在提出基于滑动窗口的LZ77算法后,两位大神Jacob Ziv与Abraham Lempel于1978年在发表的论文 [1]中提出了LZ78算法;与LZ77算法不同的是LZ78算法使用动态树状词典维护历史字符串。

【数据压缩】LZ77算法原理及实现

【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)

3. 参考资料

[1] Ziv, Jacob, and Abraham Lempel. "Compression of individual sequences via variable-rate coding." IEEE transactions on Information Theory 24.5 (1978): 530-536.

[2] Storer, James A., and Thomas G. Szymanski. "Data compression via textual substitution." Journal of the ACM (JACM) 29.4 (1982): 928-951.

[3] Welch, T. A. "A Technique for High-Performance Data Compression." Computer 17.17(1984):8-19.

[4] Jauhar Ali, Unit31_LZ78.ppt.

[5] guyb, 15-853:Algorithms in the Real World - Data Compression III.