zip压缩算法分析(2)
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zip压缩算法分析(2)
前言
在zip压缩算法分析(1)中已经分析了利用文本中短语重复的特性来进行压缩的lz77算法部分,接下来分析利用信息熵进行压缩的huffman编码算法,zip作者在这里对霍夫曼树的处理十分精彩,huffman树本身就已经很cool了,简直无法想象还能有这么流弊的改进。
gzip源码下载地址 参考资料 1 参考资料 2
huffman编码是很有名的一种压缩编码方式,网上,书上都有很多描述;其主要思想如下:
给出文本中所有字符c与各个字符出现频率f组成的字符表S[2, n],比起定长编码,一个比较明智的编码方法是将频率大的赋予较短的编码,频率小的则赋予较长的编码。那么究竟该如何分配呢?
1.编码树的概念。
为了便于解码,这里只使用前缀码(从实际含义上来讲,应该是无前缀码,但是由于权威文献中都使用这种坑爹翻译就只好随大流了),原因如图:
如果字符x的编码成为了字符y编码的一个前缀,那么很明显地,在解码时可能无法分辨是x还是y。考虑如图的一棵树:
所有字符与其在文本中出现的频率构成这棵树的所有叶节点,由叶节点到根节点的路径决定了该字符的编码,因此编码树就成为了表示前缀编码方案的一个极好的模型:对于2个不同叶节点,不可能存在一个编码为另一个编码前缀的情形。
2.Huffman编码算法
Huffman给出了一个构造最优编码树的贪心算法,其正确性证明见《算法导论》(第三版)16章第三节。算法如下:
Huffman(C) //C 为带频率的字母表n = |C|; // n = C 字符数目Q = C; //用C初始化一个最小优先队列Qfor i = 1 to n - 1 allocate a new node z z.left = x = EXTRACT-MIN(Q) //取出最小节点 z.right = y = EXTRACT-MIN(Q) z.freq = x.freq + y.freq INSERT(Q, z)return EXTRACT-MIN(Q)主要思想是用最小优先队列装载字母表,每次循环将频率最小的2个节点取出,合并为一个新节点,再插入队列,直到队列只剩一个节点为止。
霍夫曼编码的原理讲解到此打住,接下来说一说zip压缩算法为何使用 以及如何实现霍夫曼树的。
zip压缩算法分析(1)中分析的lz77压缩算法部分已经利用重复出现的短语进行了压缩,短语式重复的倾向已被破坏,无再次进行lz77的条件。但此时压缩后输出的几种形式的压缩数据:(distance, length), literal仍然在信息熵方面上有压缩的可能,就是说,不同的distance,length, literal在lz77输出流中的频率可能存在较大差距,这样就可以用huffman编码继续对其进行压缩。
huffman编码的主要部分位于tree.c文件中,既然是源码分析,接下来就上码吧:
头部注释
/* trees.c -- output deflated data using Huffman coding * trees.c -- 对deflate()处理过后的数据使用霍夫曼编码 * 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. *//* * PURPOSE * * Encode various sets of source values using variable-length * binary code trees. * 使用二进制变长编码树来编码各种值。 * * DISCUSSION * * The PKZIP "deflation" process uses several Huffman trees. The more * common source values are represented by shorter bit sequences. * PKZIP的"deflation"算法使用了一些霍夫曼树,出现频率越高的值用越短的bit序列编码。 * * Each code tree is stored in the ZIP file in a compressed form * which is itself a Huffman encoding of the lengths of * all the code strings (in ascending order by source values). * (此处更倾向于意译而非直译,因为我觉得理解这几句话比看懂这个句子更重要) * 编码树的实现方式是:按照源数据(即lz77压缩后输出的length, distance, literal) * 所有可能值从小到大顺序,将它们的霍夫曼码长(只记录长度!很神奇吧)记录在数组里。 * The actual code strings are reconstructed from the lengths in * the UNZIP process, as described in the "application note" * (APPNOTE.TXT) distributed as part of PKWARE's PKZIP program. * * REFERENCES * 一些相关算法知识的参考书籍 * Lynch, Thomas J. * Data Compression: Techniques and Applications, pp. 53-55. * Lifetime Learning Publications, 1985. ISBN 0-534-03418-7. * * Storer, James A. * Data Compression: Methods and Theory, pp. 49-50. * Computer Science Press, 1988. ISBN 0-7167-8156-5. * * Sedgewick, R. * Algorithms, p290. * Addison-Wesley, 1983. ISBN 0-201-06672-6. * * INTERFACE * 3个接口 * 1.初始化函数 * void ct_init (ush *attr, int *methodp) * Allocate the match buffer, initialize the various tables and save * the location of the internal file attribute (ascii/binary) and * method (DEFLATE/STORE) * 2.计数器函数 * void ct_tally (int dist, int lc); * Save the match info and tally the frequency counts. * 3. * long flush_block (char *buf, ulg stored_len, int eof) * Determine the best encoding for the current block: dynamic trees, * static trees or store, and output the encoded block to the zip * file. Returns the total compressed length for the file so far. * */
和上篇文章一样,分析是以3个interface函数为主要对象的,在分析源码的同时补充相关信息。
之所以这么安排是因为前面的常量和宏太尼玛多了,全部列出来,谁记得住啊?只有从对函数
的解读中才比较方便认识理解这些常量,宏。毕竟这些量说到底还是为了函数而存在的。
函数ct_init
代码:
/* =========================================================================== * Allocate the match buffer, initialize the various tables and save the * location of the internal file attribute (ascii/binary) and method * (DEFLATE/STORE). * 为匹配块收集空间,初始化变量表并存储文件属性(ASCII编码/二进制编码)与压缩方案 * (压缩/仅存储) */void ct_init(attr, methodp) ush *attr; /* pointer to internal file attribute */ int *methodp; /* pointer to compression method */{ int n; /* iterates over tree elements */ int bits; /* bit counter */ int length; /* length value */ int code; /* code value */ int dist; /* distance index */ file_type = attr; file_method = methodp; compressed_len = input_len = 0L; if (static_dtree[0].Len != 0) return; /* ct_init already called */ /* Initialize the mapping length (0..255) -> length code (0..28) * 初始化映射 0到255的length值 -> 30个length区间码 * 这个区间压缩方案实在太厉害了,详情见该函数源码下方注1。 */ length = 0; for (code = 0; code < LENGTH_CODES-1; code++) { base_length[code] = length; for (n = 0; n < (1<<extra_lbits[code]); n++) { length_code[length++] = (uch)code; } } Assert (length == 256, "ct_init: length != 256"); /* Note that the length 255 (match length 258) can be represented * in two different ways: code 284 + 5 bits or code 285, so we * overwrite length_code[255] to use the best encoding: * 刚好有一个剩下的code,分配给length 255(+ 3 = 258) */ length_code[length-1] = (uch)code; /* Initialize the mapping dist (0..32K) -> dist code (0..29) * 初始化映射 所有0 ~ 32k(16bit)distance值 -> 30个区间码, * 详情见该函数源码下方注2。 */ dist = 0; //此处的循环与前面length数据处理方法类似。 for (code = 0 ; code < 16; code++) { base_dist[code] = dist; for (n = 0; n < (1<<extra_dbits[code]); n++) { dist_code[dist++] = (uch)code; } } Assert (dist == 256, "ct_init: dist != 256"); //此处的处理很微妙,见下方注2。 dist >>= 7; /* from now on, all distances are divided by 128 */ for ( ; code < D_CODES; code++) { base_dist[code] = dist << 7; for (n = 0; n < (1<<(extra_dbits[code]-7)); n++) { dist_code[256 + dist++] = (uch)code; } } Assert (dist == 256, "ct_init: 256+dist != 512"); //初始化静态literal树中各值对应编码的长度, //静态意为所有literal字符值都给出默认的编码 /* Construct the codes of the static literal tree */ for (bits = 0; bits <= MAX_BITS; bits++) bl_count[bits] = 0; n = 0; while (n <= 143) static_ltree[n++].Len = 8, bl_count[8]++; while (n <= 255) static_ltree[n++].Len = 9, bl_count[9]++; while (n <= 279) static_ltree[n++].Len = 7, bl_count[7]++; while (n <= 287) static_ltree[n++].Len = 8, bl_count[8]++; /* Codes 286 and 287 do not exist, but we must include them in the * tree construction to get a canonical Huffman tree (longest code * all ones) */ //调用gen_codes为已经分配好长度的静态literal树生成编码, //gen_codes是一个重要函数,看懂就能理解为什么这里只需记录编码长度的分布 //而不需记录所有的编码值了。详情见后面对gen_codes的专门分析。 gen_codes((ct_data near *)static_ltree, L_CODES+1); /* The static distance tree is trivial: * 静态distance树的初始化较为简单,就不用gen_codes了, * 注意要用bi_reverse将编码方向,这是为了便于输出,详情见注3。 */ for (n = 0; n < D_CODES; n++) { static_dtree[n].Len = 5; static_dtree[n].Code = bi_reverse(n, 5); } /* Initialize the first block of the first file: * 初始化第一个文件的第一个压缩数据块,函数 init_block 详情见该函数源码下方注4。 */ init_block();}
注1 length与literal 数据的组织方式
lz77输出的3种数据类型:
length, literal, distance
length: 匹配串的长度,限制在 [MIN_MATCH = 3, MAX_MATCH = 258]区间范围内;
literal: 未匹配的字符或匹配串的首字符,按ASCII表来说取值在0~256之间
distance: 一对匹配串之间的距离
先介绍几个常量和数据块:
#define LENGTH_CODES 29/* number of length codes, not counting the special END_BLOCK code * length 区间码的个数,还有一个特殊的结束符END_BLOCK没有算上*/#define LITERALS 256/* number of literal bytes 0..255 * literal 字符的个数(ASCII表 256个值对应256个字符)*/#define L_CODES (LITERALS+1+LENGTH_CODES)/* number of Literal or Length codes, including the END_BLOCK code * literal 和 length 编码的总个数,包括结束符END_BLOCK在内。*/#define HEAP_SIZE (2*L_CODES+1)/* maximum heap size * 为霍夫曼编码提供足够的空间,装有字符信息的叶节点L_CODES个,但还需要额外的* 内部节点,最多L_CODES + 1个,详细情况后面的几个造树函数会解释。*/typedef static locallocal ct_data near dyn_ltree[HEAP_SIZE]; /* literal and length tree * length & literal动态编码树,literal 和 length 数据被存在同一个树中*/local ct_data near static_ltree[L_CODES+2];/* The static literal tree. Since the bit lengths are imposed, there is no * need for the L_CODES extra codes used during heap construction. However * The codes 286 and 287 are needed to build a canonical tree (see ct_init * below). * length & literal静态编码树,由于编码方案已经钦定了,所以不用再特意多出L_CODES空间来 * 构建最小堆进行多余的霍夫曼编码。 */
literal的值范围有ASCII编码表决定,[0, 255];
而length值范围则为MIN_MATCH = 3 ~ MAX_MATCH = 258;恰好都是256个值;
literal 与 length共存在同一棵静态编码树或动态编码树中,
以动态编码树为例,
local ct_data near dyn_ltree[HEAP_SIZE]
该树包含了所有的length与literal数据
每一个length或literal或distance数据都装在
结构体 ct_data
/* Data structure describing a single value and its code string. */typedef struct ct_data { union { ush freq; /* 出现的频率 */ ush code; /* huffman编码 */ } fc; union { ush dad; /* huffman树中该数据的父节点 */ ush len; /* huffman编码的长度 */ } dl;} ct_data;
literal直接用原本的值表示,0 ~ 255;256表示结束符END_BLOCK
那么,既然length有256种不同取值,为何只有LENGTH_CODES = 29个编码呢?
这里就要说到一个重要的概念了:
区间码
将length 分为29个区间,257 ~ 285这29个值作为区间的编号,我们只统计落在
各区间内的length数目,在把每个区间看做节点,进行霍夫曼编码。真他妈聪明。
具体的分法如下图,
图源:《Data Compression The Complete Reference》
code为各区间的标号,bits则表示区间内可以自由取值的bit位数,
例如
区间[35, 42] 或 [43 , 50]之间的值有8 = 2 ^ (bit位= 3) 个。这些多出的bit位的信息
存放在数组extra_lbits[LENGH_CODES]中。
local int near extra_lbits[LENGTH_CODES] /* extra bits for each length code */ = {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};
好,说完这些原理后再回过头来看建立length实际值到length区间映射的代码
length = 0; for (code = 0; code < LENGTH_CODES-1; code++) { base_length[code] = length; for (n = 0; n < (1<<extra_lbits[code]); n++) { length_code[length++] = (uch)code; } }
明显地,code为区间的编号,length为实际的length值,
base_length[code]为code编号区间的起始值,
extra_lbit记录该区间的bit长度
而length_code[length]记载了length值所位于的区间,
程序员前辈应该不难看懂,
1 << extra_lbits[code]表示2 ^ (extra_lbits[code]),即该区间的长度。
@注2 distance数据的组织形式**
distance 与前面length类似,也分成一个个区间来编号,
先介绍几个相关的量:
#define D_CODES 30/* number of distance codes* 30个distance区间 */local uch dist_code[512];/* distance codes. The first 256 values correspond to the distances * 3 .. 258, the last 256 values correspond to the top 8 bits of * the 15 bit distances. * distance码,distance_code[distance]为distance所在区间的编号, * 前256个下标值对应distance值3,...,258,后256个值对应大于256的 * distance值的前8bit位(二进制位)。 */ local int near base_dist[D_CODES];/* First normalized distance for each code (0 = distance of 1) * 装载各区间最左端的值,作为其“基数” */
distance的存储方式与length类似,一共划分为30个区间,
取值范围是 3 ~ MAX_DIST,其中MAX_DIST定义在前一篇文章中有描述:
#define MAX_DIST (WSIZE-MIN_LOOKAHEAD)/* In order to simplify the code, particularly on 16 bit machines, match * distances are limited to MAX_DIST instead of WSIZE. */
由于MAX_DIST十分接近lz77窗口长度W_SIZE = 32k, 于是不妨将取值范围扩大到 0 ~ 32k;
具体的划分方案如下:
图源:《Data Compression The Complete Reference》
前一段代码还比较好理解,和length一样处理:
dist = 0; //此处的循环与前面length数据处理方法类似。 for (code = 0 ; code < 16; code++) { base_dist[code] = dist; for (n = 0; n < (1<<extra_dbits[code]); n++) { dist_code[dist++] = (uch)code; } }
而后半段就不一样了:
dist >>= 7; /* from now on, all distances are divided by 128 */ for ( ; code < D_CODES; code++) { base_dist[code] = dist << 7; for (n = 0; n < (1<<(extra_dbits[code]-7)); n++) { dist_code[256 + dist++] = (uch)code; } }
为什么要从dist = 267开始将dist右移7位呢?
考察记录每个区间bit长度的数组extra_dbits[D_CODES]:
{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}
257是16号区间的开头,可以发现从该区间开始,bit长度都不低于7,进一步说,
每个区间长度都是2 ^ 7 = 128的整数倍,用二进制表示从第16号开始各区间范围:
16 [000000010 0000001, 000000011 0000000],
17 [000000011 0000001, 000000100 0000000],
18 [000000100 0000001, 000000110 0000000],
… … … … … … … … … … …
… … … … … … … … … … …
29 [011000000 0000001, 100000000 0000000]
由此可看出,只要前9位相同,就一定在同一区间内,而不用去管后7位。
注3 函数bi_reverse
该函数实现位于文件 bits.c中
/* =========================================================================== * Reverse the first len bits of a code, using straightforward code (a faster * method would use a table)将码的前len位反转,使用的是直观易懂的代码,用表实现的话会更快。 * IN assertion: 1 <= len <= 15 输入要求:1 <= len <= 15 */unsigned bi_reverse(code, len) unsigned code; /* the value to invert */ int len; /* its bit length */{ register unsigned res = 0; do { res |= code & 1; code >>= 1, res <<= 1; } while (--len > 0); return res >> 1;}
用掩码反转位,这个算法算是烂大街了,就不多说了。
注4 函数init_block
一个工具函数。
/* =========================================================================== * Initialize a new block. * 初始化一个新的数据块。 */local void init_block(){ int n; /* iterates over tree elements */ /*初始化树 */ for (n = 0; n < L_CODES; n++) dyn_ltree[n].Freq = 0; for (n = 0; n < D_CODES; n++) dyn_dtree[n].Freq = 0; for (n = 0; n < BL_CODES; n++) bl_tree[n].Freq = 0; dyn_ltree[END_BLOCK].Freq = 1; opt_len = static_len = 0L; last_lit = last_dist = last_flags = 0; flags = 0; flag_bit = 1;}
建立huffman树
高度特化的huffman树—deflate树
不妨来思考一下huffman树的形式:
对于一组给定的输入,一共有多少颗结构不同的huffman树?
事实上,huffman算法所给出的解并非唯一的,如图:
第二层的2棵子树交换位置,得
第三层的2棵子树交换位置,得
毋庸置疑,这三棵树代表的压缩后文本总长度(即各字符频率 * 深度之和)都是一样的,
第一棵树是《算法导论》这三棵树中作为例子算出来的一棵最优树,那么按照如上所示的
交换操作,可以生成许多的最优编码方案。
但是 无论如何交换,每一层的叶节点数始终不变!,
于是,zip作者提出了一种特化编码的方法,只要知道每种长度,即每层的叶节点个数,便可以
得到一颗独特的编码树,称为deflate树。建立deflate树的过程包含在函数build_tree() 中。
建树前的准备统计频率函数ct_tally
当然,在建树之前,还必须获取length, literal, distance各种值在数据流中出现的的频率,
该过程在函数 ct_tally 内,lz77部分的函数deflate()就使用了该函数来不断地更新统计数据。ct_tally将统计数据暂时放在2个buffer中 :
//表示2个buff大小的宏,//LIT_BUFSIZE: l_buf 的大小#ifndef LIT_BUFSIZE# ifdef SMALL_MEM# define LIT_BUFSIZE 0x2000# else# ifdef MEDIUM_MEM# define LIT_BUFSIZE 0x4000# else# define LIT_BUFSIZE 0x8000# endif# endif#endif//DIST_BUFSIZE: d_buff的大小#ifndef DIST_BUFSIZE# define DIST_BUFSIZE LIT_BUFSIZE#endif//l_buf 用于临时装载literal 和 length的信息,//由于literal取值范围都是 (0, 255),所以用8bit的charunsigned char l_buf[LIT_BUFSIZE];//d_buff 用于临时装载distance的信息,//由于distance区间码取值范围是 (0, 511),所以用16bit的shortunsigned short d_buff[DIST_BUFSIZE];
这2个buffer用2个counter跟踪记录运行时的数组index
static unsigned last_lit; /* running index in l_buf */static unsigned last_dist; /* running index in d_buf */
l_buf内一个值究竟表示length还是literal,使用flag_buf数组来区分,flag_buf为unsigned char *类型数组,
里面每个char值相当于8个bool值,可以记录8次。
static uch flags; /* 当前待存入flag_buf的flag */static uch flag_bit; /* flags目前正在用的bit位 *//* bits are filled in flags starting at bit 0 (least significant). * Note: these flags are overkill in the current code since we don't * take advantage of DIST_BUFSIZE == LIT_BUFSIZE. */ local uch near flag_buf[(LIT_BUFSIZE/8)];//由于每个flags可以记录8次所以除以8/* flag_buf is a bit array distinguishing literals from lengths in * l_buf, thus indicating the presence or absence of a distance. */
这里就有一个有趣的问题了:
既然literal取值在0~255, length区间码取值在257到285,那么只要依据是否大于256就可以分出
literal和length了,那么为何又要特意用一个flag_buf呢?
—注意l_buf的类型: unsigned char *,也就是说,length存入的值事实上不是区间码,而是真实
的length值,只要你仔细观察deflate() 的这一段代码:
flush = ct_tally(strstart-1-prev_match, prev_length - MIN_MATCH);
就可得知。因此,length的取值范围为0~255, 这样做也是为了满足char 8bit的大小限制,那么这时
length就和literal在取值上无区别了。
另外还要讲一下flag的运行机制:ct_tally接受2个参数:
int dist; /* 匹配串的距离 */ int lc; /* 如果dist!=0, 为匹配串的长度,lc = match length-MIN_MATCH * 否则dist==0, 为匹配失败后输出的单个字符literal*/
假设初始化后,一组连续输入的参数值从早到晚为
{ (0, literal), (0, literal) , (dist, length), (0, literal), (dist, length), ……… };
init_block函数初始化
flags = 0000 0000, flag_bit = 0000 0001;
第一次:判断为literal, 标记为true, flag_bit左移一位
flags = flags | flag_bit = 0000 000 1 , flag_bit = 0000 00 1 0;
第二次:判断为literal, 标记为true, flag_bit左移一位
flags = flags | flag_bit = 0000 00 11, flag_bit = 0000 0 1 00;
第三次:判断为匹配串, 标记为false, flag_bit左移一位
flags = flags = 0000 00 11, flag_bit = 0000 1 000;
第四次:判断为literal, 标记为true, flag_bit左移一位
flags = flags | flag_bit = 0000 1 0 11, flag_bit = 000**1** 0000;
第五次:判断为匹配串, 标记为false, flag_bit左移一位
flags = flags = 0000 1 0 11, flag_bit = 00**1** 0 0000;
………
一直到第八次操作完,算作一个周期,将flags存入flag_buf;
接下来就可以贴上ct_tally的源代码了:
/* =========================================================================== * Save the match info and tally the frequency counts. Return true if * the current block must be flushed. * 存储本次匹配的信息并更新频率统计数据,如果目前的数据块必须要刷新的话,返回true。 */int ct_tally (dist, lc) int dist; /* 匹配串的距离 */ int lc; /* 如果dist!=0, 为匹配串的长度,lc = match length-MIN_MATCH * 否则dist==0, 为匹配失败后输出的单个字符literal*/{ //l_buf 记录下lc的值 l_buf[last_lit++] = (uch)lc; if (dist == 0) { /* 是literal,只更新动态树dyn_ltree */ dyn_ltree[lc].Freq++; } else { /* 是匹配串, lc == match length - MIN_MATCH */ dist--; /* dist = match distance - 1 */ Assert((ush)dist < (ush)MAX_DIST && (ush)lc <= (ush)(MAX_MATCH-MIN_MATCH) && (ush)d_code(dist) < (ush)D_CODES, "ct_tally: bad match"); //更新两棵动态树 //LITERALS == 256 dyn_ltree[length_code[lc]+LITERALS+1].Freq++; //宏d_code用于从distance值获取其区间码,比较trivial所以就不细究了。 dyn_dtree[d_code(dist)].Freq++; //更新d_buf d_buf[last_dist++] = (ush)dist; //记录为true flags |= flag_bit; } //左移一位 flag_bit <<= 1; /* 如果flags整个byte都记录满了,即过了8次操作 */ if ((last_lit & 7) == 0) { flag_buf[last_flags++] = flags; flags = 0, flag_bit = 1; } /* Try to guess if it is profitable to stop the current block here * 判断是不是该输出这个数据块的压缩结果 */ if (level > 2 && (last_lit & 0xfff) == 0) { /* Compute an upper bound for the compressed length * 计算压缩长度的上界 */ //输出长度,按bit算 ulg out_length = (ulg)last_lit*8L; //输入长度,按byte(8 bit)算 ulg in_length = (ulg)strstart-block_start; int dcode; for (dcode = 0; dcode < D_CODES; dcode++) { //加上distance的长度:5 + extra bits //因为distance区间码为 0~29, 2^4 < 29 < 2^5,所以取基本长度5,再加上额外的bit位 out_length += (ulg)dyn_dtree[dcode].Freq*(5L+extra_dbits[dcode]); } //除以8,化为以byte为单位 out_length >>= 3; Trace((stderr,"\nlast_lit %u, last_dist %u, in %ld, out ~%ld(%ld%%) ", last_lit, last_dist, in_length, out_length, 100L - out_length*100L/in_length)); //判断条件 if (last_dist < last_lit/2 && out_length < in_length/2) return 1; } return (last_lit == LIT_BUFSIZE-1 || last_dist == DIST_BUFSIZE); /* We avoid equality with LIT_BUFSIZE because of wraparound at 64K * on 16 bit machines and because stored blocks are restricted to * 64K-1 bytes. */}
build_tree开始建deflate树
deflate树相关数据的组织形式
如前面所述,树是以ct_data结构体数组来表示的,例如2棵动态树:
#define HEAP_SIZE (2*L_CODES+1)local ct_data near dyn_ltree[HEAP_SIZE]; /* literal and length tree */local ct_data near dyn_dtree[2*D_CODES+1]; /* distance tree */
之所以数组大小比要编码的元素个数要大一倍再多一个,是因为建树时内部节点需要额外空间,这一点前面已经讲过,现在再在这里重复一遍。如果还是不能理解,可以参考下图:
其中node是build_tree函数中要用到的一个迭代器,作为树和free space 的分界线。
对于每种数据(literal & length 或 distance),都有一棵静态树(static_ltree或static_dtree)和一棵动态树(dyn_ltree或dyn_dtree),其中静态树的编码方案是预先给定的,而动态树的编码方案动态生成;从而有了这样一个聚合了所有树相关信息的结构体 tree_desc:
typedef struct tree_desc { ct_data near *dyn_tree; /* 动态树 */ ct_data near *static_tree; /* 静态树 or NULL */ int near *extra_bits; /* extra bits数组 or NULL */ int extra_base; /* base index for extra_bits */ int elems; /* 树内叶节点个数上限 */ int max_length; /* 树内元素的最长编码长度 */ int max_code; /* 最大的频率非零元素位置,也就是说右边的元素频率都为零 */} tree_desc;#define LITERALS 256/* number of literal bytes 0..255 */#define L_CODES (LITERALS+1+LENGTH_CODES)/* number of Literal or Length codes, including the END_BLOCK code */#define D_CODES 30/* number of distance codes */#define MAX_BITS 15/* All codes must not exceed MAX_BITS bits */local tree_desc near l_desc ={dyn_ltree, static_ltree, extra_lbits, LITERALS+1, L_CODES, MAX_BITS, 0};local tree_desc near d_desc ={dyn_dtree, static_dtree, extra_dbits, 0, D_CODES, MAX_BITS, 0};
huffman树的建造过程
build_tree 使用比较常规的方法建造haffman树,最小优先队列使用基于数组的最小堆实现,数组中节点heap[i]的左子节点为heap[2*i], 右子节点为heap[2*i + 1];
有一点是要提醒的:事实上,堆装载的是int值而不是ct_data结构体, 例如,若heap[i] = j,则表示堆的第i个节点对应的数据是ct_data 结构体tree[j]。也就是说,heap中只保留指向树中数据结构体的句柄,或者说序号。
static int near heap[2*L_CODES+1]
堆的结构如图:
堆的相关代码如下:
/* =========================================================================== * Remove the smallest element from the heap and recreate the heap with * one less element. Updates heap and heap_len. * 弹出最小堆的根节点,也就是目前的最小元素,将根节点换为尾节点的值, * 并调用pqdownheap及时维护堆的结构。 */#define pqremove(tree, top) \{\ top = heap[SMALLEST]; \ heap[SMALLEST] = heap[heap_len--]; \ pqdownheap(tree, SMALLEST); \}/* =========================================================================== * Compares to subtrees, using the tree depth as tie breaker when * the subtrees have equal frequency. This minimizes the worst case length. * 比较2个节点的频率:若频率相等,则比较以2个节点为根节点的2棵子树的深度。 */#define smaller(tree, n, m) \ (tree[n].Freq < tree[m].Freq || \ (tree[n].Freq == tree[m].Freq && depth[n] <= depth[m]))/* =========================================================================== * Restore the heap property by moving down the tree starting at node k, * exchanging a node with the smallest of its two sons if necessary, stopping * when the heap property is re-established (each father smaller than its * two sons).将heap[k]向下移动到适当的地方。 */local void pqdownheap(tree, k) ct_data near *tree; /* the tree to restore */ int k; /* node to move down */{ int v = heap[k]; int j = k << 1; /* left son of k */ while (j <= heap_len) { /* Set j to the smallest of the two sons: */ if (j < heap_len && smaller(tree, heap[j+1], heap[j])) j++; /* Exit if v is smaller than both sons */ if (smaller(tree, v, heap[j])) break; /* Exchange v with the smallest son */ heap[k] = heap[j]; k = j; /* And continue down the tree, setting j to the left son of k */ j <<= 1; } heap[k] = v;}
这几个操作也算是很经典的啦,我就不多说了。
接下来就是函数build_tree的代码
/* =========================================================================== * Construct one Huffman tree and assigns the code bit strings and lengths. * 构造一颗huffman树并生成编码串,记录其长度 * Update the total bit length for the current block. * IN assertion: the field freq is set for all tree elements. * 输入前:要保证属性freq(即各值对应的频率)已经统计完备 * OUT assertions: the fields len and code are set to the optimal bit length * and corresponding code. The length opt_len is updated; static_len is * also updated if stree is not null. The field max_code is set. * 输出后:树中所有节点赋得属性len(huffman编码长度)与属性code(编码比特串)。 * 最优树的总长 */local void build_tree(desc) tree_desc near *desc; /* the tree descriptor */{ ct_data near *tree = desc->dyn_tree; ct_data near *stree = desc->static_tree; int elems = desc->elems; //迭代器 int n, m; /* iterate over heap elements */ //最大的频率非零节点位置 int max_code = -1; /* largest code with non zero frequency */ //下一个可以构造内部节点的位置 int node = elems; /* next internal node of the tree */ /* Construct the initial heap, with least frequent element in * heap[SMALLEST]. The sons of heap[n] are heap[2*n] and heap[2*n+1]. * heap[0] is not used. * 初始化堆,heap[n]子节点为heap[2*n]和heap[2*n+1],heap[0]闲置。 */ //2个全局变量:heap_len: 堆长度 | heap_max: heap_len = 0, heap_max = HEAP_SIZE; for (n = 0; n < elems; n++) { if (tree[n].Freq != 0) { heap[++heap_len] = max_code = n; depth[n] = 0; } else { tree[n].Len = 0; } } /* The pkzip format requires that at least one distance code exists, * and that at least one bit should be sent even if there is only one * possible code. So to avoid special checks later on we force at least * two codes of non zero frequency. */ while (heap_len < 2) { int new = heap[++heap_len] = (max_code < 2 ? ++max_code : 0); tree[new].Freq = 1; depth[new] = 0; opt_len--; if (stree) static_len -= stree[new].Len; /* new is 0 or 1 so it does not have extra bits */ } desc->max_code = max_code; /* The elements heap[heap_len/2+1 .. heap_len] are leaves of the tree, * establish sub-heaps of increasing lengths: */ for (n = heap_len/2; n >= 1; n--) pqdownheap(tree, n); /* Construct the Huffman tree by repeatedly combining the least two * frequent nodes. */ do { //弹出2个最小节点 pqremove(tree, n); /* n = node of least frequency */ m = heap[SMALLEST]; /* m = node of next least frequency */ heap[--heap_max] = n; /* keep the nodes sorted by frequency */ heap[--heap_max] = m; /* Create a new node father of n and m * 合并n和m,在tree中生成新节点*/ tree[node].Freq = tree[n].Freq + tree[m].Freq; depth[node] = (uch) (MAX(depth[n], depth[m]) + 1); tree[n].Dad = tree[m].Dad = (ush)node;#ifdef DUMP_BL_TREE if (tree == bl_tree) { fprintf(stderr,"\nnode %d(%d), sons %d(%d) %d(%d)", node, tree[node].Freq, n, tree[n].Freq, m, tree[m].Freq); }#endif /* and insert the new node in the heap * 将tree中新节点(的位置)塞到堆里面*/ heap[SMALLEST] = node++; pqdownheap(tree, SMALLEST); } while (heap_len >= 2); heap[--heap_max] = heap[SMALLEST]; //huffman树结构已生成,开始特化成deflate树 /* At this point, the fields freq and dad are set. We can now * generate the bit lengths. * 生成编码长度记录 */ gen_bitlen((tree_desc near *)desc); /* The field len is now set, we can generate the bit codes * * 生成bit编码串 */ gen_codes ((ct_data near *)tree, max_code);}
在函数中heap的结构变化如图所示:
初始堆
huffman编码中
处理完毕
在函数中tree的结构变化如图所示:
初始树
huffman编码中
记录长度gen_bitlen
gen_bitlen函数用于记录长度
编码长度存储数组bl_count
这里要先介绍一个数组,用于存储各长度节点的数目:
//bl_count[i]表示长度为i的编码个数local ush near bl_count[MAX_BITS+1];/* number of codes at each bit length for an optimal tree */
长度限制下的长度统计
如前面所讲到的,tree_desc 内含有属性max_length, 表示huffman编码的最大长度,对于length&literal tree descripter或distance tree descripter为MAX_BITS = 15;然而在实际的编码过程中可能由于元素之间频率差距过大,导致有些节点深度大于15,这样就很尴尬了;
gen_bitlen函数使用了合并节点的策略:
将 超长节点中层数最浅且频率最大的叶节点A与 目前正常节点中层数最深且频率最小的叶节点B合并,如图:
虚线表示max_length,2个紫色叶节点分别为A, B;可见合并之后A, B的父节点占据了B原来的位置,而A的一个超长兄弟节点向上移了一位,而A原来的那个超长(或刚好深度为max_length)父节点就可以扔掉无视了。
/* =========================================================================== * Compute the optimal bit lengths for a tree and update the total bit length * for the current block. * 计算最优的bit串长度,并更新整个数据块的压缩后长度。 * IN assertion: the fields freq and dad are set, heap[heap_max] and * above are the tree nodes sorted by increasing frequency. * OUT assertions: the field len is set to the optimal bit length, the * array bl_count contains the frequencies for each bit length. * The length opt_len is updated; static_len is also updated if stree is * not null. */local void gen_bitlen(desc) tree_desc near *desc; /* the tree descriptor */{ ct_data near *tree = desc->dyn_tree; int near *extra = desc->extra_bits; int base = desc->extra_base; int max_code = desc->max_code; int max_length = desc->max_length; ct_data near *stree = desc->static_tree; int h; /* 用于遍历堆 */ int n, m; /* 用于遍历树 */ int bits; /* bit长度 */ int xbits; /* extra bits */ ush f; /* 频率 */ int overflow = 0; /* 超长的节点数 */ for (bits = 0; bits <= MAX_BITS; bits++) bl_count[bits] = 0; /* In a first pass, compute the optimal bit lengths (which may * overflow in the case of the bit length tree). * 第一次遍历,计算出最优bit长度,并记录超长节点的数量, * 如果节点超长,把它长度记为max_length。 */ tree[heap[heap_max]].Len = 0; /* root of the heap */ for (h = heap_max+1; h < HEAP_SIZE; h++) { n = heap[h]; bits = tree[tree[n].Dad].Len + 1; //统计overflow节点数目,注意:此处超长的内节点也算在内 if (bits > max_length) bits = max_length, overflow++; //如果节点超长,把它长度记为max_length。 tree[n].Len = (ush)bits; /* We overwrite tree[n].Dad which is no longer needed * 这里体现了在ct_data结构体中使用union{dad, len}的好处:当用父节点dad * 求出len后,数据成员dad就没有意义了,可以替换为len,节约空间 */ if (n > max_code) continue; /* 如果不是叶节点,不进行计数 */ //计数数组bl_count bl_count[bits]++; //由于区间码的原因,额外的扩展bit位要另外加上来。 xbits = 0; if (n >= base) xbits = extra[n-base]; f = tree[n].Freq; opt_len += (ulg)f * (bits + xbits); //静态树(静态树老兄似乎没什么存在感。。。) if (stree) static_len += (ulg)f * (stree[n].Len + xbits); } if (overflow == 0) return; Trace((stderr,"\nbit length overflow\n")); /* This happens for example on obj2 and pic of the Calgary corpus */ /* Find the first bit length which could increase: * 开始合并超长节点 */ do { bits = max_length-1; while (bl_count[bits] == 0) bits--; bl_count[bits]--; /* 将一个叶节点下移一位 */ bl_count[bits+1] += 2; /* 将一个 overflow 叶节点上移作为其兄弟节点 */ bl_count[max_length]--; /* 注意:此处max_length长度编码的计数器只减1, * 却认为超长的节点消掉了2个,原因见上方图示。 */ overflow -= 2; } while (overflow > 0); /* Now recompute all bit lengths, scanning in increasing frequency. * h is still equal to HEAP_SIZE. (It is simpler to reconstruct all * lengths instead of fixing only the wrong ones. This idea is taken * from 'ar' written by Haruhiko Okumura.) * 按照频率从小到大扫描,调整所有bit长度,h由于之前的循环递增,现在等于HEAP_SIZE。 */ for (bits = max_length; bits != 0; bits--) { n = bl_count[bits]; while (n != 0) { m = heap[--h]; if (m > max_code) continue; if (tree[m].Len != (unsigned) bits) { Trace((stderr,"code %d bits %d->%d\n", m, tree[m].Len, bits)); opt_len += ((long)bits-(long)tree[m].Len)*(long)tree[m].Freq; tree[m].Len = (ush)bits; } n--; } }}
这里的映射可能有些让人头晕;就来解释一下:
映射 i -> heap[i] 中的 i 表示最大堆中的位置,而heap[i]则表示位于堆中该位置的节点在树中的位置,
因此,heap[i] = j 意味着 树节点tree[j] 在最小堆中的位置为i;
映射 j -> tree[j] 中的 j 表示length & literal 或 distace 的某个取值,
而tree[j]则是 包含着这个取值在文本中出现的频率 / huffma编码串,在堆中的父节点 / huffman编码长度的ct_data结构体;
因此,tree[j] 表示着值 j 的所有编码信息。
deflate树的特化规则gen_len
接下来分析最有意思的一段:根据长度自动生成huffman编码的过程,
1.首先将huffman树中每层节点按值(不是编码长度,是它实际的取值)从小到大排列,
比如distance = 2, 30, 5, 8, 7, 1拥有同一编码长度,就排成1, 2, 5, 7, 8, 30;
2.排列好之后,同一层从左到右(从小到大)逐个编码加一,跨层时就把当前编码左移一位;
由于之前已经记录了每层叶节点的数目,存储在bl_count中,我们可以预先算出每层最左边的那个编码;
比如,bl_count[16] =
{ 0(第零项不用), 0, 1, 1, 0, 1, 3, 1, 5, 6 ……… }
第一层:0000000 00000000(无叶节点);加0,左移 第二层:0000000 00000000;加1,左移第三层:0000000 00000010;加1,左移第四层:0000000 00000110;加0,左移第五层:0000000 00001100;加1,左移第六层:0000000 00011010;加3,左移 第七层:0000000 00111010;
从而可以由最左边的值衍生出同一层右边的编码。
贴代码:
/* =========================================================================== * Generate the codes for a given tree and bit counts (which need not be * optimal).为作为参数传入的树生成编码 * IN assertion: the array bl_count contains the bit length statistics for * the given tree and the field len is set for all tree elements. * 输入前: 数组 bl_count 包含了所给树的所有叶节点霍夫曼编码长度的统计信息, * 并且树中所有节点都已拥有其编码串长度记录。 * OUT assertion: the field code is set for all tree elements of non * zero code length. * 输出后:根据编码串长度自动分配好编码值。 */local void gen_codes (tree, max_code) ct_data near *tree; /* the tree to decorate */ int max_code; /* largest code with non zero frequency */{ ush next_code[MAX_BITS+1]; /* next code value for each bit length */ ush code = 0; /* running code value */ int bits; /* bit index */ int n; /* code index */ /* The distribution counts are first used to generate the code values * without bit reversal. */ for (bits = 1; bits <= MAX_BITS; bits++) { next_code[bits] = code = (code + bl_count[bits-1]) << 1; } /* Check that the bit counts in bl_count are consistent. The last code * must be all ones. */ Assert (code + bl_count[MAX_BITS]-1 == (1<<MAX_BITS)-1, "inconsistent bit counts"); Tracev((stderr,"\ngen_codes: max_code %d ", max_code)); for (n = 0; n <= max_code; n++) { int len = tree[n].Len; if (len == 0) continue; /* Now reverse the bits */ tree[n].Code = bi_reverse(next_code[len]++, len); Tracec(tree != static_ltree, (stderr,"\nn %3d %c l %2d c %4x (%x) ", n, (isgraph(n) ? n : ' '), len, tree[n].Code, next_code[len]-1)); }}
小结
到此为止,huffman压缩的部分已经基本分析完毕了,然而zip压缩算法还将huffman压缩的结果进行了再次压缩并封装输出,
这些过程将在下一篇文章中分析。如果你耐着性子看到了这里,真的很感谢了。
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