Access, Rank, and Select in Grammar-compressed Strings

Abstract

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 $\O(n\log N)$ bits of space and $\O(\log N+m/\log_\sigma N)$ time to extract m = j − i + 1 consecutive symbols from S. Alternatively, we can achieve $\O(\log_\tau N+m/\log_\sigma N)$ query time using $\O(n\tau\log_\tau (N/n)\log N)$ 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 $\O(n\sigma\log N)$ bits that support the two operations in $\O(\log N)$ time. We also extend our other structure to support both operations in $\O(\log_\tau N)$ time using $\O(n\tau\sigma\log_\tau (N/n)\log N)$ 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.