Compressed Indexing with Signature Grammars

Abstract

The compressed indexing problem is to preprocess a string S of length n into a compressed representation that supports pattern matching queries. That is, given a string P of length m report all occurrences of P in S.

We present a data structure that supports pattern matching queries in $O(m + \mathsf {occ}(\lg \lg n + \lg ^\epsilon z))$ time using $O(z \lg (n / z))$ space where z is the size of the LZ77 parse of S and $\epsilon > 0$ is an arbitrarily small constant, when the alphabet is small or $z = O(n^{1 - \delta })$ for any constant $\delta > 0$. We also present two data structures for the general case; one where the space is increased by $O(z\lg \lg z)$, and one where the query time changes from worst-case to expected. These results improve the previously best known solutions. Notably, this is the first data structure that decides if P occurs in S in O(m) time using $O(z\lg (n/z))$ space.

Our results are mainly obtained by a novel combination of a randomized grammar construction algorithm with well known techniques relating pattern matching to 2D-range reporting.