On the Bit-Complexity of Lempel--Ziv Compression

Abstract

One of the most famous and investigated lossless data-compression schemes is the one introduced by Lempel and Ziv about 30 years ago [IEEE Trans. Inform. Theory, 23 (1977), pp. 337--343]. This compression scheme is known as “dictionary-based compressor” 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 the speed and effectiveness of this compression process, “we are not aware of any parsing scheme that achieves optimality$\ldots$under any constraint on the codewords other than being of equal length” [N. Rajpoot and C. Sahinalp, in Handbook of Lossless Data Compression, Academic Press, New York, 2002, p. 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 investigate...