Summary: Given \(d\) strings over the alphabet \(\{0,1,\dots,\sigma-1\}\), the classical Aho-Corasick data structure allows us to find all occ occurrences of the strings in any text \(T\) in \(O(|T|+\text{occ})\) time using \(O(m\log m)\) bits of space, where \(m\) is the number of edges in the trie containing the strings. Fix any constant \(\varepsilon\in(0,2)\). We describe a compressed solution for the problem that, provided \(\sigma\le m^\delta\) for a constant \(\delta <1\), works in \(O(|T|\frac{1}{\varepsilon}\log\frac{1}{\varepsilon})+\text{occ}\) time, which is \(O(|T|+\text{occ})\) since \(\varepsilon\) is constant, and occupies \(mH_k+ 1.443m+ \varepsilon m+ O(d\log\frac{m}{d})\) bits of space, for all \(0\le k\le\max\{0,\alpha\log_\sigma m-2\}\) simultaneously, where \(\alpha\in(0,1)\) is an arbitrary constant and \(H_k\) is the \(k\)th-order empirical entropy of the trie. Hence, we reduce the \(3.443m\) term in the space bounds of previously best succinct solutions to \((1.443+\varepsilon)m\), thus solving an open problem posed by D. Belazzougui [Lect. Notes Comput. Sci. 6129, 88–100 (2010; Zbl 1286.68521)]. Further, we notice that \(L= \log\binom{\sigma(m+1)}{m} - O(\log(\sigma m))\) is a worst-case space lower bound for any solution of the problem and, for \(d= o(m)\) and constant \(\varepsilon\), our approach allows to achieve \(L+ \varepsilon m\) bits of space, which gives an evidence that, for \(d= o(m)\), the space of our data structure is theoretically optimal up to the \(\varepsilon m\) additive term and it is hardly possible to eliminate the term \(1.443m\). In addition, we refine the space analysis of previous works by proposing a more appropriate definition for \(H_k\). We also simplify the construction for practice adapting the fixed block compression boosting technique, then implement our data structure, and conduct a number of experiments showing that it is comparable to the state of the art in terms of time and is superior in space.
For the entire collection see [Zbl 1414.68007].