Episode Matching的细粒度复杂度
The Fine-Grained Complexity of Episode Matching
AI总结：
本文研究了Episode Matching问题的细粒度复杂度，证明了除非SETH为假否则不存在$O((nm)^{1-ε})$算法，并为该问题的索引版本提出了近乎最优的时空权衡数据结构。
摘要：
给定两个字符串$S$和$P$，Episode Matching问题是找到$S$中包含$P$作为子序列的最短子串。该问题已知的最佳上界是由Das等人（1997年）提出的$\tilde O(nm)$，其中$n$和$m$分别是$S$和$P$的长度。尽管该问题已被充分研究并在数据挖掘中有许多应用，但这一上界从未被改进。在本文中，我们通过证明不存在$O((nm)^{1-ε})$算法（即使是对于二进制字符串），除非强指数时间假设（SETH）为假，来解释了这种情况的原因。然后，我们考虑了该问题的索引版本，其中$S$被预处理为一个数据结构，用于回答Episode Matching查询$P$。我们证明，对于任何$τ$，存在一个使用$O(n+\left(\frac{n}τ\right)^k)$空间的数据结构，可以在$O(k\cdot τ\cdot \log \log n)$时间内回答任意长度为$k$的$P$的Episode Matching查询。我们用一个几乎匹配的下界来补充这个上界，证明任何在$O(n^δ)$时间内回答长度为$k$的模式Episode Matching查询的数据结构，必须使用$Ω(n^{k-kδ-o(1)})$空间，除非强$k$-Set Disjointness猜想为假。最后，对于$k=2$的特殊情况，我们提出了一种使用有界整数矩阵的快速min-plus乘法的更快数据结构构造方法。
英文摘要：
Given two strings $S$ and $P$, the Episode Matching problem is to find the shortest substring of $S$ that contains $P$ as a subsequence. The best known upper bound for this problem is $\tilde O(nm)$ by Das et al. (1997) , where $n,m$ are the lengths of $S$ and $P$, respectively. Although the problem is well studied and has many applications in data mining, this bound has never been improved. In this paper we show why this is the case by proving that no $O((nm)^{1-ε})$ algorithm (even for binary strings) exists, unless the Strong Exponential Time Hypothesis (SETH) is false. We then consider the indexing version of the problem, where $S$ is preprocessed into a data structure for answering episode matching queries $P$. We show that for any $τ$, there is a data structure using $O(n+\left(\frac{n}τ\right)^k)$ space that answers episode matching queries for any $P$ of length $k$ in $O(k\cdot τ\cdot \log \log n )$ time. We complement this upper bound with an almost matching lower bound, showing that any data structure that answers episode matching queries for patterns of length $k$ in time $O(n^δ)$, must use $Ω(n^{k-kδ-o(1)})$ space, unless the Strong $k$-Set Disjointness Conjecture is false. Finally, for the special case of $k=2$, we present a faster construction of the data structure using fast min-plus multiplication of bounded integer matrices.
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