Safe and complete contig assembly via omnitigs

Contig assembly is the first stage that most assemblers solve when reconstructing a genome from a set of reads. Its output consists of contigs -- a set of strings that are promised to appear in any genome that could have generated the reads. From the introduction of contigs 20 years ago, assemblers have tried to obtain longer and longer contigs, but the following question was never solved: given a genome graph $G$ (e.g. a de Bruijn, or a string graph), what are all the strings that can be safely reported from $G$ as contigs? In this paper we finally answer this question, and also give a polynomial time algorithm to find them. Our experiments show that these strings, which we call omnitigs, are 66% to 82% longer on average than the popular unitigs, and 29% of dbSNP locations have more neighbors in omnitigs than in unitigs.Comment: Full version of the paper in the proceedings of RECOMB 201

Safety in s-t Paths, Trails and Walks

Given a directed graph G and a pair of nodes s and t, an s-t bridge of G is an edge whose removal breaks all s-t paths of G (and thus appears in all s-t paths). Computing all s-t bridges of G is a basic graph problem, solvable in linear time. In this paper, we consider a natural generalisation of this problem, with the notion of “safety” from bioinformatics. We say that a walk W is safe with respect to a set W' of s-t walks, if W is a subwalk of all walks in W'. We start by considering the maximal safe walks when consists of: all s-t paths, all s-t trails, or all s-t walks of G. We show that the solutions for the first two problems immediately follow from finding all s-t bridges after incorporating simple characterisations. However, solving the third problem requires non-trivial techniques for incorporating its characterisation. In particular, we show that there exists a compact representation computable in linear time, that allows outputting all maximal safe walks in time linear in their length. Our solutions also directly extend to multigraphs, except for the second problem, which requires a more involved approach. We further generalise these problems, by assuming that safety is defined only with respect to a subset of visible edges. Here we prove a dichotomy between the s-t paths and s-t trails cases, and the s-t walks case: the former two are NP-hard, while the latter is solvable with the same complexity as when all edges are visible. We also show that the same complexity results hold for the analogous generalisations of s-t articulation points (nodes appearing in all s-t paths). We thus obtain the best possible results for natural “safety”-generalisations of these two fundamental graph problems. Moreover, our algorithms are simple and do not employ any complex data structures, making them ideal for use in practice.Peer reviewe

An Optimal O(nm) Algorithm for Enumerating All Walks Common to All Closed Edge-covering Walks of a Graph

In this article, we consider the following problem. Given a directed graph G, output all walks of G that are sub-walks of all closed edge-covering walks of G. This problem was first considered by Tomescu and Medvedev (RECOMB 2016), who characterized these walks through the notion of omnitig. Omnitigs were shown to be relevant for the genome assembly problem from bioinformatics, where a genome sequence must be assembled from a set of reads from a sequencing experiment. Tomescu and Medvedev (RECOMB 2016) also proposed an algorithm for listing all maximal omnitigs, by launching an exhaustive visit from every edge. In this article, we prove new insights about the structure of omnitigs and solve several open questions about them. We combine these to achieve an O(nm)-time algorithm for outputting all the maximal omnitigs of a graph (with n nodes and m edges). This is also optimal, as we show families of graphs whose total omnitig length is Omega(nm). We implement this algorithm arid show that it is 9-12 times faster in practice than the one of Tomescu and Medvedev (RECOMB 2016).Peer reviewe