Papers › Sketching and Streaming for Dictionary Compression
Sketching and Streaming for Dictionary Compression
Ruben Becker, Matteo Canton, Davide Cenzato, Sung-Hwan Kim, Bojana Kodric, Nicola Prezza
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We initiate the study of sub-linear sketching and streaming techniques for estimating the output size of common dictionary compressors such as Lempel-Ziv '77, the run-length Burrows-Wheeler transform, and grammar compression. To this end, we focus on a measure that has recently gained much attention in the information-theoretic community and which approximates up to a polylogarithmic multiplicative factor the output sizes of those compressors: the normalized substring complexity function δ. We present a data sketch of O(ϵ⁻³logn + ϵ⁻¹log² n) words that allows computing a multiplicative (1±ϵ)-approximation of δ with high probability, where n is the string length. The sketches of two strings S₁,S₂ can be merged in O(ϵ⁻¹log² n) time to yield the sketch of {S₁,S₂}, speeding up by orders of magnitude tasks such as the computation of all-pairs \emph{Normalized Compression Distances} (NCD). If random access is available on the input, our sketch can be updated in O(ϵ⁻¹log² n) time for each character right-extension of the string. This yields a polylogarithmic-space algorithm for approximating δ, improving exponentially over the working space of the state-of-the-art algorithms running in nearly-linear time. Motivated by the fact that random access is not always available on the input data, we then present a streaming algorithm computing our sketch in O(√(n) ·logn) working space and O(ϵ⁻¹log² n) worst-case delay per character. We show that an implementation of our streaming algorithm can estimate {\delta} on a dataset of 189GB with a throughput of 203MB per minute while using only 5MB of RAM, and that our sketch speeds up the computation of all-pairs NCD distances by one order of magnitude, with applications to phylogenetic tree reconstruction.
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