We initiate the study of computational problems on $k$-mers (strings of length $k$) in labeled graphs. As a starting point, we consider the problem of counting the number of distinct $k$-mers found on the walks of a graph. We establish that this is $\#$P-hard, even on connected deterministic DAGs. However, in the class of deterministic Wheeler graphs (Gagie, Manzini, and Sirèn, TCS 2017), we show that distinct $k$-mers of such a graph $W=(V, E)$ can be counted using $O(|W|k)$ or $O(n^4 \log k)$ arithmetic operations, where $n=|V|$, $m=|E|$ and $|W|=n+m$. The latter result uses a new generalization of the technique of prefix doubling to Wheeler graphs. To generalize our results beyond Wheeler graphs, we discuss ways to transform a graph into a Wheeler graph in a manner that preserves the $k$-mers. As an application of our $k$-mer counting algorithms, we construct a representation of the de Bruijn graph of the $k$-mers that occupies $O(n_k + |W|k \log(\max_{1 \leq \ell \leq k} n_\ell) + σ\log m)$ bits of space, where $n_\ell$ is the number of distinct $\ell$-mers in the Wheeler graph, and $σ$ is the size of the alphabet. We show how to construct it in the same time complexity. Given that the Wheeler graph can be exponentially smaller than the de Bruijn graph, for large $k$ this provides a theoretical improvement over previous de Bruijn graph construction methods from graphs, which must spend $Ω(k)$ time per $k$-mer in the graph.

In recent years, pseudoalignment as a means for mapping reads to databases of reference genomes has become a widely-used method in studies of bacterial pathogenesis. A popular pseudoalignment criterion is thresholded union, in which a read is said to pseudoalign to a reference if the reference contains more than a percentage t of the read’s k-mers. Several pseudoalignment indexing tools that implement this and other pseudoalignment criteria are now available, including Bifrost, Themisto, and Fulgor. In this paper, we describe a scheme for single-species bacterial pseudoalignment that, instead of k-mers, uses shortest unique finimizers (Alanko et al., IEEE/ACM TCBB, 2025) as features for determining pseudoalignment. We show that this scheme, which we call Finimap, leads to a significantly lower false-positive rate than other recent "approximate pseudoalignment" methods Kaminari and Raptor, and is also faster.

We clarify the design principles and evaluation choices underlying Syotti, a robust and scalable probe-design tool developed to support large, heterogeneous bacterial datasets with minimal parameter tuning. We highlight Syotti's ability to perform simultaneous large-scale designs and its effectiveness as a reliable alternative when existing tools such as CATCH are not well suited to the problem setting.

Since their emergence some two decades ago, indexes based on the Burrows-Wheeler transform (BWT) have been intensely studied and today find wide use in genomics, where they form the basis of software tools for read alignment and \(k\)-mer lookup—routine tasks in modern data-intensive bioinformatics pipelines.

In this paper, we study the following problem: given n subsets S₁, … , S_n of an integer universe U = {0,… , u-1}, having total cardinality N = ∑_{i = 1}ⁿ |S_i|, find a prefix-free encoding enc : U → {0,1}^+ minimizing the so-called trie measure, i.e., the total number of edges in the n binary tries T₁, … , T_n, where T_i is the trie packing the encoded integers {enc(x):x ∈ S_i}. We first observe that this problem is equivalent to that of merging u sets with the cheapest sequence of binary unions, a problem which in [Ghosh et al., ICDCS 2015] is shown to be NP-hard. Motivated by the hardness of the general problem, we focus on particular families of prefix-free encodings. We start by studying the fixed-length shifted encoding of [Gupta et al., Theoretical Computer Science 2007]. Given a parameter 0 ≤ a < u, this encoding sends each x ∈ U to (x + a) mod u, interpreted as a bit-string of log u bits. We develop the first efficient algorithms that find the value of a minimizing the trie measure when this encoding is used. Our two algorithms run in O(u + Nlog u) and O(Nlog² u) time, respectively. We proceed by studying ordered encodings (a.k.a. monotone or alphabetic), and describe an algorithm finding the optimal such encoding in O(N+u³) time. Within the same running time, we show how to compute the best shifted ordered encoding, provably no worse than both the optimal shifted and optimal ordered encodings. We provide implementations of our algorithms and discuss how these encodings perform in practice.

The minimizer of a $k$-mer is the smallest $m$-mer inside the $k$-mer according to some total order $< $ of the $m$-mers. Minimizers are often used as keys in hash tables in indexing tasks in metagenomics and pangenomics. The main weakness of minimizer-based indexing is the possibility of very frequently occurring minimizers, which can slow query times down significantly. Popular minimizer alignment tools employ various and often wild heuristics as workarounds, typically by ignoring frequent minimizers or blacklisting commonly occurring patterns, to the detriment of other metrics (e.g., alignment recall, space usage, or code complexity). In this paper, we introduce frequency-bounded minimizers, which we call finimizers, for indexing sets of $k$-mers. The idea is to use an order relation $< $ for minimizer comparison that depends on the frequency of the minimizers within the indexed $k$-mers. With finimizers, the length $m$ of the $m$-mers is not fixed, but is allowed to vary depending on the context, so that the length can increase to bring the frequency down below a user-specified threshold $t$. Setting a maximum frequency solves the issue of very frequent minimizers and gives us a worst-case guarantee for the query time. We show how to implement a particular finimizer scheme efficiently using the Spectral Burrows-Wheeler transform ($SBWT$) (Alanko et al. Proc. SIAM ACDA, 2023) augmented with longest common suffix information. In experiments, we explore in detail the special case in which we set $t = 1$. This choice simplifies the index structure and makes the scheme completely parameter-free apart from the choice of $k$. A prototype implementation of this scheme exhibits $k$-mer localization times close to, and often faster than, state-of-the-art minimizer-based schemes.

The LCP array is an important tool in stringology, allowing to speed up pattern matching algorithms and enabling compact representations of the suffix tree. Recently, Conte et al. [DCC 2023] and Cotumaccio et al. [SPIRE 2023] extended the definition of this array to Wheeler DFAs and, ultimately, to arbitrary labeled graphs, proving that it can be used to efficiently solve matching statistics queries on the graph’s paths. In this paper, we provide the first efficient algorithm building the LCP array of a directed labeled graph with n nodes and m edges labeled over an alphabet of size σ. The first step is to transform the input graph G into a deterministic Wheeler pseudoforest G_{is} with O(n) edges encoding the lexicographically- smallest and largest strings entering in each node of the original graph. Using state-of-the-art algorithms, this step runs in O(min{mlog n, m+n²}) time on arbitrary labeled graphs, and in O(m) time on Wheeler DFAs. The LCP array of G stores the longest common prefixes between those strings, i.e. it can easily be derived from the LCP array of G_{is}. After arguing that the natural generalization of a compact-space LCP-construction algorithm by Beller et al. [J. Discrete Algorithms 2013] runs in time Ω(nσ) on pseudoforests, we present a new algorithm based on dynamic range stabbing building the LCP array of G_{is} in O(nlog σ) time and O(nlogσ) bits of working space. Combined with our reduction, we obtain the first efficient algorithm to build the LCP array of an arbitrary labeled graph. An implementation of our algorithm is publicly available at https://github.com/regindex/Labeled-Graph-LCP.

MOTIVATION: World Health Organization estimates that there were over 10 million cases of tuberculosis (TB) worldwide in 2019, resulting in over 1.4 million deaths, with a worrisome increasing trend yearly. The disease is caused by Mycobacterium tuberculosis (MTB) through airborne transmission. Treatment of TB is estimated to be 85% successful, however, this drops to 57% if MTB exhibits multiple antimicrobial resistance (AMR), for which fewer treatment options are available. RESULTS: We develop a robust machine-learning classifier using both linear and nonlinear models (i.e. LASSO logistic regression (LR) and random forests (RF)) to predict the phenotypic resistance of Mycobacterium tuberculosis (MTB) for a broad range of antibiotic drugs. We use data from the CRyPTIC consortium to train our classifier, which consists of whole genome sequencing and antibiotic susceptibility testing (AST) phenotypic data for 13 different antibiotics. To train our model, we assemble the sequence data into genomic contigs, identify all unique 31-mers in the set of contigs, and build a feature matrix M, where M[i, j] is equal to the number of times the ith 31-mer occurs in the jth genome. Due to the size of this feature matrix (over 350 million unique 31-mers), we build and use a sparse matrix representation. Our method, which we refer to as MTB++, leverages compact data structures and iterative methods to allow for the screening of all the 31-mers in the development of both LASSO LR and RF. MTB++ is able to achieve high discrimination (F-1 >80%) for the first-line antibiotics. Moreover, MTB++ had the highest F-1 score in all but three classes and was the most comprehensive since it had an F-1 score >75% in all but four (rare) antibiotic drugs. We use our feature selection to contextualize the 31-mers that are used for the prediction of phenotypic resistance, leading to some insights about sequence similarity to genes in MEGARes. Lastly, we give an estimate of the amount of data that is needed in order to provide accurate predictions. AVAILABILITY: The models and source code are publicly available on Github at https://github.com/M-Serajian/MTB-Pipeline.

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: Huge datasets containing whole-genome sequences of bacterial strains are now commonplace and represent a rich and important resource for modern genomic epidemiology and metagenomics. In order to efficiently make use of these datasets, efficient indexing data structures-that are both scalable and provide rapid query throughput-are paramount. RESULTS: Here, we present Themisto, a scalable colored k-mer index designed for large collections of microbial reference genomes, that works for both short and long read data. Themisto indexes 179 thousand Salmonella enterica genomes in 9 h. The resulting index takes 142 gigabytes. In comparison, the best competing tools Metagraph and Bifrost were only able to index 11 000 genomes in the same time. In pseudoalignment, these other tools were either an order of magnitude slower than Themisto, or used an order of magnitude more memory. Themisto also offers superior pseudoalignment quality, achieving a higher recall than previous methods on Nanopore read sets. AVAILABILITY AND IMPLEMENTATION: Themisto is available and documented as a C++ package at https://github.com/algbio/themisto available under the GPLv2 license.

Wheeler DFAs (WDFAs) are a sub-class of finite-state automata which is playing an important role in the emerging field of compressed data structures: as opposed to general automata, WDFAs can be stored in just$\log\sigma+O(1)$bits per edge,$\sigma$being the alphabet's size, and support optimal-time pattern matching queries on the substring closure of the language they recognize. An important step to achieve further compression is minimization. When the input$\mathcal{A}$is a general deterministic finite-state automaton (DFA), the state-of-the-art is represented by the classic Hopcroft's algorithm, which runs in$O(\vert \mathcal{A}\vert \log\vert \mathcal{A}\vert )$time. This algorithm stands at the core of the only existing minimization algorithm for Wheeler DFAs, which inherits its complexity. In this work, we show that the minimum WDFA equivalent to a given input WDFA can be computed in linear$O(\vert \mathcal{A}\vert )$time. When run on de Bruijn WDFAs built from real DNA datasets, an implementation of our algorithm reduces the number of nodes from 14% to 51% at a speed of more than 1 million nodes per second.

MOTIVATION: Bait enrichment is a protocol that is becoming increasingly ubiquitous as it has been shown to successfully amplify regions of interest in metagenomic samples. In this method, a set of synthetic probes ('baits') are designed, manufactured and applied to fragmented metagenomic DNA. The probes bind to the fragmented DNA and any unbound DNA is rinsed away, leaving the bound fragments to be amplified for sequencing. Metsky et al. demonstrated that bait-enrichment is capable of detecting a large number of human viral pathogens within metagenomic samples. RESULTS: We formalize the problem of designing baits by defining the Minimum Bait Cover problem, show that the problem is NP-hard even under very restrictive assumptions, and design an efficient heuristic that takes advantage of succinct data structures. We refer to our method as Syotti. The running time of Syotti shows linear scaling in practice, running at least an order of magnitude faster than state-of-the-art methods, including the method of Metsky et al. At the same time, our method produces bait sets that are smaller than the ones produced by the competing methods, while also leaving fewer positions uncovered. Lastly, we show that Syotti requires only 25 min to design baits for a dataset comprised of 3 billion nucleotides from 1000 related bacterial substrains, whereas the method of Metsky et al. shows clearly super-linear running time and fails to process even a subset of 17% of the data in 72 h. AVAILABILITY AND IMPLEMENTATION: https://github.com/jnalanko/syotti. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

Indexing strings via prefix (or suffix) sorting is, arguably, one of the most successful algorithmic techniques developed in the last decades. Can indexing be extended to languages? The main contribution of this paper is to initiate the study of the sub-class of regular languages accepted by an automaton whose states can be prefix-sorted. Starting from the recent notion of Wheeler graph [Gagie et al., TCS 2017]— which extends naturally the concept of prefix sorting to labeled graphs—we investigate the properties of Wheeler languages, that is, regular languages admitting an accepting Wheeler finite automaton. We first characterize this family as the natural extension of regular languages endowed with the co-lexicographic ordering: the sorted prefixes of strings belonging to a Wheeler language are partitioned into a finite number of co-lexicographic intervals, each formed by elements from a single Myhill-Nerode equivalence class. We proceed by proving several results related to Wheeler automata: (i) We show that every Wheeler NFA (WNFA) with n states admits an equivalent Wheeler DFA (WDFA) with at most 2n – 1 – |Σ| states (Σ being the alphabet) that can be computed in O(n3) time. (ii) We describe a quadratic algorithm to prefix-sort a proper superset of the WDFAs, a O(n log n)-time online algorithm to sort acyclic WDFAs, and an optimal linear-time offline algorithm to sort general WDFAs. (iii) We provide a minimization theorem that characterizes the smallest WDFA recognizing the same language of any input WDFA. The corresponding constructive algorithm runs in optimal linear time in the acyclic case, and in O(n log n) time in the general case. (iv) We show how to compute the smallest WDFA equivalent to any acyclic DFA in nearly-optimal time. Our contributions imply new results of independent interest. Contributions (i-iii) provide a new class of NFAs for which the minimization problem can be approximated within a constant factor in polynomial time. Contribution (iv) provides a provably minimum-size solution for the well-studied problem of indexing deterministicacyclic graphs for linear-time pattern matching queries.

Baier (CPM 2018) describes tunneling as a technique to further exploit redundancies in the Burrows-Wheeler Transform. In this paper we show how to retain indexed text searching on the resulting structure and generalize the concept to Wheeler graphs.

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: Markov models with contexts of variable length are widely used in bioinformatics for representing sets of sequences with similar biological properties. When models contain many long contexts, existing implementations are either unable to handle genome-scale training datasets within typical memory budgets, or they are optimized for specific model variants and are thus inflexible. RESULTS: We provide practical, versatile representations of variable-order Markov models and of interpolated Markov models, that support a large number of context-selection criteria, scoring functions, probability smoothing methods, and interpolations, and that take up to four times less space than previous implementations based on the suffix array, regardless of the number and length of contexts, and up to ten times less space than previous trie-based representations, or more, while matching the size of related, state-of-the-art data structures from Natural Language Processing. We describe how to further compress our indexes to a quantity related to the redundancy of the training data, saving up to 90% of their space on very repetitive datasets, and making them become up to 60 times smaller than previous implementations based on the suffix array. Finally, we show how to exploit constraints on the length and frequency of contexts to further shrink our compressed indexes to half of their size or more, achieving data structures that are a hundred times smaller than previous implementations based on the suffix array, or more. This allows variable-order Markov models to be used with bigger datasets and with longer contexts on the same hardware, thus possibly enabling new applications. AVAILABILITY AND IMPLEMENTATION: https://github.com/jnalanko/VOMM. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

BACKGROUND: A metagenomic sample is a set of DNA fragments, randomly extracted from multiple cells in an environment, belonging to distinct, often unknown species. Unsupervised metagenomic clustering aims at partitioning a metagenomic sample into sets that approximate taxonomic units, without using reference genomes. Since samples are large and steadily growing, space-efficient clustering algorithms are strongly needed. RESULTS: We design and implement a space-efficient algorithmic framework that solves a number of core primitives in unsupervised metagenomic clustering using just the bidirectional Burrows-Wheeler index and a union-find data structure on the set of reads. When run on a sample of total length n, with m reads of maximum length ℓ each, on an alphabet of total size σ, our algorithms take O(n(t+logσ)) time and just 2n+o(n)+O(max{ℓ σlogn,K logm}) bits of space in addition to the index and to the union-find data structure, where K is a measure of the redundancy of the sample and t is the query time of the union-find data structure. CONCLUSIONS: Our experimental results show that our algorithms are practical, they can exploit multiple cores by a parallel traversal of the suffix-link tree, and they are competitive both in space and in time with the state of the art.