Given a string S, the \emph{compressed indexing problem} is to preprocess
S into a compressed representation that supports fast \emph{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. We obtain the following time-space
trade-offs: For constant-sized alphabets; (i) O(m+occlglgn) time using
O(zlg(n/z)lglgz) space, or (ii) O(m(1+lg(n/z)lgϵz)+occ(lglgn+lgϵz)) time using O(zlg(n/z)) space. For integer
alphabets polynomially bounded by n; (iii) O(m(1+lg(n/z)lgϵz)+occ(lglgn+lgϵz)) time using O(z(lg(n/z)+lglgz)) space, or (iv) O(m+occ(lglgn+lgϵz)) time using
O(z(lg(n/z)+lgϵz)) space, where n and m are the length of
the input string and query string respectively, z is the number of phrases in
the LZ77 parse of the input string, occ is the number of occurrences of the
query in the input and ϵ>0 is an arbitrarily small constant. In
particular, (i) improves the leading term in the query time of the previous
best solution from O(mlgm) to O(m) at the cost of increasing the space by
a factor lglgz. Alternatively, (ii) matches the previous best space
bound, but has a leading term in the query time of O(m(1+lg(n/z)lgϵz)). 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