Description:
<jats:p>
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 (
<jats:italic>unitigs</jats:italic>
) in an assembly graph. While such paths constitute only partial assemblies, they are likely to be correct. More precisely, if one defines a genome assembly solution as a
<jats:italic>closed arc-covering walk</jats:italic>
of the graph, then unitigs appear in all solutions, being thus
<jats:italic>safe</jats:italic>
partial solutions. Until recently, it was open what are
<jats:italic>all</jats:italic>
the safe walks of an assembly graph. Tomescu and Medvedev (RECOMB 2016) characterized all such safe walks (
<jats:italic>omnitigs</jats:italic>
), thus giving the first safe and
<jats:italic>complete</jats:italic>
genome assembly algorithm. Even though maximal omnitig finding was later improved to quadratic time by Cairo et al. (ACM Trans. Algorithms 2019), it remained open whether the crucial linear-time feature of finding unitigs can be attained with omnitigs.
</jats:p>
<jats:p>
We answer this question affirmatively, by describing a surprising
<jats:italic>O(m)</jats:italic>
-time algorithm to
<jats:italic>identify</jats:italic>
all maximal omnitigs of a graph with
<jats:italic>n</jats:italic>
nodes and
<jats:italic>m</jats:italic>
arcs, notwithstanding the existence of families of graphs with
<jats:italic>Θ (mn)</jats:italic>
total maximal omnitig size. This is based on the discovery of a family of walks (
<jats:italic>macrotigs</jats:italic>
) with the property that all the non-trivial omnitigs are univocal extensions of subwalks of a macrotig. This has two consequences: (1) A
<jats:italic>linear-time output-sensitive</jats:italic>
algorithm enumerating all maximal omnitigs. (2) A
<jats:italic>
compact
<jats:italic>O(m)</jats:italic>
representation
</jats:italic>
of all maximal omnitigs, which allows, e.g., for
<jats:italic>O(m)</jats:italic>
-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
<jats:italic>complete</jats:italic>
genome assembly programs, as has been the case for other key Bioinformatics problems.
</jats:p>