Fingerprints in compressed strings

In this paper we show how to construct a data structure for a string S of size N compressed into a context-free grammar of size n that supports efficient Karp-Rabin fingerprint queries to any substring of S. That is, given indices i and j, the answer to a query is the fingerprint of the substring S[i,j]. We present the first O(n) space data structures that answer fingerprint queries without decompressing any characters. For Straight Line Programs (SLP) we get O(log⁡N) query time, and for Linear SLPs (an SLP derivative that captures LZ78 compression and its variations) we get O(log⁡log⁡N) query time. We extend the result to solve the longest common extension problem in query time O(log⁡Nlog⁡ℓ) and O(log⁡ℓlog⁡log⁡ℓ+log⁡log⁡N) for SLPs and Linear SLPs, respectively. Here, ℓ denotes the length of the LCE.