence alignment (MSA). Such founder sequences have the feature that each row of the MSA is a recombination of the founders. Several linear time dynamic programming algorithms have been previously devised to optimize segmentations that induce founder blocks that then can be concatenated into a set of founder sequences. All possible concatenation orders can be expressed as a founder block graph. We observe a key property of such graphs: if the node labels (founder segments) do not repeat in the paths of the graph, such graphs can be indexed for efficient string matching. We call such graphs segment repeat-free founder block graphs. We give a linear time algorithm to construct a segment repeat-free founder block graph given an MSA. The algorithm combines techniques from the founder segmentation algorithms (Cazaux et al. SPIRE 2019) and fully-functional bidirectional Burrows-Wheeler index (Belazzougui and Cunial, CPM 2019). We derive a succinct index structure to support queries of arbitrary length in the paths of the graph. Experiments on an MSA of SAR-CoV-2 strains are reported. An MSA of size $410\times 29811$ is compacted in one minute into a segment repeat-free founder block graph of 3900 nodes and 4440 edges. The maximum length and total length of node labels is 12 and 34968, respectively. The index on the graph takes only $3\%$ of the size of the MSA.

The boom of genomic sequencing makes compression of sets of sequences inescapable. This underlies the need for multi-string indexing data structures that helps compressing the data. The most prominent example of such data structures is the Burrows-Wheeler Transform (BWT), a reversible permutation of a text that improves its compressibility. A similar data structure, the eXtended Burrows-Wheeler Transform (XBW), is able to index a tree labelled with alphabet symbols. A link between a multi-string BWT and the Aho-Corasick automaton has already been found and led to a way to build a XBW from a multi-string BWT. We exhibit a stronger link between a multi-string BWT and a XBW by using the order of the concatenation in the multi-string. This bijective link has several applications: first, it allows one to build one data structure from the other; second, it enables one to compute an ordering of the input strings that optimises a Run-Length measure (i.e., the compressibility) of the BWT or of the XBW.

Recent large-scale community sequencing efforts allow at an unprecedented level of detail the identification of genomic regions that show signatures of natural selection. Traditional methods for identifying such regions from individuals' haplotype data, however, require excessive computing times and therefore are not applicable to current datasets. In 2019, Cunha et al. (Proceedings of BSB 2019) suggested the maximal perfect haplotype block as a very simple combinatorial pattern, forming the basis of a new method to perform rapid genome-wide selection scans. The algorithm they presented for identifying these blocks, however, had a worst-case running time quadratic in the genome length. It was posed as an open problem whether an optimal, linear-time algorithm exists. In this paper we give two algorithms that achieve this time bound, one conceptually very simple one using suffix trees and a second one using the positional Burrows-Wheeler Transform, that is very efficient also in practice.

MOTIVATION: The visualization and interpretation of evolutionary spatiotemporal scenarios is broadly and increasingly used in infectious disease research, ecology or agronomy. Using probabilistic frameworks, well-known tools can infer from molecular data ancestral traits for internal nodes in a phylogeny, and numerous phylogenetic rendering tools can display such evolutionary trees. However, visualizing such ancestral information and its uncertainty on the tree remains tedious. For instance, ancestral nodes can be associated to several geographical annotations with close probabilities and thus, several migration or transmission scenarios exist. RESULTS: We expose a web-based tool, named AQUAPONY, that facilitates such operations. Given an evolutionary tree with ancestral (e.g. geographical) annotations, the user can easily control the display of ancestral information on the entire tree or a subtree, and can view alternative phylogeographic scenarios along a branch according to a chosen uncertainty threshold. AQUAPONY interactively visualizes the tree and eases the objective interpretation of evolutionary scenarios. AQUAPONY's implementation makes it highly responsive to user interaction, and instantaneously updates the tree visualizations even for large trees (which can be exported as image files). AVAILABILITY AND IMPLEMENTATION: AQUAPONY is coded in JavaScript/HTML, available under Cecill license, and can be freely used at http://www.atgc-montpellier.fr/aquapony/.

DNA sequencing technologies have tremendously increased their throughput, and hence complicated DNA assembly. Numerous assembly programs use de Bruijn graphs (dBG) built from short reads to merge these into contigs, which represent putative DNA segments. In a dBG of order k, nodes are substrings of length k of reads (or k-mers), while arcs are their k+1-mers. As analysing reads often require to index all their substrings, it is interesting to exhibit algorithms that directly build a dBG from a pre-existing index, and especially a contracted dBG, where non-branching paths are condensed into single nodes. Here, we exhibit linear time algorithms for constructing the full or contracted dBGs from suffix trees, suffix arrays, and truncated suffix trees. With the latter the construction uses a space that is linear in the size of the dBG. Finally, we also provide algorithms to dynamically update the order of the graph without reconstructing it.

A superstring of a set of words P = {s_1, ..., s_p } is a string that contains each word of P as substring. Given P, the well known Shortest Linear Superstring problem (SLS), asks for a shortest superstring of P. In a variant of SLS, called Multi-SLS, each word s_i comes with an integer m(i), its multiplicity, that sets a constraint on its number of occurrences, and the goal is to find a shortest superstring that contains at least m(i) occurrences of s_i. Multi-SLS generalizes SLS and is obviously as hard to solve, but it has been studied only in special cases (with words of length 2 or with a fixed number of words). The approximability of Multi-SLS in the general case remains open. Here, we study the approximability of Multi-SLS and that of the companion problem Multi-SCCS, which asks for a shortest cyclic cover instead of shortest superstring. First, we investigate the approximation of a greedy algorithm for maximizing the compression offered by a superstring or by a cyclic cover: the approximation ratio is 1/2 for Multi-SLS and 1 for Multi-SCCS. Then, we exhibit a linear time approximation algorithm, Concat-Greedy, and show it achieves a ratio of 4 regarding the superstring length. This demonstrates that for both measures Multi-SLS belongs to the class of APX problems.

Given a threshold L and a set R = {R_1, ..., R_m} of m strings (haplotype sequences), each having length n, the minimum segmentation problem for founder reconstruction is to partition [1,n] into set P of disjoint segments such that each segment [a,b] in P has length at least L and the number d(a,b)=|{R_i[a,b] : 1 <= i <= m}| of distinct substrings at segment [a,b] is minimized over [a,b] in P. The distinct substrings in the segments represent founder blocks that can be concatenated to form max{d(a,b) : [a,b] in P} founder sequences representing the original R such that crossovers happen only at segment boundaries. We give an optimal O(mn) time algorithm to solve the problem, improving over earlier O(mn^2). This improvement enables to exploit the algorithm on a pan-genomic setting of input strings being aligned haplotype sequences of complete human chromosomes, with a goal of finding a representative set of references that can be indexed for read alignment and variant calling. We implemented the new algorithm and give some experimental evidence on the practicality of the approach on this pan-genomic setting.

For analysing text algorithms, for computing superstrings, or for testing random number generators, one needs to compute all overlaps between any pairs of words in a given set. The positions of overlaps of a word onto itself, or of two words, are needed to compute the absence probability of a word in a random text, or the numbers of common words shared by two random texts. In all these contexts, one needs to compute or to query overlaps between pairs of words in a given set. For this sake, we designed COvI, a compressed overlap index that supports multiple queries on overlaps: like computing the correlation of two words, or listing pairs of words whose longest overlap is maximal among all possible pairs. COvI stores overlaps in a hierarchical and non-redundant manner. We propose an implementation that can handle datasets of millions of words and still answer queries efficiently. Comparison with a baseline solution - called FullAC - relying on the Aho-Corasick automaton shows that COvI provides significant advantages. For similar construction times, COvI requires half the memory FullAC, and still solves complex queries much faster.

A superstring of a set of words is a string that contains each input word as a substring. Given such a set, the Shortest Superstring Problem (SSP) asks for a superstring of minimum length. SSP is an important theoretical problem related to the Asymmetric Travelling Salesman Problem, and also has practical applications in data compression and in bioinformatics. Indeed, it models the question of assembling a genome from a set of sequencing reads. Unfortunately, SSP is known to be NP-hard even on a binary alphabet and also hard to approximate with respect to the superstring length or to the compression achieved by the superstring. Even the variant in which all words share the same length r, called r-SSP, is NP-hard whenever r>2. Numerous involved approximation algorithms achieve approximation ratio above 2 for the superstring, but remain difficult to implement in practice. In contrast the greedy conjecture asked in 1988 whether a simple greedy algorithm achieves ratio of 2 for SSP. Here, we present a novel approach to bound the superstring approximation ratio with the compression ratio, which, when applied to the greedy algorithm, shows a 2 approximation ratio for 3-SSP, and also that greedy achieves ratios smaller than 2. This leads to a new version of the greedy conjecture.

For a set of input words, finding a superstring (a string containing each word of the set as a substring) of minimal length is hard. Most approximation algorithms solve the Shortest Cyclic Cover problem before merging the cyclic strings into a linear superstring. A cyclic cover is a set of cyclic strings in which the input words occur as a substring. We investigate a variant of the Shortest Cyclic Cover problem for the case of DNA. Because the two strands that compose DNA have a reverse complementary sequence, and because the sequencing process often overlooks the strand of a read, each read or its reverse complement must occur as a substring in a cyclic cover. We exhibit a linear time algorithm based on graphs for solving the Shortest DNA Cyclic Cover problem and propose compressed data structures for storing the underlying graphs. All results and algorithms can be adapted to the case where strings are simply reversed but not complemented (e.g. in pattern recognition).

BACKGROUND: Next Generation Sequencing (NGS) has dramatically enhanced our ability to sequence genomes, but not to assemble them. In practice, many published genome sequences remain in the state of a large set of contigs. Each contig describes the sequence found along some path of the assembly graph, however, the set of contigs does not record all the sequence information contained in that graph. Although many subsequent analyses can be performed with the set of contigs, one may ask whether mapping reads on the contigs is as informative as mapping them on the paths of the assembly graph. Currently, one lacks practical tools to perform mapping on such graphs. RESULTS: Here, we propose a formal definition of mapping on a de Bruijn graph, analyse the problem complexity which turns out to be NP-complete, and provide a practical solution. We propose a pipeline called GGMAP (Greedy Graph MAPping). Its novelty is a procedure to map reads on branching paths of the graph, for which we designed a heuristic algorithm called BGREAT (de Bruijn Graph REAd mapping Tool). For the sake of efficiency, BGREAT rewrites a read sequence as a succession of unitigs sequences. GGMAP can map millions of reads per CPU hour on a de Bruijn graph built from a large set of human genomic reads. Surprisingly, results show that up to 22 % more reads can be mapped on the graph but not on the contig set. CONCLUSIONS: Although mapping reads on a de Bruijn graph is complex task, our proposal offers a practical solution combining efficiency with an improved mapping capacity compared to assembly-based mapping even for complex eukaryotic data.