Parameterized DAWGs: efficient constructions and bidirectional pattern searches

Abstract

Two strings $x$ and $y$ over $\Sigma \cup \Pi$ of equal length are said to \emph{parameterized match} (\emph{p-match}) if there is a renaming bijection $f:\Sigma \cup \Pi \to \Sigma \cup \Pi$ that is identity on $\Sigma$ and transforms $x$ to $y$ (or vice versa). The \emph{p-matching} problem is to look for substrings in a text that p-match a given pattern. In this paper, we propose \emph{parameterized suffix automata} (\emph{p-suffix automata}) and \emph{parameterized directed acyclic word graphs} (\emph{PDAWGs}) which are the p-matching versions of suffix automata and DAWGs. While suffix automata and DAWGs are equivalent for standard strings, we show that p-suffix automata can have $\Theta (n^2)$ nodes and edges but PDAWGs have only $O(n)$ nodes and edges, where $n$ is the length of an input string. We also give an $O(n|\Pi |\log (|\Pi |+|\Sigma |))$-time $O(n)$-space algorithm that builds the PDAWG in a left-to-right online manner. As a byproduct, it is shown that the \emph{parameterized suffix tree} for the reversed string can also be built in the same time and space, in a right-to-left online manner. This duality also leads us to two further efficient algorithms for p-matching: Given the parameterized suffix tree for the reversal of the input string $T$, one can build the PDAWG of $T$ in $O(n)$ time in an offline manner; One can perform \emph{bidirectional} p-matching in $O(m\log (|\Pi |+|\Sigma |)+occ)$ time using $O(n)$ space, where $m$ denotes the pattern length and $occ$ is the number of pattern occurrences in the text $T$.

28 pages, 7 figures

Cited by in corpus (1)

• The Parameterized Suffix Tray