【数据压缩】LZ78算法原理及实现

1. 原理

压缩

LZ78算法的压缩过程非常简单。在压缩时维护一个动态词典Dictionary，其包括了历史字符串的index与内容；压缩情况分为三种：

1. 若当前字符c未出现在词典中，则编码为(0, c)；
2. 若当前字符c出现在词典中，则与词典做最长匹配，然后编码为(prefixIndex,lastChar)，其中，prefixIndex为最长匹配的前缀字符串，lastChar为最长匹配后的第一个字符；
3. 为对最后一个字符的特殊处理，编码为(prefixIndex,)。

如果对于上述压缩的过程稍感费解，下面给出三个例子。例子一，对于字符串“ABBCBCABABCAABCAAB”压缩编码过程如下：

1. A is not in the Dictionary; insert it 2. B is not in the Dictionary; insert it 3. B is in the Dictionary. BC is not in the Dictionary; insert it. 4. B is in the Dictionary. BC is in the Dictionary. BCA is not in the Dictionary; insert it. 5. B is in the Dictionary. BA is not in the Dictionary; insert it. 6. B is in the Dictionary. BC is in the Dictionary. BCA is in the Dictionary. BCAA is not in the Dictionary; insert it. 7. B is in the Dictionary. BC is in the Dictionary. BCA is in the Dictionary. BCAA is in the Dictionary. BCAAB is not in the Dictionary; insert it.

例子二，对于字符串“BABAABRRRA”压缩编码过程如下：

1.  B is not in the Dictionary; insert it 2. A is not in the Dictionary; insert it 3. B is in the Dictionary. BA is not in the Dictionary; insert it. 4. A is in the Dictionary. AB is not in the Dictionary; insert it. 5. R is not in the Dictionary; insert it. 6. R is in the Dictionary. RR is not in the Dictionary; insert it. 7. A is in the Dictionary and it is the last input character; output a pair containing its index: (2, )

例子三，对于字符串“AAAAAAAAA”压缩编码过程如下：

1.  A is not in the Dictionary; insert it 2. A is in the Dictionary AA is not in the Dictionary; insert it 3. A is in the Dictionary. AA is in the Dictionary. AAA is not in the Dictionary; insert it. 4. A is in the Dictionary. AA is in the Dictionary. AAA is in the Dictionary and it is the last pattern; output a pair containing its index: (3, )

解压缩

解压缩能更根据压缩编码恢复出（压缩时的）动态词典，然后根据index拼接成解码后的字符串。为了便于理解，我们拿上述例子一中的压缩编码序列(0, A) (0, B) (2, C) (3, A) (2, A) (4, A) (6, B)来分解解压缩步骤，如下图所示：

前后拼接后，解压缩出来的字符串为“ABBCBCABABCAABCAAB”。

LZ系列压缩算法

LZ系列压缩算法均为LZ77与LZ78的变种，在此基础上做了优化。

• LZ77：LZSS、LZR、LZB、LZH；
• LZ78：LZW、LZC、LZT、LZMW、LZJ、LZFG。

其中，LZSS与LZW为这两大阵容里名气最响亮的算法。LZSS是由Storer与Szymanski [2]改进了LZ77：增加最小匹配长度的限制，当最长匹配的长度小于该限制时，则不压缩输出，但仍然滑动窗口右移一个字符。Google开源的Snappy压缩算法库大体遵循LZSS的编码方案，在其基础上做了一些工程上的优化。

2. 实现

Python 3.5实现LZ78算法：

# -*- coding: utf-8 -*- # A simplified implementation of LZ78 algorithm # @Time : 2017/1/13 # @Author : rain def compress(message): tree_dict, m_len, i = {}, len(message), 0 while i < m_len: # case I if message[i] not in tree_dict.keys(): yield (0, message[i]) tree_dict[message[i]] = len(tree_dict) + 1 i += 1 # case III elif i == m_len - 1: yield (tree_dict.get(message[i]), '') i += 1 else: for j in range(i + 1, m_len): # case II if message[i:j + 1] not in tree_dict.keys(): yield (tree_dict.get(message[i:j]), message[j]) tree_dict[message[i:j + 1]] = len(tree_dict) + 1 i = j + 1 break # case III elif j == m_len - 1: yield (tree_dict.get(message[i:j + 1]), '') i = j + 1 def uncompress(packed): unpacked, tree_dict = '', {} for index, ch in packed: if index == 0: unpacked += ch tree_dict[len(tree_dict) + 1] = ch else: term = tree_dict.get(index) + ch unpacked += term tree_dict[len(tree_dict) + 1] = term return unpacked if __name__ == '__main__': messages = ['ABBCBCABABCAABCAAB', 'BABAABRRRA', 'AAAAAAAAA'] for m in messages: pack = compress(m) unpack = uncompress(pack) print(unpack == m)