Access, Rank, and Select in Grammar-compressed Strings

Given a string S of length N on a fixed alphabet of σ symbols, a grammar compressor produces a context-free grammar G of size n that generates S and only S. In this paper we describe data structures to support the following operations on a grammar-compressed string: access(S,i,j) (return substring S[i,j]), rank c (S,i) (return the number of occurrences of symbol c before position i in S), and select c (S,i) (return the position of the ith occurrence of c in S). Our main result for access is a method that requires \O(nlogN) bits of space and \O(logN+m/logσN) time to extract m = j − i + 1 consecutive symbols from S. Alternatively, we can achieve \O(logτN+m/logσN) query time using \O(nτlogτ(N/n)logN) bits of space, matching a lower bound stated by Verbin and Yu for strings where N is polynomially related to n when τ = log ε N. For rank and select we describe data structures of size \O(nσlogN) bits that support the two operations in \O(logN) time. We also extend our other structure to support both operations in \O(logτN) time using \O(nτσlogτ(N/n)logN) bits of space. When τ = log ε N the query time is O(logN/loglogN) and we provide a hardness result showing that significantly improving this would imply a major breakthrough on a hard graph-theoretical problem.