Indexing labeled graphs for pattern matching is a central challenge of pangenomics. Equi et al. (2022) [14] developed the Elastic Founder Graph (EFG) representing an alignment of m sequences of length n, drawn from alphabet Σ plus the special gap character: the paths spell the original sequences or their recombination. By enforcing the semi-repeat-free property, the EFG admits a polynomial-space index for linear-time pattern matching, breaking through the conditional lower bounds on indexing labeled graphs (Equi et al. [13]). In this work, we improve the space of the EFG index answering pattern matching queries in linear time, from linear in the length of all strings spelled by three consecutive node labels, to linear in the size of the edge labels. Then, we develop linear-time construction algorithms optimizing for different metrics: we improve the existing linearithmic construction algorithms to O(mn), by solving the novel exclusive ancestor set problem on trees; we propose, for the simplified gapless setting, an O(mn)-time solution minimizing the maximum block height, that we generalize by substituting block height with prefix-aware height. Finally, to show the versatility of the framework, we develop a BWT-based EFG index and study how to encode and perform document listing queries on a set of paths of the graphs, reporting which paths present a given pattern as a substring. We propose the EFG framework as an improved and enhanced version of the framework for the gapless setting, along with construction methods that are valid in any setting concerned with the segmentation of aligned sequences.

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: 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.

Crochemore et al. gave in WABI 2017 an algorithm that from a set of input strings finds all pairs of strings that have Hamming distance at most a given threshold. The proposed algorithm first finds all long enough exact matches between the strings, and sorts these into pairs whose coordinates also match. Then the remaining pairs are verified for the Hamming distance threshold. The algorithm was shown to work in average linear time, under some constraints and assumptions. We show that one can use the Positional Burrows–Wheeler Transform (PBWT) by Durbin (Bioinformatics, 2014) to directly find all exact matches whose coordinates also match. The same structure also extends to verifying the pairs for the Hamming distance threshold. The same analysis as for the algorithm of Crochemore et al. applies. As a side result, we show how to extend PBWT for non-binary alphabets. The new operations provided by PBWT find other applications in similar tasks as those considered here.

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.

Transcript prediction can be modeled as a graph problem where exons are modeled as nodes and reads spanning two or more exons are modeled as exon chains. Pacific Biosciences third-generation sequencing technology produces significantly longer reads than earlier second-generation sequencing technologies, which gives valuable information about longer exon chains in a graph. However, with the high error rates of third-generation sequencing, aligning long reads correctly around the splice sites is a challenging task. Incorrect alignments lead to spurious nodes and arcs in the graph, which in turn lead to incorrect transcript predictions. We survey several approaches to find the exon chains corresponding to long reads in a splicing graph, and experimentally study the performance of these methods using simulated data to allow for sensitivity/precision analysis. Our experiments show that short reads from second-generation sequencing can be used to significantly improve exon chain correctness either by error-correcting the long reads before splicing graph creation, or by using them to create a splicing graph on which the long-read alignments are then projected. We also study the memory and time consumption of various modules, and show that accurate exon chains lead to significantly increased transcript prediction accuracy. AVAILABILITY: The simulated data and in-house scripts used for this article are available at http://www.cs.helsinki.fi/group/gsa/exon-chains/exon-chains-bib.tar.bz2.