Two strings x and y over \Sigma \cup \Pi of equal length are said to parameterized match (p-match) if there is a renaming bijection f:\Sigma \cup \Pi \rightarrow \Sigma \cup \Pi that is identity on \Sigma and transforms x to y (or vice versa). The p-matching problem is to look for substrings in a text that p-match a given pattern. In this paper, we propose parameterized suffix automata (p-suffix automata) and parameterized directed acyclic word graphs (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 O(n |\Pi| \log (|\Pi| + |\Sigma|))-time O(n)-space algorithm that builds the PDAWG in a left-to-right online manner. We then show that an implicit representation for the PDAWG can be built in O(n \log (|\Pi| + |\Sigma|)) time and O(n) space from left to right. As a byproduct, it is shown that the parameterized suffix tree for the reversed string can also be built in the same time and space, in a right-to-left online manner. We also discuss parameterized compact DAWGs.
Two strings x and y over \Sigma \cup \Pi of equal length are said to parameterized match (p-match) if there is a renaming bijection f:\Sigma \cup \Pi \rightarrow \Sigma \cup \Pi that is identity on \Sigma and transforms x to y (or vice versa). The p-matching problem is to look for substrings in a text that p-match a given pattern. In this paper, we propose parameterized suffix automata (p-suffix automata) and parameterized directed acyclic word graphs (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 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 parameterized suffix tree for the reversed string can also be built in the same time and space, in a right-to-left online manner. We also discuss parameterized compact DAWGs.
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We then show that an implicit representation for the PDAWG can be built in O(n \log (|\Pi| + |\Sigma|)) time and O(n) space from left to right.