Bit-Optimal Lempel-Ziv compression

One of the most famous and investigated lossless data-compression scheme is the one introduced by Lempel and Ziv about 40 years ago [23]. This compression scheme is known as ”dictionary-based compression” and consists of squeezing an input string by replacing some of its substrings with (shorter) codewords which are actually pointers to a dictionary of phrases built as the string is processed. Surprisingly enough, although many fundamental results are nowadays known about upper bounds on the speed and effectiveness of this compression process (see e.g. [12, 16] and references therein), “we are not aware of any parsing scheme that achieves optimality when the LZ77-dictionary is in use under any constraint on the codewords other than being of equal length” [16, pag. 159]. Here optimality means to achieve the minimum number of bits in compressing each individual input string, without any assumption on its generating source. In this paper we provide the first LZ-based compressor which computes the bit-optimal parsing of any input string in efficient time and optimal space, for a general class of variable-length codeword encodings which encompasses most of the ones typically used in data compression and in the design of search engines and compressed indexes [14, 17, 22].