漫长的路径使得模式计数变得困难,而茂密的树木使其更加困难

V'it Jel'inek, Michal Opler, Jakub Pek'arek
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引用次数: 0

摘要

我们研究称为#PPM的计数问题,其输入是一对排列$\pi$和$\tau$(分别称为模式和文本),任务是找到$\tau$与$\pi$具有相同相对顺序的子序列的个数。对于长度为$k$的模式和长度为$n$的文本,一个简单的蛮力方法在时间$O(n^{k+1})$上解决了#PPM,而Berendsohn、Kozma和Marx最近表明,在指数时间假设(ETH)下,对于任何函数$f$,它都不能在时间$f(k) n^{o(k/\log k)}$上解决。在本文中,我们考虑了#PPM的限制,称为$\mathcal{C}$ -模式#PPM,其中模式$\pi$必须属于遗传排列类$\mathcal{C}$。我们的目标是确定$\mathcal{C}$的结构属性,这些属性决定了$\mathcal{C}$ -Pattern #PPM的复杂性。我们主要关注两种结构性质,即长路径性质(LPP)和深树性质(DTP)。假设ETH,我们得到这些结果:1。如果$C$有LPP,则$\mathcal{C}$ -Pattern #PPM无法及时求解$f(k)n^{o(\sqrt{k})}$对于任何函数$f$;如果$C$有DTP,那么$\mathcal{C}$ -模式#PPM无法及时解决$f(k)n^{o(k/\log^2 k)}$对于任何函数$f$。此外,当$\mathcal{C}$是所谓的单调网格类之一时,我们表明,如果$\mathcal{C}$有LPP而没有DTP,则$\mathcal{C}$ -Pattern #PPM可以及时求解$f(k)n^{O(\sqrt k)}$。特别地,上面的下界紧挨着指数中的多对数项。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Long paths make pattern-counting hard, and deep trees make it harder
We study the counting problem known as #PPM, whose input is a pair of permutations $\pi$ and $\tau$ (called pattern and text, respectively), and the task is to find the number of subsequences of $\tau$ that have the same relative order as $\pi$. A simple brute-force approach solves #PPM for a pattern of length $k$ and a text of length $n$ in time $O(n^{k+1})$, while Berendsohn, Kozma and Marx have recently shown that under the exponential time hypothesis (ETH), it cannot be solved in time $f(k) n^{o(k/\log k)}$ for any function $f$. In this paper, we consider the restriction of #PPM, known as $\mathcal{C}$-Pattern #PPM, where the pattern $\pi$ must belong to a hereditary permutation class $\mathcal{C}$. Our goal is to identify the structural properties of $\mathcal{C}$ that determine the complexity of $\mathcal{C}$-Pattern #PPM. We focus on two such structural properties, known as the long path property (LPP) and the deep tree property (DTP). Assuming ETH, we obtain these results: 1. If $C$ has the LPP, then $\mathcal{C}$-Pattern #PPM cannot be solved in time $f(k)n^{o(\sqrt{k})}$ for any function $f$, and 2. if $C$ has the DTP, then $\mathcal{C}$-Pattern #PPM cannot be solved in time $f(k)n^{o(k/\log^2 k)}$ for any function $f$. Furthermore, when $\mathcal{C}$ is one of the so-called monotone grid classes, we show that if $\mathcal{C}$ has the LPP but not the DTP, then $\mathcal{C}$-Pattern #PPM can be solved in time $f(k)n^{O(\sqrt k)}$. In particular, the lower bounds above are tight up to the polylog terms in the exponents.
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