A tight upper bound on the size of the antidictionary of a binary string

An antidictionary is a set of words that never appear in a binary string. In 2000, Crochemore et al. (2000) presented a compression algorithm of binary text using antidictionary called DCA. Their coding algorithm has been tested on the Calgary Corpus, and their experimental results show that we get compression ratios equivalent to those of most common compressors such as pkzip. Recently, an online source coding scheme based on DCA is presented to apply for compressing losslessly ECG (ElectroCardioGram) in Ota and Morita (2004). Experimental results show that their algorithm achieved 10% smaller compression ratio than LZ ones. In this article, we present au upper bound of the size of the antidictionary of a binary string. The upper bound we obtained is stronger than that in Crochemore et al. (1998). And it is tight in the sense there exists a string to attain the bound. We also proved that the antidictionary of a binary string is always smaller than or equal to that of the dictionary of the same string. Moreover, we give an algorithm to reconstruct the dictionary from the antidictionary. This article is organized as follows. Section 2 gives definitions on antidictionary with some examples. In Sections 3 and 4, we investigate the size of the antidictionary of a given string and derive a tight upper bound on its size. Section 5 presents an algorithm to reconstruct the dictionary from the antidictionary of a given string and Section 6 summarizes our results.