Parameterized Pattern Matching - Succinctly

The fields of succinct data structures and compressed text indexing have seen quite a bit of progress over the last 15 years. An important achievement, primarily using techniques based on the Burrows-Wheeler Transform (BWT), was obtaining the full functionality of suffix tree in the optimal number of bits. A crucial property that allows the use of BWT for designing compressed indexes is order-preserving suffix links. Specifically, the relative ordering between two suffixes in the subtree of an internal node is same as that of the suffixes obtained by truncating the first character of the two suffixes. Unfortunately, in many variants of the text-indexing problem, this property does not hold, for e.g., parameterized pattern matching, 2D pattern matching, etc. Consequently, the compressed indexes based on BWT do not directly apply. Furthermore, a compressed index for any of these variants has been elusive throughout the advancement of the field of succinct data structures. We achieve a positive breakthrough on one such problem, namely the Parameterized Pattern Matching problem. In parameterized matching, a pattern matches some location in the text iff there is a oneto-one correspondence between the alphabet symbols of the pattern to those of the text. More specifically, assume that the text T contains n characters from a static alphabet Σs and a parameterized alphabet Σp, where Σs ∩ Σp = ∅ and |Σs ∪ Σp| = σ. A pattern P matches a substring S of T iff the static characters match exactly, and there exists a one-to-one function that renames the parameterized characters in S to that in P . Previous indexing solution [Baker, STOC 1993], known as Parameterized Suffix Tree, requires Θ(n logn) bits of space, and can find all occ occurrences of P in O(|P | log σ + occ) time. In this paper, we present the first succinct index that occupies n logσ+O(n) bits and answers queries in O((|P |+occ · log n) log σ log log σ) time. We also present a compact index that occupies O(n log σ) bits and answers queries in O(|P | log σ + occ · logn) time.