Some experimentation showed that having these different ranges was indeed more efficient than having just a single offset length, but it was hard to tell just what the optimal sizes were for each range. I kept it to only three different ranges because initially I didn’t want the number of identifier symbols to be too large, but after merging the identifiers into the literals, I had a bit more leeway in how many more ranges I could add.

So…why not add a range for every bit length? I set it up so that 256 would correspond to a 6-bit offset, 257 indicated a 7-bit offset, 258 is an 8-bit offset, etc., all the way up to 24-bit offsets. This also had the property that, except for the bottom range, an ‘n’-bit offset could be stored in ‘n-1′ bits, since the uppermost bit would always be ’1′ and could be thrown away (if it was ’0′, it wouldn’t be considered an ‘n’-bit offset, since it fits in a smaller range). Some testing against a set of data files showed that this did indeed improve the compression efficiency and produced smaller files.

With all of these possible bit values and lengths though, there was still the open question of what should be considered reasonable values for things like the default history buffer size and match length. Unfortunately, the answer is that it…depends. I used a shell script called ‘explode’ to run files through the compressor with all possible combinations of a set of buffer sizes and match lengths to see which would produce the smallest files, and the results varied a lot depending on the type and size of input file. Increasing the match length did not necessarily help, since it increased the average size of the length symbols and didn’t necessarily find enough long matches to cancel that out. Increasing the buffer size generally improves compression, but greatly increases memory usage and slows down compression. After some more experimentation with the ‘explode’ script, I settled on defaults of 17 bits for the buffer size, and a match length of 130.

Another idea I’d remembered hearing about was how the best match at the current byte might not necessarily be the most efficient match. It might be more efficient to emit the current byte as a literal instead if the next byte is the start of an even longer match. It was only an intuitive feeling though, so I implemented this and tested it and it did indeed seem to give a consistent improvement in compression efficiency. As an example, in one text document the phrase ‘edge of the dock’ was compressed like so:

Literal: 'e' (101) (4 bits)
Literal: 'd' (100) (6 bits)
Literal: 'g' (103) (8 bits)
10-bit offset: 544   Length: 3 'e o' (16 bits)
 8-bit offset: 170   Length: 6 'f the ' (17 bits)
10-bit offset: 592   Length: 3 ' do' (16 bits)
Literal: 'c' (99) (6 bits)
Literal: 'k' (107) (7 bits)

but with the new test, it generated the following instead:

Literal: 'e' (101) (4 bits)
Literal: 'd' (100) (6 bits)
Literal: 'g' (103) (8 bits)
Literal: 'e' (101) (4 bits) (forced, match len=3)
 8-bit offset: 170   Length: 8 ' of the ' (19 bits)
10-bit offset: 592   Length: 3 ' do' (16 bits)
Literal: 'c' (99) (6 bits)
Literal: 'k' (107) (7 bits)

The ‘forced’ literal normally would have been part of the first match, but by emitting it as a literal instead it was able to find a more efficient match and only two offset/length tokens were needed instead of three, for a difference of 80 bits for the original versus 70 bits for the improved match. Doing these extra tests does slow down compression a fair bit though, so I made it an optional feature, enabled on the command line.

At this point though, it’s getting harder and harder to extract gains in compression efficiency, as it starts devolving into a whole bunch of special cases. For example, increasing the buffer size sometimes makes compression worse, as in the following example:

'diff' output between two runs:
 17-bit offset: 87005   Length: 10 'with the t' (26 bits)
 14-bit offset: 10812   Length: 3 'arp' (18 bits)
-13-bit offset: 7705   Length: 3 ', w' (17 bits)
-13-bit offset: 5544   Length: 8 'ould you' (19 bits)
+18-bit offset: 131750   Length: 4 ', wo' (41 bits)
+13-bit offset: 5544   Length: 7 'uld you' (19 bits)
 16-bit offset: 50860   Length: 7 '?  You ' (22 bits)
 17-bit offset: 73350   Length: 10 'take that ' (26 bits)

The compressor looks for the longest matches, and in the ‘+’ run it found a longer match, but at a larger offset than in the ‘-’ run. In this case, 18-bit offsets are rare enough that their symbol has been pushed low in the Huffman tree and the bitstring is very long, making it even less efficient to use a long offset, and in the end a whopping 24 bits are completely wasted. Detecting these kinds of cases requires a bunch of extra tests though, and this is just one example.

So, I think that’s about all I’m going to do for attempting to improve the compression efficiency. How does it do overall? Well, that 195kB text file that originally compressed to 87.4kB and then made it down to 84.2kB can now be compressed down, with harder searching on and optimal buffer and match length sizes determined, to 77.9kB. That’s even lower than ‘gzip -9′ at 81.1kB!

It’s not all good news, though. If I take the Canterbury Corpus and test against it, the original total size is 2810784 bytes, ‘gzip -9′ reduces them to a total of 730732 bytes (26.0%), and at the default settings, my compressor gets…785421 bytes (27.9%). If I enable the extra searching and find optimal compression parameters for each file via ‘explode’, I can get it down to 719246 bytes (25.6%), but that takes a lot of effort. Otherwise, at the default settings, some of the files are smaller than gzip and others are larger; typically I do worse on the smaller files where there hasn’t really been much of a chance for the Huffman trees to adapt yet, and the Excel spreadsheet in particular does really poorly with my compressor, for some reason I’d have to investigate further.

But I’m not going to. No, the main remaining problem was one of speed…

Unfortunately, that method led to horribly imbalanced trees, since it only considered nearby nodes when rebalancing, when changing the frequency of a symbol can actually affect the relationship of symbols on relatively distant parts of the tree as well. As an example, here’s what a 4-bit length tree wound up looking like with my original adaptive method:

Lengths tree:
    Leaf node 0: Count=2256 BitString=1
    Leaf node 1: Count=1731 BitString=001
    Leaf node 2: Count=1268 BitString=0001
    Leaf node 3: Count=853 BitString=00001
    Leaf node 4: Count=576 BitString=000001
    Leaf node 5: Count=405 BitString=0000001
    Leaf node 6: Count=313 BitString=00000001
    Leaf node 7: Count=215 BitString=000000000
    Leaf node 8: Count=108 BitString=0000000011
    Leaf node 9: Count=81 BitString=00000000101
    Leaf node 10: Count=47 BitString=000000001001
    Leaf node 11: Count=22 BitString=00000000100001
    Leaf node 12: Count=28 BitString=0000000010001
    Leaf node 13: Count=15 BitString=000000001000000
    Leaf node 14: Count=9 BitString=000000001000001
    Leaf node 15: Count=169 BitString=01
    Avg bits per symbol = 3.881052

If you take the same data and manually construct a Huffman tree the proper way, you get a much more balanced tree without the ludicrously long strings:

    Leaf node 0: Count=2256 BitString=10
    Leaf node 1: Count=1731 BitString=01
    Leaf node 2: Count=1268 BitString=111
    Leaf node 3: Count=853 BitString=001
    Leaf node 4: Count=576 BitString=1100
    Leaf node 5: Count=405 BitString=0001
    Leaf node 6: Count=313 BitString=11011
    Leaf node 7: Count=215 BitString=00001
    Leaf node 8: Count=108 BitString=000001
    Leaf node 9: Count=81 BitString=000000
    Leaf node 10: Count=47 BitString=1101000
    Leaf node 11: Count=22 BitString=110100110
    Leaf node 12: Count=28 BitString=11010010
    Leaf node 13: Count=15 BitString=1101001111
    Leaf node 14: Count=9 BitString=1101001110
    Leaf node 15: Count=169 BitString=110101
    Avg bits per symbol = 2.969368

That’s nearly a bit per symbol better, which may not sound like much but with the original method there was barely any compression happening at all, whereas a proper tree achieves just over 25% compression.

So, I simply dumped my original adaptive method and made it construct a Huffman tree in the more traditional way, pairing the highest count nodes in a sorted list. To keep it adaptive, it still does the count check against the parent’s parent’s other child, and when it crosses the threshold it simply rebuilds the entire Huffman tree from scratch based on the current symbol counts. This involves a lot more CPU work, but as we’ll see later, performance bottlenecks aren’t necessarily where you think they are…

My trees also differ from traditional ones in that they prepopulate the tree with all possible symbols with a count of zero, whereas usually you only insert nodes into a Huffman tree if they have a count greater than zero. This is slightly suboptimal, but it avoids a chicken-and-egg problem with the decoder not knowing what symbol a bitstring corresponds to if it doesn’t exist in the tree yet because it’s the first time the symbol has been seen.

Knowing that, and with the improved Huffman trees, another thing became clear: using Huffman trees for the offsets wasn’t really doing much good at all. With most files, the offset values are too evenly distributed, and many are never used at all, and all those zero-count entries would get pushed down the tree and become longer strings, so the first time an offset got used it would often have a string longer than its basic bit length, causing file growth instead of compression. I instead just ripped those trees out and emitted plain old integer values for the offsets.

The way I was constructing my trees also had another limitation: the total number of symbols had to be a power of two. With the proper construction method, an arbitrary number of symbols could be specified, and that allowed another potential optimization: merging the identifier tree and the literals tree. The identifier token in the output stream guaranteed that there would always be at least 1 wasted non-data bit per token, and often two. Merging it with the literals would increase the size of literal symbols, but the expectation is that the larger literal size would on average still be smaller than the sum of the identifier symbols and smaller literal symbols, on average, especially as more ‘special case’ symbols are added. Instead of reading an identifier symbol and deciding what to do based on that, the decoder would read a ‘literal’ symbol, and if it was in the range 0-255, it was indeed a literal byte value and interpreted that way, but if it was 256 or above, it would be treated as having a following offset/length pair.

The range of offsets to handle would also have to change, but that’s for next time… With the Huffman tree improvements, my 195kB test file that compressed to 87.4kB before now compressed to 84.2kB. Still not as good as gzip, but getting there.

Looking at it now it’s kind of embarrassing, since of course it’s riddled with inexperience. No comments at all, leaving me mystified at what some of the code was even doing in the first place, unnecessary global variables, little error checking, poor header/module separation, unnecessary exposure of class internals, poor const correctness, and so on. It kind of irks my pride to leave something in such a poor state though, so I quickly resolved to at least clean it up a bit.

Of course, I have to understand it first, and I started to remember more about it as I looked over the code. It’s a fairly basic combination of both LZ77 pattern matching and Huffman coding, like the ubiquitous Zip method, but the twist I wanted to explore was in making the Huffman trees adaptive, so that the symbols would shift around the tree to automatically adjust as their frequency changed within the input stream. There were two parameters that controlled compression efficiency: history buffer size, and maximum pattern length. The history size controls how far back it would look for matching patterns, and the length controlled the upper limit on the length of a match that could be found.

Compression proceeded by moving through the input file byte by byte, looking for the longest possible exact byte match between the data ahead of the current position and the data in the history buffer just behind the current position. If a match could not be found, it would emit the current byte as a literal and move one byte ahead, and if a match was found, it would emit a token with the offset and length of the match in the history buffer. To differentiate between these cases, it would first emit an ‘identifier’ token with one of four possible values: one for a literal, which would then be followed by the 8-bit value of the literal, and three for offset and length values, with three different possible bit lengths for the offset so that closer matches took fewer bits. Only matches of length 3 or longer are considered, since two-byte matches would likely have an identifier+offset+length string longer than just emitting the two bytes as literals. In summary, the four possible types of bit strings you’d see in the output were:

    | ident 0 | 8-bit literal |

    | ident 1 | 'x'-bit offset    | length |

    | ident 2 | 'y'-bit offset        | length |

    | ident 3 | 'z'-bit offset            | length |

And then I used a lot of Huffman trees. Each of these values were then run through a Huffman tree to generate the actual symbol emitted to the output stream, with separate trees for the identifier token, the literals, the lengths, and the three offset types. HUFFMAN TREES EVERYWHERE! The compression parameters were also written to a header in the file, so the decoder would know what history buffer size to use and maximum length allowed.

It worked…okay… I’ve lost my original test files, but on one example text file of 195kB, my method compresses it down to 87.4kB, while ‘gzip -9′ manages 81.1kB. Not really competitive, but not too bad for a completely amateur attempt either. There’s still plenty of room for improvement, which will come…next time.

It was mainly meant for a (now postponed) trip so I could use VMware Fusion effectively, but the difference was immediately noticeable when I went to fire up WoW as well. Normally, running WoW on my laptop grinds and chugs and stutters a lot, mainly because I always have Firefox open as well, and together the two just use up too much memory. Now though, it’s smooth as silk, with WoW loading only a little bit slower than it does on my desktop machine.

It also updated MythTV, which was a bit of a surprise and I needed to go get a newer build of the OS X frontend. That took a while to get working because it would just suddenly stop running immediately after launching it, until I figured out that I had to run the main executable directly with a ‘-r’ option to reset which theme it wanted to use.

The most obvious change in 7 is in the taskbar. It uses large icons instead of a small icon and window title, all open windows of the same type are always consolidated into a single entry on the taskbar (previously it would only consolidate them once it started running out of room), and you can pin a running program to the taskbar in order to launch it again later. It’s basically a lot more like OS X’s Dock now.

The Explorer has also changed a bit. There’s no more ‘Explore’ option off the computer icon’s context menu, which is kind of annoying. And they’ve removed the tree view from the Explorer windows (but you can reenable it in the folder options), instead using the sidebar to emphasize a bunch of standard locations like your home directory, your music directory, network servers, etc. Which is also a lot like how Finder works…

Otherwise, things have gone fairly smoothly, and I haven’t really had any problems that I can attribute directly to Windows 7 itself. I still have to poke around and explore what else might be new, though.

It actually went a bit smoother than expected, with only two major hiccups. The first was when I went to install a new 120mm fan to improve airflow, when I suddenly realized that I didn’t know which direction it was supposed to face… Fortunately it’s the same style as a couple other fans in the system, so I was able to deduce the direction from that, and confirmed it with a sheet of paper. The second was getting the BIOS settings right, since this motherboard doesn’t seem to do a very good job of autodetecting all the right settings. It only took a little bit of fiddling to get it up to the proper 2.83GHz speed, though.

It’s practically an entirely new machine at this point, with a new CPU (Q9550 quad-core, replacing an E6600 dual-core), more memory (from 2 to 6GB), bigger drives (1.25TB total), and a new video card (Radeon 4870). It is actually missing something, though — I yanked out the Audigy 2 sound card I’d been using before. Creative’s support for newer OSes with older cards has been a bit lacking, so I’ll take a chance on the onboard sound for now.

Now I just have to put some OSes back on it…

I only want to crack the case open once, so I have to install everything at the same time. But if I install the other hard drives, I’ll lose the OS and have to reinstall it. But Win7 isn’t ready yet for me, so I’d have to either reinstall Vista or restore the old install from the current drive, and there’s not much point in wasting time on that when I’d have to do a new install in a week now anyway.

So, for now the parts all sit on my kitchen table, taunting me, tempting me…

So, tomorrow I say goodbye to my old friend as I drop him off downtown. It may be old, but it served me well for an awfully long time.