Constructing Antidictionaries in Output-Sensitive Space (1902.04785v1)

Abstract: 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 $\Sigma$, 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 $y=y_1#y_2#\ldots#y_k$, $#\notin\Sigma$. 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 $\Omega(n)$ space for $n=|y|$ using any of the plenty available $\mathcal{O}(n)$-time algorithms. This is because an $\Omega(n)$-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.