Time-space trade-offs for lempel-ziv compressed indexing

Given a string S, the compressed indexing problem is to preprocess S into a compressed representation that supports fast substring queries. The goal is to use little space relative to the compressed size of S while supporting fast queries. We present a compressed index based on the Lempel-Ziv 1977 compression scheme. Let n, and z denote the size of the input string, and the compressed LZ77 string, respectively. We obtain the following time-space trade-offs. Given a pattern string P of length m, we can solve the problem in (i) O (m + occ lg lg n) time using O(z lg(n/z) lg lg z) space, or (ii) (m (1 + lgϵ z/lg(n/z) + occ(lg lg n + lgϵ z)) time using O(z lg (n/z)) space, for any 0 <ϵ <1 In particular, (i) improves the leading term in the query time of the previous best solution from O(m lg m) to O(m) at the cost of increasing the space by a factor lg lg z. Alternatively, (ii) matches the previous best space bound, but has a leading term in the query time of O(m(1 + lgϵ z/lg(n/z))). However, for any polynomial compression ratio, i.e., z = O(n1-δ), for constant δ > 0, this becomes O(m). Our index also supports extraction of any substring of length ℓ in O(ℓ + lg(n/z)) time. Technically, our results are obtained by novel extensions and combinations of existing data structures of independent interest, including a new batched variant of weak prefix search.