输出敏感空间中反字典的构造
Constructing Antidictionaries in Output-Sensitive Space
摘要：
如果一个词$x$的所有固有因子都出现在$y$中，则该词在一个词$y$中不存在，则该词被称为最小的。给定一个集合的$k$字$y_1,y_2,\ldots,y_k$在一个字母表$Σ$，我们被要求计算的集合$\mathrm{M}^{\ell}_{y_{1}\#\ldots\#y_{k}}$的最小缺席的字的长度最多$\ell$的字$y=y_1\#y_2\#\ldots\#y_k$，$\#\notinΣ$。在数据压缩中，这相当于计算$k$文档的反字典。在生物信息学中，它对应于计算$k$染色体基因组中不存在的单词。这种计算通常需要$Ω(n)$空间，用于使用任何大量可用的$\mathcal{O}(n)$时间算法的$n=|y|$。这是因为$Ω(n)$大小的文本索引是在$y$上构造的，这对于大型$n$来说是不切实际的。我们使用输出敏感空间增量地进行相同的计算。当$||\mathrm{M}^{\ell}_{y_{1}\#\ldots\#y_{N}}||=o(n)$用于所有$N\in[1,k]$时，该目标是合理的。例如，在人类基因组中，$n \approx 3\times 10^9$而不是$||\mathrm{M}^{12}_{y_{1}\#\ldots\#y_{k}}|| \approx 10^6$。我们考虑一个常数大小的字母表来说明我们的结果。我们表明，所有的$\mathrm{M}^{\ell}_{y_{1}},\ldots,\mathrm{M}^{\ell}_{y_{1}\#\ldots\#y_{k}}$可以计算在$\mathcal{O}(kn+\sum^{k}_{N=1}||\mathrm{M}^{\ell}_{y_{1}\#\ldots\#y_{N}}||)$总时间使用$\mathcal{O}(\mathrm{MaxIn}+\mathrm{MaxOut})$空间，其中$\mathrm{MaxIn}$是最长的词在$\{y_1,\ldots,y_{k}\}$和$\mathrm{MaxOut}=\max\{||\mathrm{M}^{\ell}_{y_{1}\#\ldots\#y_{N}}||:N\in[1,k]\}$的长度。概念验证的实验结果也证实了我们的理论研究结果和证明我们的贡献。
英文摘要：
A word $x$ that is absent from a word $y$ is called minimal if all its proper factors occur in $y$. Given a collection of $k$ words $y_1,y_2,\ldots,y_k$ over an alphabet $Σ$, we are asked to compute the set $\mathrm{M}^{\ell}_{y_{1}\#\ldots\#y_{k}}$ of minimal absent words of length at most $\ell$ of word $y=y_1\#y_2\#\ldots\#y_k$, $\#\notinΣ$. In data compression, this corresponds to computing the antidictionary of $k$ documents. In bioinformatics, it corresponds to computing words that are absent from a genome of $k$ chromosomes. This computation generally requires $Ω(n)$ space for $n=|y|$ using any of the plenty available $\mathcal{O}(n)$-time algorithms. This is because an $Ω(n)$-sized text index is constructed over $y$ which can be impractical for large $n$. We do the identical computation incrementally using output-sensitive space. This goal is reasonable when $||\mathrm{M}^{\ell}_{y_{1}\#\ldots\#y_{N}}||=o(n)$, for all $N\in[1,k]$. For instance, in the human genome, $n \approx 3\times 10^9$ but $||\mathrm{M}^{12}_{y_{1}\#\ldots\#y_{k}}|| \approx 10^6$. We consider a constant-sized alphabet for stating our results. We show that all $\mathrm{M}^{\ell}_{y_{1}},\ldots,\mathrm{M}^{\ell}_{y_{1}\#\ldots\#y_{k}}$ can be computed in $\mathcal{O}(kn+\sum^{k}_{N=1}||\mathrm{M}^{\ell}_{y_{1}\#\ldots\#y_{N}}||)$ total time using $\mathcal{O}(\mathrm{MaxIn}+\mathrm{MaxOut})$ space, where $\mathrm{MaxIn}$ is the length of the longest word in $\{y_1,\ldots,y_{k}\}$ and $\mathrm{MaxOut}=\max\{||\mathrm{M}^{\ell}_{y_{1}\#\ldots\#y_{N}}||:N\in[1,k]\}$. Proof-of-concept experimental results are also provided confirming our theoretical findings and justifying our contribution.
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