05-13-15 - Skewed Pareto Chart

It's hard to see just the decomp speed in the normal Pareto Chart. It gets squished down over at the far-right Y-intercept.

The obvious fix is just to magnify the right side. This is a linear scaling of the data; *1 on the far left, *10 on the far right :

The far-left is still proportional to the compression ratio, the far right is proportional to the decompression speed. The compressor lines are still speedups vs. memcpy, but the memcpy baseline is now sloped.

I'm not really sure how I feel about the warped chart vs unwarped.

The Pareto curves are in fact sigmoids (tanh's).

speedup = 1 / (1/compression_ratio + disk_speed / decompress_speed)

speedup = 1 / (1/compression_ratio + exp( log_disk_speed ) / decompress_speed)

(here they're warped sigmoids because of the magnification; the ones back here in the LZNA post are true sigmoids)

I believe (but have not proven) that a principle of the Pareto Frontier is that the maximum of all compressors should also be a sigmoid.

max_speedup(disk_speed) = MAX{c}( speedup[compressor c](disk_speed) );

One of the nice things about these charts is it makes it easy to see where some compressors are not as good as possible. If we fit a sigmoid over the top of all the curves :

We can easily see that LZHLW and LZNIB are not touching the curve. They're not as good as they should be in space/speed. Even thought nothing beats them at the moment (that I know of), they are algorithmically short of what's possible.

There are two things that constrain compressors from being better in a space/speed way. There's 1. what is our current best known algorithm. And then there's 2. what is possible given knowledge of all possible algorithms. #2 is the absolute limit and eventually it runs into a thermodynamic limit. In a certain amount of cpu time (cpu bit flips, which increase entropy), how much entropy can you take out of a a given data stream. You can't beat that limit no matter how good your algorithm is. So our goal in compression is always to just find improvements in the algorithms to edge closer to that eventual limit.

Anyway. I think I know how to fix them, and hopefully they'll be up at the gray line soon.

No comments:

old rants