Linear Time Construction of Indexable Founder Block Graphs

Peer reviewe

Linear time minimum segmentation enables scalable founder reconstruction

Background: We study a preprocessing routine relevant in pan-genomic analyses: consider a set of aligned haplotype sequences of complete human chromosomes. Due to the enormous size of such data, one would like to represent this input set with a few founder sequences that retain as well as possible the contiguities of the original sequences. Such a smaller set gives a scalable way to exploit pan-genomic information in further analyses (e.g. read alignment and variant calling). Optimizing the founder set is an NP-hard problem, but there is a segmentation formulation that can be solved in polynomial time, defined as follows. Given a threshold L and a set R={R1,...,Rm} 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]P has length at least L and the number d(a,b)=|{Ri[a,b]:1im}| of distinct substrings at segment [a,b] is minimized over [a,b]P. The distinct substrings in the segments represent founder blocks that can be concatenated to form max{d(a,b):[a,b]P} founder sequences representing the original R such that crossovers happen only at segment boundaries. Results: We give an O(mn) time (i.e. linear time in the input size) algorithm to solve the minimum segmentation problem for founder reconstruction, improving over an earlier O(mn2). Conclusions: Our improvement enables to apply the formulation on an input of thousands of complete human chromosomes. We implemented the new algorithm and give experimental evidence on its practicality. The implementation is available in https://github.com/tsnorri/founder-sequences.Peer reviewe

Elastic-Degenerate String Matching with 1 Error

An elastic-degenerate string is a sequence of $n$ finite sets of strings of total length $N$, introduced to represent a set of related DNA sequences, also known as a pangenome. The ED string matching (EDSM) problem consists in reporting all occurrences of a pattern of length $m$ in an ED text. This problem has recently received some attention by the combinatorial pattern matching community, culminating in an $\tilde{\mathcal{O}}(nm^{\omega-1})+\mathcal{O}(N)$-time algorithm [Bernardini et al., SIAM J. Comput. 2022], where $\omega$ denotes the matrix multiplication exponent and the $\tilde{\mathcal{O}}(\cdot)$ notation suppresses polylog factors. In the $k$-EDSM problem, the approximate version of EDSM, we are asked to report all pattern occurrences with at most $k$ errors. $k$-EDSM can be solved in $\mathcal{O}(k^2mG+kN)$ time, under edit distance, or $\mathcal{O}(kmG+kN)$ time, under Hamming distance, where $G$ denotes the total number of strings in the ED text [Bernardini et al., Theor. Comput. Sci. 2020]. Unfortunately, $G$ is only bounded by $N$, and so even for $k=1$, the existing algorithms run in $\Omega(mN)$ time in the worst case. In this paper we show that $1$-EDSM can be solved in $\mathcal{O}((nm^2 + N)\log m)$ or $\mathcal{O}(nm^3 + N)$ time under edit distance. For the decision version, we present a faster $\mathcal{O}(nm^2\sqrt{\log m} + N\log\log m)$-time algorithm. We also show that $1$-EDSM can be solved in $\mathcal{O}(nm^2 + N\log m)$ time under Hamming distance. Our algorithms for edit distance rely on non-trivial reductions from $1$-EDSM to special instances of classic computational geometry problems (2d rectangle stabbing or 2d range emptiness), which we show how to solve efficiently. In order to obtain an even faster algorithm for Hamming distance, we rely on employing and adapting the $k$-errata trees for indexing with errors [Cole et al., STOC 2004].Comment: This is an extended version of a paper accepted at LATIN 202

A Rearrangement Distance for Fully-Labelled Trees

The problem of comparing trees representing the evolutionary histories of cancerous tumors has turned out to be crucial, since there is a variety of different methods which typically infer multiple possible trees. A departure from the widely studied setting of classical phylogenetics, where trees are leaf-labelled, tumoral trees are fully labelled, i.e., every vertex has a label. In this paper we provide a rearrangement distance measure between two fully-labelled trees. This notion originates from two operations: one which modifies the topology of the tree, the other which permutes the labels of the vertices, hence leaving the topology unaffected. While we show that the distance between two trees in terms of each such operation alone can be decided in polynomial time, the more general notion of distance when both operations are allowed is NP-hard to decide. Despite this result, we show that it is fixed-parameter tractable, and we give a 4-approximation algorithm when one of the trees is binary

Development of efficient De Bruijn graph-based algorithms for genome assembly

Programa Oficial de Doutoramento en Computación. 5009V01[Abstract] During the last two decades, thanks to the development of new sequencing techniques, the study of the genome has become very popular in order to discover the genetic variation present in both humans and other organisms. The predominant mode of genome analysis is through the assembly of reads in one or multiple chains for as long as possible. The most traditional way of assembly is the one that involves reads from a single genome. In this field, in the last decade, third-generation readings have emerged with new challenges for which there are no efficient solutions. The first contribution that has been made in this thesis is Compact-Flye, a tool for the efficient assembly of third-generation reads on the Flye algorithm. This tool is based on the ingenious use of compact data structures to improve typical assembly steps such as counting and indexing k-mers. Apart from the assembly of a genome, there are techniques that seek to assemble all the genomes contained in a given sample. This assembly is known as multiple sequence assembly or haplotype reconstruction, a subject also treated in this thesis. Our first approach to solving this has been viaDBG, which is the first solution based on de Bruijn graphs that offers results comparable to current techniques in viral genome assembly while maintaining the efficiency of these graphs. Our second contribution is ViQUF, which is a natural improvement on its predecessor. ViQUF completely changes the algorithm of viaDBG but continues to be based on the same structures, although with some variations that allow it not only to improve results in terms of time and quality, but also to provide additionalinformation such as an estimate of the relative presence of each species in the sample.[Resumen] Durante las últimas dos décadas, gracias al desarrollo de nuevas técnias secuenciación, el estudio del genoma ha ganado mucha popularidad de cara a conocer la variación genética presente tanto seres humanos como otros organismos. El modo predominante de análisis del genoma es a través del ensamblaje de lecturas en una o múltiples cadenas lo más largas posibles. La manera más tradicional de ensamblaje es el que implica lecturas provenientes de un solo genoma. En este campo, en la última década han surgido las lecturas de tercera generación con nuevos retos para los que no existen soluciones eficientes. La primera aportación que se ha realizado en esta tesis es Compact-Flye una herramienta para el ensamblaje eficiente de lecturas de tercera generación sobre el algoritmo Flye. Esta herramienta está basada en el uso igenioso de estructuras compactas de datos para mejorar etapas típicas del ensamblaje como el conteo e indexación de k-mers. Al margen del ensamblaje de un genoma existen técnicas que buscan ensamblar todos los genomas contenidos en una muestra determinada. Este ensamblaje es conocido como ensamblaje múltiple de secuencias o reconstrucción de haplotipos, tema también tratado en esta tesis. Nuestra primera aproximación para la resolución de este ha sido viaDBG, que es la primera solución basada en grafos de de Bruijn que ofrece resultados comparables a las técnicas vigentes en ensamblaje de genomas víricos, mientras que mantiene la eficiencia de estos grafos. Nuestra segunda aportación es ViQUF, que es una mejora natural de su predecesor. ViQUF cambia totalmente la algoritmia de viaDBG, pero sigue cimentándose en las mismas estructuras aunque con alguna variación que le permite no solo mejorar resultados en tiempo y calidad. Sino que además le permite aportar más información como estimaciones relativa de cada especie en la muestra.[Resumo] Durante as dúas últimas décadas, grazas ao desenvolvemento de novas técnicas de secuenciación, o estudo do xenoma fíxose moi popular para descubrir a variación xenética presente tanto nos humanos como noutros organismos. O modo predominante de análise do xenoma é a través da ensamblaxe de lecturas nunha ou varias cadeas o maior tempo posible. A forma máis tradicional de ensamblar é a que implica lecturas dun só xenoma. Neste campo, na última década xurdiron lecturas de terceira xeración con novos retos para os que non existen solucións eficientes. A primeira contribución que se fixo nesta tese é Compact-Flye, unha ferramenta para a montaxe eficiente de lecturas de terceira xeración sobre o algoritmo Flye. Esta ferramenta baséase no uso intelixente de estruturas de datos compactas para mellorar os pasos típicos de montaxe, como contar e indexar k-mers. Ademais da montaxe dun xenoma, existen técnicas que buscan ensamblar todos os xenomas contidos nunha determinada mostra. Este conxunto coñécese como conxunto de secuencias múltiples ou reconstrución de haplotipos, tema tamén tratado nesta tesis. O noso primeiro enfoque para resolver isto foi viaDBG, que é a primeira solución baseada en gráficos de Bruijn que ofrece resultados comparables ás técnicas actuais de ensamblaxe de xenoma viral, mantendo a eficiencia destes gráficos. A nosa segunda incorporación é ViQUF, que é unha mellora natural con respecto ao seu predecesor. ViQUF cambia completamente o algoritmo de viaDBG pero segue baseándose nas mesmas estruturas, aínda que con algunha variación que lle permite non só mellorar os resultados en tempo e calidade. Pero tamén permite achegar máis información como estimacións relativas de cada especie da mostra.Xunta de Galicia; ED431G 2019/01Xunta de Galicia; ED431C 2021/53Xunta de Galicia; IG240.2020.1.185Xunta de Galicia; IN852A 2018/14Quiero agradecer al Centro de Investigación de Galicia “CITIC”, financiado por la Xunta de Galicia y la Unión Europea (European Regional Development Fund- Galicia 2014-2020 Program), con la beca ED431G 2019/01. También agradecer a la Xunta de Galicia/FEDER-UE que ha financiado esta tesis a través de las becas [ED431C 2021/53; IG240.2020.1.185; IN852A 2018/14]; al Ministerio de Ciencia e Innovación con las becas [TIN2016- 78011-C4-1-R; FPU17/02742; PID2019-105221RB-C41; PID2020-114635RB-I00]; y a la academia de Finlandia [grants 308030 and 323233 (LS)]

LIPIcs, Volume 244, ESA 2022, Complete Volume

LIPIcs, Volume 244, ESA 2022, Complete Volum