The Parameterized Suffix Tray

Let $Σ$ and $Π$ be disjoint alphabets, respectively called the static alphabet and the parameterized alphabet. Two strings $x$ and $y$ over $Σ\cup Π$ of equal length are said to parameterized match (p-match) if there exists a renaming bijection $f$ on $Σ$ and $Π$ which is identity on $Σ$ and maps the characters of $x$ to those of $y$ so that the two strings become identical. The indexing version of the problem of finding p-matching occurrences of a given pattern in the text is a well-studied topic in string matching. In this paper, we present a state-of-the-art indexing structure for p-matching called the parameterized suffix tray of an input text $T$, denoted by $\mathsf{PSTray}(T)$. We show that $\mathsf{PSTray}(T)$ occupies $O(n)$ space and supports pattern matching queries in $O(m + \log (σ+π) + \mathit{occ})$ time, where $n$ is the length of $T$, $m$ is the length of a query pattern $P$, $π$ is the number of distinct symbols of $|Π|$ in $T$, $σ$ is the number of distinct symbols of $|Σ|$ in $T$ and $\mathit{occ}$ is the number of p-matching occurrences of $P$ in $T$. We also present how to build $\mathsf{PSTray}(T)$ in $O(n)$ time from the parameterized suffix tree of $T$.