Parameterized DAWGs: Efficient constructions and bidirectional pattern searches

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