Compressed Indexing with Signature Grammars

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 + 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.