Genome Assembly, from Practice to Theory: Safe, Complete and Linear-Time

Genome assembly asks to reconstruct an unknown string from many shorter substrings of it. Even though it is one of the key problems in Bioinformatics, it is generally lacking major theoretical advances. Its hardness stems both from practical issues (size and errors of real data), and from the fact that problem formulations inherently admit multiple solutions. Given these, at their core, most state-of-the-art assemblers are based on finding non-branching paths (unitigs) in an assembly graph. If one defines a genome assembly solution as a closed arc-covering walk of the graph, then unitigs appear in all solutions, being thus safe partial solutions. All all such safe walks were recently characterized as omnitigs, leading to the first safe and complete genome assembly algorithm. Even if omnitig finding was improved to quadratic time, it remained open whether the crucial linear-time feature of finding unitigs can be attained with omnitigs. We describe a surprising $O(m)$-time algorithm to identify all maximal omnitigs of a graph with $n$ nodes and $m$ arcs, notwithstanding the existence of families of graphs with $\Theta(mn)$ total maximal omnitig size. This is based on the discovery of a family of walks (macrotigs) with the property that all the non-trivial omnitigs are univocal extensions of subwalks of a macrotig, with two consequences: (1) A linear-time output-sensitive algorithm enumerating all maximal omnitigs. (2) A compact $O(m)$ representation of all maximal omnitigs, which allows, e.g., for $O(m)$-time computation of various statistics on them. Our results close a long-standing theoretical question inspired by practical genome assemblers, originating with the use of unitigs in 1995. We envision our results to be at the core of a reverse transfer from theory to practical and complete genome assembly programs, as has been the case for other key Bioinformatics problems.