New analysis of the asymptotic behavior of the Lempel-Ziv compression algorithm

We give a new analysis and proof of the Normal limiting distribution of the number of phrases in the 1978 Lempel-Ziv compression algorithm on random sequences built from a memoriless source. This work is a follow-up of our last paper on this subject in 1995. The analysis stands on the asymptotic behavior of a DST obtained by the insertion of random sequences. Our proofs are augmented of new results on moment convergence, moderate and large deviations, redundancy analysis. Key-words: data compression, digital search tree, generating functions, depoissonization, renewal processes in ria -0 04 76 90 2, v er si on 1 27 A pr 2 01 0 Nouvelle analyse du comportement asymptotique de l’algorithme de compression de Lempel-Ziv Résumé : Nous donnons une nouvelle analyse et preuve de la convergence vers la loi normale des performances de l’algorithme de compression de Lempel et Ziv 1978 sur des séquences aléatoires construites à partir d’une source sans mémoire. Ce travail est une continuation de notre papier sur le même sujet de 1995. L’analyse repose sur le comportement asymptotique d’un arbre digital de recherche obtenu par l’insertion de séquences aléatoires. Nos preuves sont augmentées de nouveaux résultats sur la convergence des moments, les moyennes et grandes déviations et l’analyse de la redondance. Mots-clés : compression de données, arbre digital de recherche, fonctions génératrices, dépoissonisation, processus à renouvellement in ria -0 04 76 90 2, v er si on 1 27 A pr 2 01 0 New analysis of Lempel-Ziv algorithm 3 1 Ziv Lempel compression algorithm We investigate the number of phrases generated by the compression algorithm Lempel-Ziv 1978 [?]. The algorithm consists in arranging the text to be compressed in consecutive phrases such that the next phrase is the unique prefix of the remaining text that has a previous phrase as largest prefix. In other words the next phrase can be identified by its prefix in the previously listed phrases plus one symbol: namely one pointer and a symbol. It is convenient to see the phrases organized in a digital search tree. Imagine a root. The first phrase is the first symbol, say a pending on the root. Depending if the next symbol is again a a, then the next phrase will be with two symbols, the last symbol being appended to the node with a. The new phrase can be read from the root to the new symbol. If the next symbol was different of a, say b, then the new phrase is simply b (the largest phrase prefix is therefore empty) and is directly appended to root. Let a text w written on alphabet A, and T (w) is the digital tree created by the algorithm. To each node in T (w) corresponds a phrase in the parsing algorithm. Let L(w) be the path length of T (w): the sum of all the node distances to the root. We should have L(w) = |w|. To make this rigorous we have to assume that the text w ends with an extra symbol that never occurs before in the text. The knowledge of T (w) without the node sequence order is not sufficient to recover the original text w. But if we know the sequence order of node creation in the tree, then we can reconstruct the original text w. The compression code C(w) is in fact a description of T (w), node by node in their order of creation, each node being identified by a pointer to its parent node in the tree and the symbol that label the edge that link the parent to the node. Since this description contains the sequence order of node creation, then the original text can be recovered. Every node points to parents that have been inserted before, therefore pointer for the kth node costs at most dlog2 ke, and the next symbol costs dlog2 |A|e. A negligible economy can be done by fusionning the two informations in a pointer of cost dlog2(k|A|)e. The compressed code has therefore length