Efficient Dynamic Dictionary Matching with DAWGs and AC-automata

The dictionary matching is a task to find all occurrences of patterns in a set $D$ (called a dictionary) on a text $T$. The Aho-Corasick-automaton (AC-automaton) is a data structure which enables us to solve the dictionary matching problem in $O(d\log \sigma )$ preprocessing time and $O(n\log \sigma +occ)$ matching time, where $d$ is the total length of the patterns in $D$, $n$ is the length of the text, $\sigma$ is the alphabet size, and $occ$ is the total number of occurrences of all the patterns in the text. The dynamic dictionary matching is a variant where patterns may dynamically be inserted into and deleted from $D$. This problem is called semi-dynamic dictionary matching if only insertions are allowed. In this paper, we propose two efficient algorithms. For a pattern of length $m$, our first algorithm supports insertions in $O(m\log \sigma +\log d\mathrm{/}\log \log d)$ time and pattern matching in $O(n\log \sigma +occ)$ time for the semi-dynamic setting and supports both insertions and deletions in $O(\sigma m+\log d\mathrm{/}\log \log d)$ time and pattern matching in $O(n(\log d\mathrm{/}\log \log d+\log \sigma )+occ(\log d\mathrm{/}\log \log d))$ time for the dynamic setting by some modifications. This algorithm is based on the directed acyclic word graph. Our second algorithm, which is based on the AC-automaton, supports insertions in $O(m\log \sigma +u_f+u_o)$ time for the semi-dynamic setting and supports both insertions and deletions in $O(\sigma m+u_f+u_o)$timeforthedynamicsetting,where$u_f$and$u_o$ respectively denote the numbers of states in which the failure function and the output function need to be updated. This algorithm performs pattern matching in $O(n\log \sigma +occ)$ time for both settings. Our algorithm achieves optimal update time for AC-automaton based methods over constant-size alphabets, since any algorithm which explicitly maintains the AC-automaton requires $\mathrm{\Omega }(m+u_f+u_o)$ update time.