Efficient Fully-Compressed Sequence Representations

Abstract

We present a data structure that stores a sequence $s[1..n]$ over alphabet $[1..σ]$ in $n\Ho(s) + o(n)(\Ho(s){+}1)$ bits, where $\Ho(s)$ is the zero-order entropy of $s$. This structure supports the queries \access, \rank\ and \select, which are fundamental building blocks for many other compressed data structures, in worst-case time $\Oh{\lg\lgσ}$ and average time $\Oh{\lg \Ho(s)}$. The worst-case complexity matches the best previous results, yet these had been achieved with data structures using $n\Ho(s)+o(n\lgσ)$ bits. On highly compressible sequences the $o(n\lgσ)$ bits of the redundancy may be significant compared to the the $n\Ho(s)$ bits that encode the data. Our representation, instead, compresses the redundancy as well. Moreover, our average-case complexity is unprecedented. Our technique is based on partitioning the alphabet into characters of similar frequency. The subsequence corresponding to each group can then be encoded using fast uncompressed representations without harming the overall compression ratios, even in the redundancy. The result also improves upon the best current compressed representations of several other data structures. For example, we achieve $(i)$ compressed redundancy, retaining the best time complexities, for the smallest existing full-text self-indexes; $(ii)$ compressed permutations $π$ with times for $π()$ and $\pii()$ improved to loglogarithmic; and $(iii)$ the first compressed representation of dynamic collections of disjoint sets. We also point out various applications to inverted indexes, suffix arrays, binary relations, and data compressors. ...