如何快速解码Huffman编码？
13
- Jichao
2
4恭喜你写出了格式良好的代码，这让WinAPI代码更加易于理解。 - Manuel
2FYI：现代解码器在当前普通PC上可以达到每秒约1GB的解压速度（或更快）：您必须将哈夫曼编码长度限制为11/12，使用解压缩表，并使用多个哈夫曼流，以最小化数据依赖性。 - geza
6个回答
22
优化二叉树方法的一种方式是使用查找表。您可以按照这样的方式排列表格，以便您可以直接查找特定的编码位模式，从而允许任何代码的最大可能位宽。
由于大多数代码不使用完整的最大宽度，它们在表格中的多个位置被包含 - 每个未使用位组合一个位置。该表格指示要从输入中丢弃多少位以及解码输出。
如果最长的代码太长，因此表格不切实际，那么妥协方案是使用具有更小固定宽度下标查找的树。例如，您可以使用256项表来处理字节。如果输入代码超过8位，则表条目指示解码不完整，并将您引导到处理下一个最多8位的表中。更大的表格会以内存换速度 - 256个项目可能太小。
我相信这种通用方法称为“前缀表”，这就是BobMcGees引用代码所做的事情。可能的区别在于，某些压缩算法需要在解压缩期间更新前缀表 - 这对于简单的Huffman不是必需的。我IRC是在一本关于位图图形文件格式（其中包括GIF）的书中第一次看到它，之前是专利恐慌。
从二叉树模型中预计算出一个完整的查找表、哈希表等价物或小表格树应该很容易。二叉树仍然是代码工作方式的关键表示（心理模型）- 这个查找表只是一种优化实现它的方式。
- user180247
1
2+1 这是我在 JPEG 解码器中使用的方法，我直接从图像中的 DHT 段构建查找表，无需二叉树麻烦。 - matja
6
为什么不看一下GZIP源代码，特别是unpack.c中的Huffman解压缩代码？它正在做你所做的事情，但速度更快得多。从我所看到的内容来看，它使用了一个查找数组以及位移/掩码操作来加速运行。不过代码非常密集。编辑：这里是完整的源代码。
/* unpack.c -- decompress files in pack format.
* Copyright (C) 1992-1993 Jean-loup Gailly
* This is free software; you can redistribute it and/or modify it under the
* terms of the GNU General Public License, see the file COPYING.
*/
#ifdef RCSID
static char rcsid[] = "$Id: unpack.c,v 1.4 1993/06/11 19:25:36 jloup Exp $";
#endif
#include "tailor.h"
#include "gzip.h"
#include "crypt.h"
#define MIN(a,b) ((a) <= (b) ? (a) : (b))
/* The arguments must not have side effects. */
#define MAX_BITLEN 25
/* Maximum length of Huffman codes. (Minor modifications to the code
* would be needed to support 32 bits codes, but pack never generates
* more than 24 bits anyway.)
*/
#define LITERALS 256
/* Number of literals, excluding the End of Block (EOB) code */
#define MAX_PEEK 12
/* Maximum number of 'peek' bits used to optimize traversal of the
* Huffman tree.
*/
local ulg orig_len; /* original uncompressed length */
local int max_len; /* maximum bit length of Huffman codes */
local uch literal[LITERALS];
/* The literal bytes present in the Huffman tree. The EOB code is not
* represented.
*/
local int lit_base[MAX_BITLEN+1];
/* All literals of a given bit length are contiguous in literal[] and
* have contiguous codes. literal[code+lit_base[len]] is the literal
* for a code of len bits.
*/
local int leaves [MAX_BITLEN+1]; /* Number of leaves for each bit length */
local int parents[MAX_BITLEN+1]; /* Number of parents for each bit length */
local int peek_bits; /* Number of peek bits currently used */
/* local uch prefix_len[1 << MAX_PEEK]; */
#define prefix_len outbuf
/* For each bit pattern b of peek_bits bits, prefix_len[b] is the length
* of the Huffman code starting with a prefix of b (upper bits), or 0
* if all codes of prefix b have more than peek_bits bits. It is not
* necessary to have a huge table (large MAX_PEEK) because most of the
* codes encountered in the input stream are short codes (by construction).
* So for most codes a single lookup will be necessary.
*/
#if (1<<MAX_PEEK) > OUTBUFSIZ
error cannot overlay prefix_len and outbuf
#endif
local ulg bitbuf;
/* Bits are added on the low part of bitbuf and read from the high part. */
local int valid; /* number of valid bits in bitbuf */
/* all bits above the last valid bit are always zero */
/* Set code to the next 'bits' input bits without skipping them. code
* must be the name of a simple variable and bits must not have side effects.
* IN assertions: bits <= 25 (so that we still have room for an extra byte
* when valid is only 24), and mask = (1<<bits)-1.
*/
#define look_bits(code,bits,mask) \
{ \
while (valid < (bits)) bitbuf = (bitbuf<<8) | (ulg)get_byte(), valid += 8; \
code = (bitbuf >> (valid-(bits))) & (mask); \
}
/* Skip the given number of bits (after having peeked at them): */
#define skip_bits(bits) (valid -= (bits))
#define clear_bitbuf() (valid = 0, bitbuf = 0)
/* Local functions */
local void read_tree OF((void));
local void build_tree OF((void));
/* ===========================================================================
* Read the Huffman tree.
*/
local void read_tree()
{
int len; /* bit length */
int base; /* base offset for a sequence of leaves */
int n;
/* Read the original input size, MSB first */
orig_len = 0;
for (n = 1; n <= 4; n++) orig_len = (orig_len << 8) | (ulg)get_byte();
max_len = (int)get_byte(); /* maximum bit length of Huffman codes */
if (max_len > MAX_BITLEN) {
error("invalid compressed data -- Huffman code > 32 bits");
}
/* Get the number of leaves at each bit length */
n = 0;
for (len = 1; len <= max_len; len++) {
leaves[len] = (int)get_byte();
n += leaves[len];
}
if (n > LITERALS) {
error("too many leaves in Huffman tree");
}
Trace((stderr, "orig_len %ld, max_len %d, leaves %d\n",
orig_len, max_len, n));
/* There are at least 2 and at most 256 leaves of length max_len.
* (Pack arbitrarily rejects empty files and files consisting of
* a single byte even repeated.) To fit the last leaf count in a
* byte, it is offset by 2. However, the last literal is the EOB
* code, and is not transmitted explicitly in the tree, so we must
* adjust here by one only.
*/
leaves[max_len]++;
/* Now read the leaves themselves */
base = 0;
for (len = 1; len <= max_len; len++) {
/* Remember where the literals of this length start in literal[] : */
lit_base[len] = base;
/* And read the literals: */
for (n = leaves[len]; n > 0; n--) {
literal[base++] = (uch)get_byte();
}
}
leaves[max_len]++; /* Now include the EOB code in the Huffman tree */
}
/* ===========================================================================
* Build the Huffman tree and the prefix table.
*/
local void build_tree()
{
int nodes = 0; /* number of nodes (parents+leaves) at current bit length */
int len; /* current bit length */
uch *prefixp; /* pointer in prefix_len */
for (len = max_len; len >= 1; len--) {
/* The number of parent nodes at this level is half the total
* number of nodes at parent level:
*/
nodes >>= 1;
parents[len] = nodes;
/* Update lit_base by the appropriate bias to skip the parent nodes
* (which are not represented in the literal array):
*/
lit_base[len] -= nodes;
/* Restore nodes to be parents+leaves: */
nodes += leaves[len];
}
/* Construct the prefix table, from shortest leaves to longest ones.
* The shortest code is all ones, so we start at the end of the table.
*/
peek_bits = MIN(max_len, MAX_PEEK);
prefixp = &prefix_len[1<<peek_bits];
for (len = 1; len <= peek_bits; len++) {
int prefixes = leaves[len] << (peek_bits-len); /* may be 0 */
while (prefixes--) *--prefixp = (uch)len;
}
/* The length of all other codes is unknown: */
while (prefixp > prefix_len) *--prefixp = 0;
}
/* ===========================================================================
* Unpack in to out. This routine does not support the old pack format
* with magic header \037\037.
*
* IN assertions: the buffer inbuf contains already the beginning of
* the compressed data, from offsets inptr to insize-1 included.
* The magic header has already been checked. The output buffer is cleared.
*/
int unpack(in, out)
int in, out; /* input and output file descriptors */
{
int len; /* Bit length of current code */
unsigned eob; /* End Of Block code */
register unsigned peek; /* lookahead bits */
unsigned peek_mask; /* Mask for peek_bits bits */
ifd = in;
ofd = out;
read_tree(); /* Read the Huffman tree */
build_tree(); /* Build the prefix table */
clear_bitbuf(); /* Initialize bit input */
peek_mask = (1<<peek_bits)-1;
/* The eob code is the largest code among all leaves of maximal length: */
eob = leaves[max_len]-1;
Trace((stderr, "eob %d %x\n", max_len, eob));
/* Decode the input data: */
for (;;) {
/* Since eob is the longest code and not shorter than max_len,
* we can peek at max_len bits without having the risk of reading
* beyond the end of file.
*/
look_bits(peek, peek_bits, peek_mask);
len = prefix_len[peek];
if (len > 0) {
peek >>= peek_bits - len; /* discard the extra bits */
} else {
/* Code of more than peek_bits bits, we must traverse the tree */
ulg mask = peek_mask;
len = peek_bits;
do {
len++, mask = (mask<<1)+1;
look_bits(peek, len, mask);
} while (peek < (unsigned)parents[len]);
/* loop as long as peek is a parent node */
}
/* At this point, peek is the next complete code, of len bits */
if (peek == eob && len == max_len) break; /* end of file? */
put_ubyte(literal[peek+lit_base[len]]);
Tracev((stderr,"%02d %04x %c\n", len, peek,
literal[peek+lit_base[len]]));
skip_bits(len);
} /* for (;;) */
flush_window();
Trace((stderr, "bytes_out %ld\n", bytes_out));
if (orig_len != (ulg)bytes_out) {
error("invalid compressed data--length error");
}
return OK;
}
- BobMcGee
4
典型的解压Huffman编码的方法是使用二叉树。您将代码插入树中，以便代码中的每个位表示一个分支，要么向左（0），要么向右（1），在叶子中包含解码字节（或任何其他值）。
然后，解码只是读取编码内容中的位，对于每个位遍历树。当到达叶子时，发出已解码的值，并继续读取，直到输入用尽。
更新：this page 描述了该技术，并具有华丽的图形。
然后，解码只是读取编码内容中的位，对于每个位遍历树。当到达叶子时，发出已解码的值，并继续读取，直到输入用尽。
更新：this page 描述了该技术，并具有华丽的图形。
- unwind
8
4链式结构最适合构建树，但我认为数组在执行查找时更好。它不必完全填充，但将16位键映射到一个65536节点的数组中应该能够良好地执行。注：我从未编写过解压缩程序。 - Potatoswatter
@Potatoswatter：那么，当你有一个缓冲区保存着不同长度（在位级别上）的哈夫曼编码时，如何将其转换为数组索引呢？基于树的方法的重点在于它告诉你何时停止读取位。也许我只是忽略了什么，因为我已经很久没有写过哈夫曼解码器了。 - unwind
1通常情况下，您会逐个单词地将输入读入变量中，逻辑上与掩码进行AND运算以读取接下来的几位，然后在完成这些位的操作后向右移动一次（跟踪剩余有效位数）。每次跨越一个字边界时需要进行一些额外的工作。虽然比逐位读取更多代码，但速度可以显著提高。 - Mike Seymour
你所要做的就是填充查找表，其中包含所有可能的组合。因此，在一个四位表中只有3位用于字母A，您可以在附加位的所有潜在组合中找到A（010 - Code，0100 = A，0101 = A）。您使用两个表，一个用于符号（或下一个要使用的查找表），另一个用于实际使用的位数。因此，0100 =（A，3），0101 =（A，3）。这样就不需要AND操作。 - Martin Kersten
一个C++的两级表查找已经被实现为实用类，可以在这里找到：https://github.com/mdejong/MetalHuffman - MoDJ
显示剩余3条评论
2
您可以在通常的Huffmann树查找上执行一种批量查找：
1.选择位深度（称其为深度n）；这是速度，内存和构建表所需的时间之间的权衡； 2.为长度为n的所有2 ^ n位字符串构建查找表。每个条目可以编码多个完整令牌;通常还会有一些剩余的位，只是Huffman代码的前缀：对于这些位，将其链接到进一步的查找表以获取该代码; 3.构建进一步的查找表。总表数最多比Huffmann树中编码的条目少一个。
选择深度是4的倍数，例如深度8，非常适合位移操作。
后记：这与potatoswatter在unwind的答案中提出的想法以及Steve314的答案不同，因为它使用了多个表：这意味着所有n位查找都得到了利用，因此应该更快，但使表格构建和查找显着棘手，并且对于给定深度将消耗更多的空间。
1.选择位深度（称其为深度n）；这是速度，内存和构建表所需的时间之间的权衡； 2.为长度为n的所有2 ^ n位字符串构建查找表。每个条目可以编码多个完整令牌;通常还会有一些剩余的位，只是Huffman代码的前缀：对于这些位，将其链接到进一步的查找表以获取该代码; 3.构建进一步的查找表。总表数最多比Huffmann树中编码的条目少一个。
选择深度是4的倍数，例如深度8，非常适合位移操作。
后记：这与potatoswatter在unwind的答案中提出的想法以及Steve314的答案不同，因为它使用了多个表：这意味着所有n位查找都得到了利用，因此应该更快，但使表格构建和查找显着棘手，并且对于给定深度将消耗更多的空间。
- Charles Stewart
0
为什么不在同一源代码模块中使用解压算法呢？它看起来是一个不错的算法。
- wallyk
1
3这不是那位发帖者提到的解压缩器吗？他说需要6个小时才能解压缩这个3MB的文件？ - Michael Burr
0
其他答案都是正确的，但这里是我最近用Rust编写的一些代码，以使思想具体化。这是关键程序：
fn decode( &self, input: &mut InpBitStream ) -> usize
{
let mut sym = self.lookup[ input.peek( self.peekbits ) ];
if sym >= self.ncode
{
sym = self.lookup[ sym - self.ncode + ( input.peek( self.maxbits ) >> self.peekbits ) ];
}
input.advance( self.nbits[ sym ] as usize );
sym
}
棘手的部分是设置查找表，可以在这个完整的 RFC 1951 Rust 解码器中查看 BitDecoder::setup_code：
// RFC 1951 inflate ( de-compress ).
pub fn inflate( data: &[u8] ) -> Vec<u8>
{
let mut inp = InpBitStream::new( &data );
let mut out = Vec::new();
let _chk = inp.get_bits( 16 ); // Checksum
loop
{
let last = inp.get_bit();
let btype = inp.get_bits( 2 );
match btype
{
2 => { do_dyn( &mut inp, &mut out ); }
1 => { do_fixed( &mut inp, &mut out ); }
0 => { do_copy( &mut inp, &mut out ); }
_ => { }
}
if last != 0 { break; }
}
out
}
fn do_dyn( inp: &mut InpBitStream, out: &mut Vec<u8> )
{
let n_lit_code = 257 + inp.get_bits( 5 );
let n_dist_code = 1 + inp.get_bits( 5 );
let n_len_code = 4 + inp.get_bits( 4 );
let mut len = LenDecoder::new( inp, n_len_code );
let mut lit = BitDecoder::new( n_lit_code );
len.get_lengths( inp, &mut lit.nbits );
lit.init();
let mut dist = BitDecoder::new( n_dist_code );
len.get_lengths( inp, &mut dist.nbits );
dist.init();
loop
{
let x = lit.decode( inp );
match x
{
0..=255 => { out.push( x as u8 ); }
256 => { break; }
_ =>
{
let mc = x - 257;
let length = MATCH_OFF[ mc ] + inp.get_bits( MATCH_EXTRA[ mc ] as usize );
let dc = dist.decode( inp );
let distance = DIST_OFF[ dc ] + inp.get_bits( DIST_EXTRA[ dc ] as usize );
copy( out, distance, length );
}
}
}
} // end do_dyn
fn copy( out: &mut Vec<u8>, distance: usize, mut length: usize )
{
let mut i = out.len() - distance;
while length > 0
{
out.push( out[ i ] );
i += 1;
length -= 1;
}
}
/// Decode length-limited Huffman codes.
struct BitDecoder
{
ncode: usize,
nbits: Vec<u8>,
maxbits: usize,
peekbits: usize,
lookup: Vec<usize>
}
impl BitDecoder
{
fn new( ncode: usize ) -> BitDecoder
{
BitDecoder
{
ncode,
nbits: vec![0; ncode],
maxbits: 0,
peekbits: 0,
lookup: Vec::new()
}
}
/// The key routine, will be called many times.
fn decode( &self, input: &mut InpBitStream ) -> usize
{
let mut sym = self.lookup[ input.peek( self.peekbits ) ];
if sym >= self.ncode
{
sym = self.lookup[ sym - self.ncode + ( input.peek( self.maxbits ) >> self.peekbits ) ];
}
input.advance( self.nbits[ sym ] as usize );
sym
}
fn init( &mut self )
{
let ncode = self.ncode;
let mut max_bits : usize = 0;
for bp in &self.nbits
{
let bits = *bp as usize;
if bits > max_bits { max_bits = bits; }
}
self.maxbits = max_bits;
self.peekbits = if max_bits > 8 { 8 } else { max_bits };
self.lookup.resize( 1 << self.peekbits, 0 );
// Code below is from rfc1951 page 7
let mut bl_count : Vec<usize> = vec![ 0; max_bits + 1 ]; // the number of codes of length N, N >= 1.
for i in 0..ncode { bl_count[ self.nbits[i] as usize ] += 1; }
let mut next_code : Vec<usize> = vec![ 0; max_bits + 1 ];
let mut code = 0;
bl_count[0] = 0;
for i in 0..max_bits
{
code = ( code + bl_count[i] ) << 1;
next_code[ i + 1 ] = code;
}
for i in 0..ncode
{
let len = self.nbits[ i ] as usize;
if len != 0
{
self.setup_code( i, len, next_code[ len ] );
next_code[ len ] += 1;
}
}
}
// Decoding is done using self.lookup ( see decode ). To keep the lookup table small,
// codes longer than 8 bits are looked up in two peeks.
fn setup_code( &mut self, sym: usize, len: usize, mut code: usize )
{
if len <= self.peekbits
{
let diff = self.peekbits - len;
for i in code << diff .. (code << diff) + (1 << diff)
{
// bits are reversed to match InpBitStream::peek
let r = reverse( i, self.peekbits );
self.lookup[ r ] = sym;
}
} else {
// Secondary lookup required.
let peekbits2 = self.maxbits - self.peekbits;
// Split code into peekbits portion ( key ) and remainder ( code).
let diff1 = len - self.peekbits;
let key = code >> diff1;
code &= ( 1 << diff1 ) - 1;
// Get the secondary lookup.
let kr = reverse( key, self.peekbits );
let mut base = self.lookup[ kr ];
if base == 0 // Secondary lookup not yet allocated for this key.
{
base = self.lookup.len();
self.lookup.resize( base + ( 1 << peekbits2 ), 0 );
self.lookup[ kr ] = self.ncode + base;
} else {
base -= self.ncode;
}
// Set the secondary lookup values.
let diff = self.maxbits - len;
for i in code << diff .. (code << diff) + (1<<diff)
{
let r = reverse( i, peekbits2 );
self.lookup[ base + r ] = sym;
}
}
}
} // end impl BitDecoder
struct InpBitStream<'a>
{
data: &'a [u8],
pos: usize,
buf: usize,
got: usize, // Number of bits in buffer.
}
impl <'a> InpBitStream<'a>
{
fn new( data: &'a [u8] ) -> InpBitStream
{
InpBitStream { data, pos: 0, buf: 1, got: 0 }
}
fn peek( &mut self, n: usize ) -> usize
{
while self.got < n
{
if self.pos < self.data.len()
{
self.buf |= ( self.data[ self.pos ] as usize ) << self.got;
}
self.pos += 1;
self.got += 8;
}
self.buf & ( ( 1 << n ) - 1 )
}
fn advance( &mut self, n:usize )
{
self.buf >>= n;
self.got -= n;
}
fn get_bit( &mut self ) -> usize
{
if self.got == 0 { self.peek( 1 ); }
let result = self.buf & 1;
self.advance( 1 );
result
}
fn get_bits( &mut self, n: usize ) -> usize
{
let result = self.peek( n );
self.advance( n );
result
}
fn get_huff( &mut self, mut n: usize ) -> usize
{
let mut result = 0;
while n > 0
{
result = ( result << 1 ) + self.get_bit();
n -= 1;
}
result
}
fn clear_bits( &mut self )
{
self.got = 0;
}
} // end impl InpBitStream
/// Decode code lengths.
struct LenDecoder
{
plenc: u8, // previous length code ( which can be repeated )
rep: usize, // repeat
bd: BitDecoder,
}
/// Decodes an array of lengths. There are special codes for repeats, and repeats of zeros.
impl LenDecoder
{
fn new( inp: &mut InpBitStream, n_len_code: usize ) -> LenDecoder
{
let mut result = LenDecoder { plenc: 0, rep:0, bd: BitDecoder::new( 19 ) };
// Read the array of 3-bit code lengths from input.
for i in 0..n_len_code
{
result.bd.nbits[ CLEN_ALPHABET[i] as usize ] = inp.get_bits(3) as u8;
}
result.bd.init();
result
}
// Per RFC1931 page 13, get array of code lengths.
fn get_lengths( &mut self, inp: &mut InpBitStream, result: &mut Vec<u8> )
{
let n = result.len();
let mut i = 0;
while self.rep > 0 { result[i] = self.plenc; i += 1; self.rep -= 1; }
while i < n
{
let lenc = self.bd.decode( inp ) as u8;
if lenc < 16
{
result[i] = lenc;
i += 1;
self.plenc = lenc;
} else {
if lenc == 16 { self.rep = 3 + inp.get_bits(2); }
else if lenc == 17 { self.rep = 3 + inp.get_bits(3); self.plenc=0; }
else if lenc == 18 { self.rep = 11 + inp.get_bits(7); self.plenc=0; }
while i < n && self.rep > 0 { result[i] = self.plenc; i += 1; self.rep -= 1; }
}
}
} // end get_lengths
} // end impl LenDecoder
/// Reverse a string of bits.
pub fn reverse( mut x:usize, mut bits: usize ) -> usize
{
let mut result: usize = 0;
while bits > 0
{
result = ( result << 1 ) | ( x & 1 );
x >>= 1;
bits -= 1;
}
result
}
fn do_copy( inp: &mut InpBitStream, out: &mut Vec<u8> )
{
inp.clear_bits(); // Discard any bits in the input buffer
let mut n = inp.get_bits( 16 );
let _n1 = inp.get_bits( 16 );
while n > 0 { out.push( inp.data[ inp.pos ] ); n -= 1; inp.pos += 1; }
}
fn do_fixed( inp: &mut InpBitStream, out: &mut Vec<u8> ) // RFC1951 page 12.
{
loop
{
// 0 to 23 ( 7 bits ) => 256 - 279; 48 - 191 ( 8 bits ) => 0 - 143;
// 192 - 199 ( 8 bits ) => 280 - 287; 400..511 ( 9 bits ) => 144 - 255
let mut x = inp.get_huff( 7 );
if x <= 23
{
x += 256;
} else {
x = ( x << 1 ) + inp.get_bit();
if x <= 191 { x -= 48; }
else if x <= 199 { x += 88; }
else { x = ( x << 1 ) + inp.get_bit() - 256; }
}
match x
{
0..=255 => { out.push( x as u8 ); }
256 => { break; }
_ => // 257 <= x && x <= 285
{
x -= 257;
let length = MATCH_OFF[x] + inp.get_bits( MATCH_EXTRA[ x ] as usize );
let dcode = inp.get_huff( 5 );
let distance = DIST_OFF[dcode] + inp.get_bits( DIST_EXTRA[dcode] as usize );
copy( out, distance, length );
}
}
}
} // end do_fixed
// RFC 1951 constants.
pub static CLEN_ALPHABET : [u8; 19] = [ 16, 17, 18, 0, 8, 7, 9, 6, 10, 5, 11, 4, 12, 3, 13, 2, 14, 1, 15 ];
pub static MATCH_OFF : [usize; 30] = [ 3,4,5,6, 7,8,9,10, 11,13,15,17, 19,23,27,31, 35,43,51,59,
67,83,99,115, 131,163,195,227, 258, 0xffff ];
pub static MATCH_EXTRA : [u8; 29] = [ 0,0,0,0, 0,0,0,0, 1,1,1,1, 2,2,2,2, 3,3,3,3, 4,4,4,4, 5,5,5,5, 0 ];
pub static DIST_OFF : [usize; 30] = [ 1,2,3,4, 5,7,9,13, 17,25,33,49, 65,97,129,193, 257,385,513,769,
1025,1537,2049,3073, 4097,6145,8193,12289, 16385,24577 ];
pub static DIST_EXTRA : [u8; 30] = [ 0,0,0,0, 1,1,2,2, 3,3,4,4, 5,5,6,6, 7,7,8,8, 9,9,10,10, 11,11,12,12, 13,13 ];
- George Barwood
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