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-1. Compression algorithm (deflate)\r\rThe deflation algorithm used by gzip (also zip and zlib) is a variation of\rLZ77 (Lempel-Ziv 1977, see reference below). It finds duplicated strings in\rthe input data. The second occurrence of a string is replaced by a\rpointer to the previous string, in the form of a pair (distance,\rlength). Distances are limited to 32K bytes, and lengths are limited\rto 258 bytes. When a string does not occur anywhere in the previous\r32K bytes, it is emitted as a sequence of literal bytes. (In this\rdescription, `string' must be taken as an arbitrary sequence of bytes,\rand is not restricted to printable characters.)\r\rLiterals or match lengths are compressed with one Huffman tree, and\rmatch distances are compressed with another tree. The trees are stored\rin a compact form at the start of each block. The blocks can have any\rsize (except that the compressed data for one block must fit in\ravailable memory). A block is terminated when deflate() determines that\rit would be useful to start another block with fresh trees. (This is\rsomewhat similar to the behavior of LZW-based _compress_.)\r\rDuplicated strings are found using a hash table. All input strings of\rlength 3 are inserted in the hash table. A hash index is computed for\rthe next 3 bytes. If the hash chain for this index is not empty, all\rstrings in the chain are compared with the current input string, and\rthe longest match is selected.\r\rThe hash chains are searched starting with the most recent strings, to\rfavor small distances and thus take advantage of the Huffman encoding.\rThe hash chains are singly linked. There are no deletions from the\rhash chains, the algorithm simply discards matches that are too old.\r\rTo avoid a worst-case situation, very long hash chains are arbitrarily\rtruncated at a certain length, determined by a runtime option (level\rparameter of deflateInit). So deflate() does not always find the longest\rpossible match but generally finds a match which is long enough.\r\rdeflate() also defers the selection of matches with a lazy evaluation\rmechanism. After a match of length N has been found, deflate() searches for\ra longer match at the next input byte. If a longer match is found, the\rprevious match is truncated to a length of one (thus producing a single\rliteral byte) and the process of lazy evaluation begins again. Otherwise,\rthe original match is kept, and the next match search is attempted only N\rsteps later.\r\rThe lazy match evaluation is also subject to a runtime parameter. If\rthe current match is long enough, deflate() reduces the search for a longer\rmatch, thus speeding up the whole process. If compression ratio is more\rimportant than speed, deflate() attempts a complete second search even if\rthe first match is already long enough.\r\rThe lazy match evaluation is not performed for the fastest compression\rmodes (level parameter 1 to 3). For these fast modes, new strings\rare inserted in the hash table only when no match was found, or\rwhen the match is not too long. This degrades the compression ratio\rbut saves time since there are both fewer insertions and fewer searches.\r\r\r2. Decompression algorithm (inflate)\r\r2.1 Introduction\r\rThe real question is, given a Huffman tree, how to decode fast. The most\rimportant realization is that shorter codes are much more common than\rlonger codes, so pay attention to decoding the short codes fast, and let\rthe long codes take longer to decode.\r\rinflate() sets up a first level table that covers some number of bits of\rinput less than the length of longest code. It gets that many bits from the\rstream, and looks it up in the table. The table will tell if the next\rcode is that many bits or less and how many, and if it is, it will tell\rthe value, else it will point to the next level table for which inflate()\rgrabs more bits and tries to decode a longer code.\r\rHow many bits to make the first lookup is a tradeoff between the time it\rtakes to decode and the time it takes to build the table. If building the\rtable took no time (and if you had infinite memory), then there would only\rbe a first level table to cover all the way to the longest code. However,\rbuilding the table ends up taking a lot longer for more bits since short\rcodes are replicated many times in such a table. What inflate() does is\rsimply to make the number of bits in the first table a variable, and set it\rfor the maximum speed.\r\rinflate() sends new trees relatively often, so it is possibly set for a\rsmaller first level table than an application that has only one tree for\rall the data. For inflate, which has 286 possible codes for the\rliteral/length tree, the size of the first table is nine bits. Also the\rdistance trees have 30 possible values, and the size of the first table is\rsix bits. Note that for each of those cases, the table ended up one bit\rlonger than the ``average'' code length, i.e. the code length of an\rapproximately flat code which would be a little more than eight bits for\r286 symbols and a little less than five bits for 30 symbols. It would be\rinteresting to see if optimizing the first level table for other\rapplications gave values within a bit or two of the flat code size.\r\r\r2.2 More details on the inflate table lookup\r\rOk, you want to know what this cleverly obfuscated inflate tree actually \rlooks like. You are correct that it's not a Huffman tree. It is simply a \rlookup table for the first, let's say, nine bits of a Huffman symbol. The \rsymbol could be as short as one bit or as long as 15 bits. If a particular \rsymbol is shorter than nine bits, then that symbol's translation is duplicated\rin all those entries that start with that symbol's bits. For example, if the \rsymbol is four bits, then it's duplicated 32 times in a nine-bit table. If a \rsymbol is nine bits long, it appears in the table once.\r\rIf the symbol is longer than nine bits, then that entry in the table points \rto another similar table for the remaining bits. Again, there are duplicated \rentries as needed. The idea is that most of the time the symbol will be short\rand there will only be one table look up. (That's whole idea behind data \rcompression in the first place.) For the less frequent long symbols, there \rwill be two lookups. If you had a compression method with really long \rsymbols, you could have as many levels of lookups as is efficient. For \rinflate, two is enough.\r\rSo a table entry either points to another table (in which case nine bits in \rthe above example are gobbled), or it contains the translation for the symbol \rand the number of bits to gobble. Then you start again with the next \rungobbled bit.\r\rYou may wonder: why not just have one lookup table for how ever many bits the \rlongest symbol is? The reason is that if you do that, you end up spending \rmore time filling in duplicate symbol entries than you do actually decoding. \rAt least for deflate's output that generates new trees every several 10's of \rkbytes. You can imagine that filling in a 2^15 entry table for a 15-bit code \rwould take too long if you're only decoding several thousand symbols. At the \rother extreme, you could make a new table for every bit in the code. In fact,\rthat's essentially a Huffman tree. But then you spend two much time \rtraversing the tree while decoding, even for short symbols.\r\rSo the number of bits for the first lookup table is a trade of the time to \rfill out the table vs. the time spent looking at the second level and above of\rthe table.\r\rHere is an example, scaled down:\r\rThe code being decoded, with 10 symbols, from 1 to 6 bits long:\r\rA: 0\rB: 10\rC: 1100\rD: 11010\rE: 11011\rF: 11100\rG: 11101\rH: 11110\rI: 111110\rJ: 111111\r\rLet's make the first table three bits long (eight entries):\r\r000: A,1\r001: A,1\r010: A,1\r011: A,1\r100: B,2\r101: B,2\r110: -> table X (gobble 3 bits)\r111: -> table Y (gobble 3 bits)\r\rEach entry is what the bits decode to and how many bits that is, i.e. how \rmany bits to gobble. Or the entry points to another table, with the number of\rbits to gobble implicit in the size of the table.\r\rTable X is two bits long since the longest code starting with 110 is five bits\rlong:\r\r00: C,1\r01: C,1\r10: D,2\r11: E,2\r\rTable Y is three bits long since the longest code starting with 111 is six \rbits long:\r\r000: F,2\r001: F,2\r010: G,2\r011: G,2\r100: H,2\r101: H,2\r110: I,3\r111: J,3\r\rSo what we have here are three tables with a total of 20 entries that had to \rbe constructed. That's compared to 64 entries for a single table. Or \rcompared to 16 entries for a Huffman tree (six two entry tables and one four \rentry table). Assuming that the code ideally represents the probability of \rthe symbols, it takes on the average 1.25 lookups per symbol. That's compared\rto one lookup for the single table, or 1.66 lookups per symbol for the \rHuffman tree.\r\rThere, I think that gives you a picture of what's going on. For inflate, the \rmeaning of a particular symbol is often more than just a letter. It can be a \rbyte (a "literal"), or it can be either a length or a distance which \rindicates a base value and a number of bits to fetch after the code that is \radded to the base value. Or it might be the special end-of-block code. The \rdata structures created in inftrees.c try to encode all that information \rcompactly in the tables.\r\r\rJean-loup Gailly Mark Adler\rjloup@gzip.org madler@alumni.caltech.edu\r\r\rReferences:\r\r[LZ77] Ziv J., Lempel A., ``A Universal Algorithm for Sequential Data\rCompression,'' IEEE Transactions on Information Theory, Vol. 23, No. 3,\rpp. 337-343.\r\r``DEFLATE Compressed Data Format Specification'' available in\rftp://ds.internic.net/rfc/rfc1951.txt\r
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projects / apple / security.git / blobdiff