图中精确模式匹配的复杂性：确定性与Z形匹配
On the Complexity of Exact Pattern Matching in Graphs: Determinism and Zig-Zag Matching
摘要：
标记图中的精确模式匹配是在图$G=(V,E)$中寻找与给定模式$P[1..m]$拼写相同字符串的路径的问题。这个基本问题可以在计算生物学中变异图上更复杂的操作、图数据库中的查询操作和异构网络分析中找到，其中某些路径的节点必须匹配一系列标签或类型。在我们最近的工作中，我们描述了一个条件下界，指出在标记图中的精确模式匹配问题不能在小于二次时间内解决，即，对于任何常数$ε>0$，$O(|E|^{1 - ε} \, m)$时间或$O(|E| \, m^{1 - ε})$时间，除非强指数时间假设（SETH）是假的。即使节点标签和模式$P$是从一个二进制字母表，和$G$被限制为无向图的最大程度的3或有向无环图的最大总和的入度和出度3。它是左开放发生在无向图的最大程度2，即，当图案可以在（循环的）双向串中具有Z字形匹配时。此外，约简创建了一个非确定性的有向无环图，如果确定性会使问题更容易解决，则它是开放的。在这项工作中，我们通过正交向量假设（OV）表明，即使在这些限制的情况下，相同的条件下界也成立。
英文摘要：
Exact pattern matching in labeled graphs is the problem of searching paths of a graph $G=(V,E)$ that spell the same string as the given pattern $P[1..m]$. This basic problem can be found at the heart of more complex operations on variation graphs in computational biology, query operations in graph databases, and analysis of heterogeneous networks, where the nodes of some paths must match a sequence of labels or types. In our recent work we described a conditional lower bound stating that the exact pattern matching problem in labeled graphs cannot be solved in less than quadratic time, namely, $O(|E|^{1 - ε} \, m)$ time or $O(|E| \, m^{1 - ε})$ time for any constant $ε>0$, unless the Strong Exponential Time Hypothesis (SETH) is false. The result holds even if node labels and pattern $P$ are drawn from a binary alphabet, and $G$ is restricted to undirected graphs of maximum degree three or directed acyclic graphs of maximum sum of indegree and outdegree three. It was left open what happens on undirected graphs of maximum degree two, i.e., when the pattern can have a zig-zag match in a (cyclic) bidirectional string. Also, the reduction created a non-determistic directed acyclic graph, and it was left open if determinism would make the problem easier. In this work, we show through the Orthogonal Vectors hypothesis (OV) that the same conditional lower bound holds even for these restricted cases.
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