关于重复序列的连续可压缩性测度
Towards a Definitive Compressibility Measure for Repetitive Sequences
摘要：
不像统计压缩，香农熵是一个明确的下限，没有这样明确的措施存在的重复序列的可压缩性。由于统计熵不捕获重复性，因此经常使用Lempel-Ziv解析的大小$z$等专门测量来估计它。最小双向宏方案的大小$b \le z$更好地捕获通过复制-粘贴过程可以实现的内容，尽管它是NP完全计算的，并且在符号追加时不是单调的。最近，引入了一种更有原则的度量，即最小字符串\emph{attractor}的大小$γ$。度量$γ\le b$下界所有先前的相关度量，但长度为$n$的字符串可以在空间$O(γ\log\frac{n}γ)$内表示和有效索引，这也是大多数度量的上界。虽然$γ$肯定是比$b$更好的重复性度量，但它在计算上也是NP完全的，并且不是单调的，并且不知道是否总是可以在$o(γ\log n)$空间中表示字符串。在本文中，我们研究了一个更小的措施，$δ\le γ$，它可以在线性时间计算，是单调的，并允许编码的每一个字符串在$O(δ\log\frac{n}δ)$空间，因为$z = O(δ\log\frac{n}δ)$。我们表明，$δ$更好地捕捉重复字符串的可压缩性。具体地说，我们证明了（1）$δ$可以严格小于$γ$，最多为对数因子;（2）存在需要$Ω(δ\log\frac{n}δ)$空间进行编码的字符串族，因此该空间对于每个$n$和$δ$都是最优的;（3）可以构建大小为$O(δ\log\frac{n}δ)$的游程长度上下文无关文法，而最小的（非游程长度）语法可以大到$Θ(\log n/\log\log n)$倍;（4）在$O(δ\log\frac{n}δ)$空间内，我们不仅可以.
英文摘要：
Unlike in statistical compression, where Shannon's entropy is a definitive lower bound, no such clear measure exists for the compressibility of repetitive sequences. Since statistical entropy does not capture repetitiveness, ad-hoc measures like the size $z$ of the Lempel--Ziv parse are frequently used to estimate it. 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 monotonic upon symbol appends. Recently, a more principled measure, the size $γ$ of the smallest string \emph{attractor}, was introduced. The measure $γ\le b$ lower bounds all the previous relevant ones, yet length-$n$ strings can be represented and efficiently indexed within space $O(γ\log\frac{n}γ)$, which also upper bounds most measures. While $γ$ is certainly a better measure of repetitiveness than $b$, it is also NP-complete to compute and not monotonic, and it is unknown if one can always represent a string in $o(γ\log n)$ space. In this paper, we study an even smaller measure, $δ\le γ$, which can be computed in linear time, is monotonic, and allows encoding every string in $O(δ\log\frac{n}δ)$ space because $z = O(δ\log\frac{n}δ)$. We show that $δ$ better captures the compressibility of repetitive strings. Concretely, we show that (1) $δ$ can be strictly smaller than $γ$, by up to a logarithmic factor; (2) there are string families needing $Ω(δ\log\frac{n}δ)$ space to be encoded, so this space is optimal for every $n$ and $δ$; (3) one can build run-length context-free grammars of size $O(δ\log\frac{n}δ)$, whereas the smallest (non-run-length) grammar can be up to $Θ(\log n/\log\log n)$ times larger; and (4) within $O(δ\log\frac{n}δ)$ space we can not only...
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。
