通过统一的SPQR树框架在线性时间内识别气泡状子图
Identifying bubble-like subgraphs in linear-time via a unified SPQR-tree framework
AI总结：
本文提出首个线性时间算法识别所有snarls和ultrabubbles，同时利用SPQR树分解和反馈弧问题，为计算生物学提供了新的理论基础。
摘要：
计算生物学中的基本算法问题是在给定有向或双向输入图中找到所有类型（superbubbles, snarls, ultrabubbles）的子图。这些对应于遗传变异的区域，在分析基因组集合时很有用。本文首次提出了线性时间算法，用于识别所有snarls和所有ultrabubbles，解决了自2018年以来未解决的问题。snarls的算法依赖于一种新的线性大小表示，以顶点数量为基准。我们利用众所周知的SPQR树分解，该分解编码了双连通图的所有2分离器。经过几次动态规划风格的树遍历后，我们维护关键属性（如无环性），以决定给定的2分离器是否定义需要报告的子图。线性时间复杂度的关键成分是，通过计算所有弧的移除使图无环（称为反馈弧）的问题，可以同时测试线性数量的子图的无环性。因此，我们证明了对于双向图的一个基本结果，可能对独立研究感兴趣：所有反馈弧可以在线性时间内计算，而对于一般情况，这至少与矩阵乘法一样难，假设k-Clique猜想成立。我们的结果形成一个统一的框架，还提供了完全不同的线性时间算法来寻找所有superbubbles。尽管一些结果技术上复杂，但基本思想是概念上简单的，可能扩展到其他气泡状子图。更广泛地说，我们的工作为计算生物学的理论基础做出了贡献，并推动了使用SPQR树分解作为通用工具设计高效算法的研究，超越了其传统角色在图绘制中的应用。
英文摘要：
A fundamental algorithmic problem in computational biology is to find all subgraphs of a given type (superbubbles, snarls, and ultrabubbles) in a directed or bidirected input graph. These correspond to regions of genetic variation and are useful in analyzing collections of genomes. We present the first linear-time algorithms for identifying all snarls and all ultrabubbles, resolving problems open since 2018. The algorithm for snarls relies on a new linear-size representation of all snarls with respect to the number of vertices in the graph. We employ the well-known SPQR-tree decomposition, which encodes all 2-separators of a biconnected graph. After several dynamic-programming-style traversals of this tree, we maintain key properties (such as acyclicity) that allow us to decide whether a given 2-separator defines a subgraph to be reported. A crucial ingredient for linear-time complexity is that acyclicity of linearly many subgraphs can be tested simultaneously via the problem of computing all arcs in a directed graph whose removal renders it acyclic (so-called feedback arcs). As such, we prove a fundamental result for bidirected graphs, that may be of independent interest: all feedback arcs can be computed in linear time for tipless bidirected graphs, while in general this is at least as hard as matrix multiplication, assuming the k-Clique Conjecture. Our results form a unified framework that also yields a completely different linear-time algorithm for finding all superbubbles. Although some of the results are technically involved, the underlying ideas are conceptually simple, and may extend to other bubble-like subgraphs. More broadly, our work contributes to the theoretical foundations of computational biology and advances a growing line of research that uses SPQR-tree decompositions as a general tool for designing efficient algorithms, beyond their traditional role in graph drawing.
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