Arithmetic coding is a type of lossless compression, where more frequently used bytes are replaced with fewer bits than less frequent.

Arithmetic coding is commonly associated with Huffman coding, but it is not correct – Huffman coding splits input string into smaller and smaller chunks until chunk can be replaced with smaller entity. Usually Huffman encoding can not reach Shannon limit (theoretical compression limit for given data) for reasonable number of iterations, while arithmetic coding – which transfers the whole string into packed binary stream at once – approaches that limit much faster.

I work with Greylock search engine where there is a major task to implement full-text search index with as small disk footprint as possible. In our tests 200k of uncompressed input events occupy about 220Mb of disk space and whole index takes close to 450Mb. Index includes original content, common index and per-author index (this basically doubles the original index). Elastic search with morphology takes about 539Mb, without morphology – 317Mb, Sphinx (+ mysql to store original content) is about 250Mb, but this index is heavily truncated, there is still a notion of stop-words and morphology is broken.

We compress our binary index entries as well as original content using lz4hc or gzip, but I wondered whether arithmetic coding can do better than that. I skipped Huffman encoder because of its theoretical problems approaching Shannon limit (not talking about practical issues), and decided to check rANS entropy coder – this is an Asymmetric Numeral entropy Systems coder which combines speed of Huffman coding with compression rate of arithmetic coding. Well, at least that’s what its author says in the paper.

Here is an image from the paper showing Huffman coder approaching Shannon limit

It happend to be worse than any common compression both for binary and text data.

Text data from original library: 768771 bytes

bz2: 232598

gzip: 312281

lz4: 363236

rans: 435113

Binary data from our index: 9678

bz2: 3652

gzip: 3665

lz4: 5262

rans: 7795

I do not drop idea of using arithmetic coding to pack binary data since Dropbox showed with its Lepton project that jpeg images can be losslessly compressed by 20% using tricky DCT indexes sorting and prediction (about 3-4% among 20%) and VP8 arithmetic coding for these indexes.

Naive comparison of Lepton and rANS yields dramatic difference:

Original jpeg file sile: 71517

Lepton encoded size: 55700

gzip: 71171

bz2: 71586

lz4: 71414

rANS encoded size: 71336

This is of course unfair comparison, since Lepton encodes not binary steam of the original image, but specially crafted DCT indexes.

Since our indexes are basically vectors of document IDs which are in fact partially increasing timestamp and partially sequence number, it might be a good idea to use timestamp compression algorithm described in Facebook’s Gorilla timeseries database – I wrote about it previously, but that’s another story.