10-24-11 - LZ Optimal Parse with A Star Part 1

First two notes that aren't about the A-Star parse :

1. All good LZ encoders these days that aren't optimal parsers use complicated heuristics to try to bias the parse towards a cheaper one. The LZ parse is massively redundant (many parses can produce the same original data) and the cost is not the same. But in the forward normal parse you can't say for sure which decision is cheapest, so they just make some guesses.

For example the two key crucial heuristics in LZMA are :

// LZMA lastOffset heuristic :
  if (repLen >= 2 && (
        (repLen + 1 >= mainLen) ||
        (repLen + 2 >= mainLen && mainDist >= (1 << 9)) ||
        (repLen + 3 >= mainLen && mainDist >= (1 << 15))))
  {
    // take repLen instead of mainLen
  }

// LZMA lazy heuristic :

        // Truncate current match if match at next position will be better (LZMA's algorithm)
        if (nextlen >= prevlen && nextdist < prevdist/4 ||
            nextlen == prevlen + 1 && !ChangePair(prevdist, nextdist) ||
            nextlen > prevlen + 1 ||
            nextlen + 1 >= prevlen && prevlen >= MINLEN && ChangePair(nextdist, prevdist))
        {
             return MINLEN-1;
        } else {
             return prevlen;
        }

One is choosing a "repeat match" over a normal match, and the next is choosing a lazy match (literal then match) over greedy (immediate match).

My non-optimal LZ parser has to make similar decisions; what I did was set up a matrix and train it. I made four categories for a match based on what the offset is : {repeat, low, medium, high } , so to decide between two matches I do :

  cat1 = category of match 1 offset
  cat2 = category of match 2 offset
  diff = len1 - len2

  return diff >= c_threshold[cat1][cat2]
  

so c_threshold is a 4x4 matrix of integers. The values of threshold are in the range [0,3] so it's not too many to just enumerate all possibilities of threshold and see what's best.

Anyway, the thing about these heuristics is that they bias the parse in a certain way. They assume a certain cost tradeoff between literals vs. repeat matches vs. normal matches, or whatever. When the heuristic doesn't match the file they do badly. Otherwise they do amazingly well. One solution might be having several heuristics trained on different files and choose the one that creates the smallest output.

Also I should note - it's not trivial to tell when you have the heuristic wrong for the file. The problem is that there's a feedback loop between the parse heuristic and the statistical coder. That causes you to get into a local minimum, and you might not see that there's a better global minimum which is only available if you make a big jump in parsing style.

Repeating that more explicitly : the statistical coder will adapt to your parse heuristic; say you have a heuristic that prefers low offsets (like the LZMA heuristic above), that will cause your parse to select more low offsets, that will cause your statistical backend to code low offsets smaller. That's good, that's what you want, that's the whole point of the heuristic, it skews the parse intentionally to get the statistics going in the right direction. The issue then is that if you try to evaluate an alternative parse using the statistics that you have, it will look bad, because your statistics are trained for the parse you have.

2. It's crazy that LZ compression can do so well with so much redundancy in the code stream.

Think about it this way. Enumerate all the compressed bit streams that are created by encoding all raw bit streams up to length N.

Start with the shortest compressed bit stream. See what raw bits that decodes to. Now there may be several more (longer) compressed bit streams that decode to that same raw bit stream. Mark those all as unused.

Move to the next shortest compressed bit stream. First if there is a shorter unused one, move it into that slot. Then mark all other output streams that make the same raw bits as unused, and repeat.

For example a file like "aaaaaaa" has one encoding that's {a, offset -1 length 6} ; but there are *tons* of other encodings, such as {a,a,[-1,5]} or {a,[-1,3],[-1,3]} or {a,[-1,2],[-2,4]} etc. etc.

All the encodings except the shortest are useless. We don't want them. But even the ones that are not the best are quite short - they are small encodings, and small encodings take up lots of code space (see Kraft Inequality for example - one bit shorter means you take up twice the code space). So these useless but short encodings are real waste. In particular, there are other raw strings that don't have such a short encoding that would like to have that output length.

Anyhoo, surprisingly (just like with video coding) it seems to be good to add even *more* code stream redundancy by using things like the "repeat matches". I don't think I've ever seen an analysis of just how much wasted code space there is in the LZ output, I'm curious how much there is.

Hmm we didn't actually get to the A Star. Next time.