Optimal Substring Equality Queries with Applications to Sparse Text Indexing

We consider the problem of encoding a string of length n from an integer alphabet of size sigma so access, substring equality, and Longest Common Extension (LCE) queries can be answered efficiently. We describe a new space-optimal data structure supporting logarithmic-time queries. Access and substring equality query times can furthermore be improved to the optimal O(1) if O(log n) additional precomputed words are allowed in the total space. Additionally, we provide in-place algorithms for converting between the string and our data structure. Using this new string representation, we obtain the first in-place subquadratic algorithms for several string-processing problems in the restore model: The input string is rewritable and must be restored before the computation terminates. In particular, we describe the first in-place subquadratic Monte Carlo solutions to the sparse suffix sorting, sparse LCP array construction, and suffix selection problems. With the sole exception of suffix selection, our algorithms are also the first running in sublinear time for small enough sets of input suffixes. Combining these solutions, we obtain the first sublinear-time Monte Carlo algorithm for building the sparse suffix tree in compact space. We also show how to build a correct version of our data structure using small working space. This leads to the first Las Vegas in-place algorithm computing the full LCP array in 0(n log n) time w.h.p. and to the first Las Vegas in-place algorithms solving the sparse suffix sorting and sparse LCP array construction problems in O(n(1.5)root log sigma) time w.h.p.