jeudi 18 août 2016

LZMA compression explained

This summer vacation's project was completed almost on schedule: write a LZMA encoder, whilst enjoying vacation - that is, work early in the morning and late in the evening when everybody else is sleeping; and have fun (bike, canoe, visiting caves and amazing dinosaurs fac-similes, enjoying special beers, ...) the rest of the day.

Well, "schedule" is a bit overstretched, because with a topic as tricky as data compression, it is difficult to tell when and even whether you will succeed...

LZMA is a compression format invented by Igor Pavlov, which combines a LZ77 compression and range encoding.

With LZ77, imagine you are copying a text, character by character, but want to take some shortcuts. You send either single characters, or a pair of numbers (distance, length) meaning "please copy 'length' characters, starting back 'distance' characters in the copied text, from the point where the cursor is right now". That's it!
LZ77 is a well covered subject and is the first stage of most compression algorithms. Basically you can pick and choose an implementation, depending on the final compression size.

Range encoding is a fascinating way of compressing a message of any nature. Say you want to send a very large number N, but with less digits. It's possible - if some of the digits (0 to 9), appear more frequently, and some, less. The method is the following.
You begin with a range, say [0, 999[.
You subdivide it in ten intervals, corresponding to the digits 0 to 9, and calibrated depending on their probability of occurrence, p0 .. p9. The first digit of N is perhaps 3, and its corresponding interval is, say, [295, 405[.
Then, you continue with the second digit by subdividing [295, 405[ in ten intervals. If the second digit is 0, you have perhaps now [295, 306[, representing the partial message "30". You see, of course, that if you want to stick with integers (with computers you don't have infinite precision anyway), you lose quickly precision when you set up the ten intervals with the probabilities p0 .. p9. The solution is to append from time to time a 0 to the interval, when the width is too small. So, if you decide to multiply everything by 10 each time the width is less than 100, then the interval for "30" will be now [2950, 3060[.
Some n digits to be encoded later (after n subdivisions and some x10 when needed) your interval will perhaps look like [298056312, 298056701[. The bounds become larger and larger - second problem. Solution: you see that the leftmost digits won't change anymore. You can get rid of them and send them as a chunk of the compressed message. The compression will be better when symbols are much more frequent than others: the closer the probability is to 1, the more the range width will be preserved. If the probability was exacly 1, the width wouldn't change at all and this trivial message with only the same symbol wouln't take any space in its compressed form! It is an absurd case, but it shows why compression methods such as LZMA are extremely good for very redundant data.
That's how the basic range encoding works.
Then, a funny thing is that you can encode a mix of different alphabets (say digits '0' to '9' and letters 'A' to 'Z') or even the same alphabet, but with different probabilities depending on the context, provided the decoder knows what to use when. That's all for range encoding (you find a more detailed description in the original article [1]).

LZMA's range encoder works exclusively on a single, binary alphabet (0's and 1's), so the range is always divided in two parts. But it works with lots of contextual probabilities. With some parameters you can have millions of different probabilities in the model! The probabilities are not known in advance, so in this respect LZMA is a purely adaptive compression method: the encoder and the decoder adapt the probabilities as the symbols are sent and received. After each bit encoded, sent, received, decoded, the entire probability set is (and has to be) exactly in the same state by the encoder and by the decoder.

Developing an encoder from scratch, even if you have open-source code to reproduce, is fun, but debugging it is a pain. A bug feels like when something doesn't work in a PhD work in maths. No way to get help from anybody or by browsing the Web. By nature, the compressed data will not contain any redundancy that would help you fixing bugs. The decoder is confused on faulty compressed data and cannot say why. For range encoding, it is worse: as in the example, digits sent have nothing to do with the message to be encoded. The interval subdivision, the shipping of the leading interval digits, and the appending of trailing '0', occur in a way which is completely asynchronous. So, the good tactic is, as elsewhere, to simplify and divide the issues to the simplest.
First, manage to encode an empty message (wow!). It seems trivial, but the range encoder works like a pipeline; you need to initialize it and flush it correctly. Then, an empty message and the end-of-stream marker. And so on.
Another source of help for LZMA is the probability set: it needs to be identical at every point as said before.

The results of this effort in a few numbers:
  • LZMA.Encoding, started July 28th, first working version August 16th (revision 457).
  • Less than 450 lines - including lots of comments and some debugging code to be removed!
  • 5 bugs had to be fixed.

To my (of course biased) opinion, this is the first LZMA encoder that a normal human can understand by reading the source code.

Zip-Ada's Zip.Compress makes use of LZMA encoding since revision 459.

The source code is available here (main SourceForge SVN repository) or here (GitHub mirror).

Back to vacation topic (which is what you do often when you're back from vacation): a tourist info sign was just perfect for a 32x32 pixels "info" icon for the AZip archive manager.

Click to enlarge
The beautiful sign

By the way, some other things are beautiful in this town (St-Ursanne at the Doubs river)...

[1] G. N. N. Martin, Range encoding: an algorithm for removing redundancy
   from a digitized message, Video & Data Recording Conference,
   Southampton, UK, July 24-27, 1979.

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