Compressed Representations of Sequences and Full-Text Indexes

Paolo Ferragina, Giovanni Manzini, Veli Mäkinen, and Gonzalo Navarro

Abstract. Given a sequence S=s_1 s_2 ... s_n of integers smaller than r = O(polylog(n)), we show how S can be represented using nH_0(S) + o(n) bits, so that we can know any s_i, as well as answer rank and select queries on S, in constant time. H_0(S) is the zero-order empirical entropy of S and nH_0(S) provides an Information Theoretic lower bound to the bit storage of any sequence S via a fixed encoding of its symbols. This extends previous results on binary sequences, and improves previous results on general sequences where those queries are answered in O(log r) time. For larger r, we can still represent S in nH_0(S) + o(n log r) bits and answer queries in O(log r / loglog n) time.

Another contribution of this paper is to show how to combine our compressed representation of integer sequences with an existing compression boosting technique to design compressed full-text indexes that scale well with the size of the input alphabet A. Namely, we design a variant of the FM-index that indexes a string T[1,n] within n H_k(T) + o(n) bits of storage, where H_k(T) is the k-th order empirical entropy of T. This space bound holds simultaneously for all k<=a log_{|A|} n, constant 0< a < 1, and |A| = O(polylog(n)). This index counts the occurrences of an arbitrary pattern P[1,p] as a substring of T in O(p) time; it locates each pattern occurrence in O(log^{1+e} n) time, for any constant 0< e <1; and it reports a text substring of length L in O(L + log^{1+e} n) time.

Compared to all previous works, our index is the first one that removes the alphabet-size dependance from all query times, in particular counting time is linear in the pattern length. Still, our index uses essentially the same space of the k-th order entropy of the text T, which is the best space obtained in previous work. We can also handle larger alphabets of size |A|=O(n^b), for any 0< b <1, by paying o(n log |A|) extra space and by multiplying all query times by O(log |A| / loglog n).