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arXiv 2007.08357cs.DS
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亚线性空间中的子串复杂度

Substring Complexity in Sublinear Space

Giulia Bernardini, Gabriele Fici, Paweł Gawrychowski, Solon P. Pissis

备注：
Accepted to ISAAC 2023. Abstract abridged to satisfy arXiv requirements
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AI总结：

本文针对重复字符串的可压缩性度量$\delta$，提出在亚线性空间中高效计算$\delta$的算法，分别给出比较模型和字RAM模型下的时间-空间权衡界限。

摘要：

Shannon熵是统计压缩的确定性下界。遗憾的是，对于重复字符串的可压缩性，并不存在这样明确的度量。因此，人们采用临时度量来估计字符串的重复性，例如Lempel-Ziv解析的大小$z$或Burrows-Wheeler变换的等字母游程数$r$。一个较新的度量是最小字符串吸引子的大小$\gamma$。设$T$为长度为$n$的字符串。$T$的字符串吸引子是$T$的一组位置，这些位置捕获了$T$的所有子串的出现。不幸的是，Kempa和Prezza [STOC 2018]表明计算$\gamma$是NP困难的。Kociumaka等人 [LATIN 2020]考虑了一种新的可压缩性度量，该度量基于函数$S_T(k)$，它计算$T$的长度为$k$的不同子串的数量，也称为$T$的子串复杂度。这个新度量定义为$\delta= \sup\{S_T(k)/k, k\geq 1\}$，并且是之前考虑的所有相关临时度量的下界。特别是，$\delta\leq \gamma$总是成立，并且$\delta$可以在$\mathcal{O}(n)$时间内使用$\Theta(n)$工作空间计算。Kociumaka等人表明，可以构造一个大小为$\mathcal{O}(\delta\log \frac{n}{\delta})$的$T$的表示，支持对$T$的高效直接访问和高效模式匹配查询。鉴于对于高度可压缩的字符串，$\delta$显著小于$n$，自然提出以下问题：我们能否使用亚线性工作空间高效计算$\delta$？我们通过展示以下计算$\delta$的界限来解决这一算法挑战：在比较模型中，对于任何$b\in[1,n]$，使用$\mathcal{O}(b)$空间，时间为$\mathcal{O}(\frac{n^3\log b}{b^2})$；或在字RAM模型中，对于任何$b\in[\sqrt{n},n]$，使用$\tilde{\mathcal{O}}(b)$空间，时间为$\tilde{\mathcal{O}}(n^2/b)$。

英文摘要：

Shannon's entropy is a definitive lower bound for statistical compression. Unfortunately, no such clear measure exists for the compressibility of repetitive strings. Thus, ad hoc measures are employed to estimate the repetitiveness of strings, e.g., the size $z$ of the Lempel-Ziv parse or the number $r$ of equal-letter runs of the Burrows-Wheeler transform. A more recent one is the size $γ$ of a smallest string attractor. Let $T$ be a string of length $n$. A string attractor of $T$ is a set of positions of $T$ capturing the occurrences of all the substrings of $T$. Unfortunately, Kempa and Prezza [STOC 2018] showed that computing $γ$ is NP-hard. Kociumaka et al. [LATIN 2020] considered a new measure of compressibility that is based on the function $S_T(k)$ counting the number of distinct substrings of length $k$ of $T$, also known as the substring complexity of $T$. This new measure is defined as $δ= \sup\{S_T(k)/k, k\geq 1\}$ and lower bounds all the relevant ad hoc measures previously considered. In particular, $δ\leq γ$ always holds and $δ$ can be computed in $\mathcal{O}(n)$ time using $Θ(n)$ working space. Kociumaka et al. showed that one can construct an $\mathcal{O}(δ\log \frac{n}δ)$-sized representation of $T$ supporting efficient direct access and efficient pattern matching queries on $T$. Given that for highly compressible strings, $δ$ is significantly smaller than $n$, it is natural to pose the following question: Can we compute $δ$ efficiently using sublinear working space? We address this algorithmic challenge by showing the following bounds to compute $δ$: $\mathcal{O}(\frac{n^3\log b}{b^2})$ time using $\mathcal{O}(b)$ space, for any $b\in[1,n]$, in the comparison model; or $\tilde{\mathcal{O}}(n^2/b)$ time using $\tilde{\mathcal{O}}(b)$ space, for any $b\in[\sqrt{n},n]$, in the word RAM model.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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