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. While such paths constitute only partial assemblies, they are likely to be correct. More precisely, 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. Until recently, it was open what are all the safe walks of an assembly graph. Tomescu and Medvedev (RECOMB 2016) characterized all such safe walks (omnitigs), thus giving the first safe and complete 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.

We answer this question affirmatively, by describing 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 Θ (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. This has 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.

基因组组装要求从未知字符串的许多较短子字符串中重建它。尽管它是生物信息学的关键问题之一，但它普遍缺乏重大的理论进展。它的难度既源于实际问题（真实数据的大小和错误），也源于问题表述本质上允许多种解决方案这一事实。鉴于这些，大多数最先进的汇编器的核心都是基于在汇编图中查找非分支路径（ unitigs ）。虽然此类路径仅构成部分程序集，但它们很可能是正确的。更准确地说，如果将基因组组装解决方案定义为图形的封闭弧覆盖步行，则单元出现在所有解决方案中，因此是安全的部分解决方案。直到最近，什么是装配图的所有安全路径仍然是开放的。Tomescu 和 Medvedev（RECOMB 2016）描述了所有此类安全行走（omnitigs）的特征，从而给出了第一个安全且完整的基因组组装算法。尽管最大全向发现后来被 Cairo 等人改进为二次时间。（ACM Trans. Algorithms 2019），寻找unitigs的关键线性时间特征是否可以通过omnitigs实现仍然悬而未决。

我们肯定地回答了这个问题，通过描述一个令人惊讶的O(m)时间算法来识别具有n 个节点和m条弧的图的所有最大全向图，尽管存在具有θ (mn)总最大全向图尺寸的图族。这是基于对一系列行走 ( macrotigs ) 的发现，其属性是所有非平凡的 omnitigs 都是 Macrotig 的子行走的明确延伸。这有两个结果：(1)枚举所有最大全向的线性时间输出敏感算法。(2)所有最大全能的紧凑O(m)表示，例如允许对它们进行O(m)时间计算的各种统计数据。我们的结果解决了一个长期存在的理论问题，该问题受到实用基因组组装程序的启发，起源于 1995 年使用的unitigs。我们设想我们的结果将成为从理论到实际和完整的基因组组装程序反向转移的核心，正如其他关键生物信息学问题的情况也是如此。