$k$

Toward a Definitive Compressibility Measure for Repetitive Sequences.

IEEE Trans. Inf. Theory(2023)

引用 6|浏览21
暂无评分
摘要
While the $k$ th order empirical entropy is an accepted measure of the compressibility of individual sequences on classical text collections, it is useful only for small values of $k$ and thus fails to capture the compressibility of repetitive sequences. In the absence of an established way of quantifying the latter, ad-hoc measures like the size $z$ of the Lempel–Ziv parse are frequently used to estimate repetitiveness. The size $b \le z$ of the smallest bidirectional macro scheme captures better what can be achieved via copy-paste processes, though it is NP-complete to compute, and it is not monotone upon appending symbols. Recently, a more principled measure, the size $\gamma $ of the smallest string attractor , was introduced. The measure $\gamma \le b$ lower-bounds all the previous relevant ones, while length- $n$ strings can be represented and efficiently indexed within space $O\left({\gamma \log \frac {n}{\gamma }}\right)$ , which also upper-bounds many measures, including $z$ . Although $\gamma $ is arguably a better measure of repetitiveness than $b$ , it is also NP-complete to compute and not monotone, and it is unknown if one can represent all strings in $o(\gamma \log n)$ space. In this paper, we study an even smaller measure, $\delta \le \gamma $ , which can be computed in linear time, is monotone, and allows encoding every string in $O\left({\delta \log \frac {n}{\delta }}\right)$ space because $z = O\left({\delta \log \frac {n}{\delta }}\right)$ . We argue that $\delta $ better captures the compressibility of repetitive strings. Concretely, we show that (1) $\delta $ can be strictly smaller than $\gamma $ , by up to a logarithmic factor; (2) there are string families needing $\Omega \left({\delta \log \frac {n}{\delta }}\right)$ space to be encoded, so this space is optimal for every $n$ and $\delta $ ; (3) one can build run-length context-free grammars of size $O\left({\delta \log \frac {n}{\delta }}\right)$ , whereas the smallest (non-run-length) grammar can be up to $\Theta (\log n/\log \log n)$ times larger; and (4) within $O\left({\delta \log \frac {n}{\delta }}\right)$ space, we can not only represent a string but also offer logarithmic-time access to its symbols, computation of substring fingerprints, and efficient indexed searches for pattern occurrences. We further refine the above results to account for the alphabet size $\sigma $ of the string, showing that $\Theta \left({\delta \log \frac {n\log \sigma }{\delta \log n}}\right)$ space is necessary and sufficient to represent the string and to efficiently support access, fingerprinting, and pattern matching queries.
更多
查看译文
关键词
Data compression,Lempel–Ziv parse,repetitive sequences,string attractors,substring complexity
AI 理解论文
溯源树
样例
生成溯源树,研究论文发展脉络