Efficient algorithms for Lempel-Ziv encoding

Abstract

We consider several basic problems for texts and show that if the input texts are given by their Lempel-Ziv codes then the problems can be solved deterministically in polynomial time in the case when the original (uncompressed) texts are of exponential size. The growing importance of massively stored information requires new approaches to algorithms for compressed texts without decompressing. Denote by LZ(ω) the version of a string ω produced by Lempel-Ziv encoding algorithm. For given compressed strings LZ(T), LZ(P) we give the first known deterministic polynomial time algorithms to compute compressed representations of the set of all occurrences of the pattern P in T, all periods of T, all palindromes of T, and all squares of T. Then we consider several classical language recognition problems:

• regular language recognition: given LZ(T) and a language L described by a regular expression, test if T ε L,

• extended regular language recognition: given LZ(T) and a language L described by a LZ-compressed regular expression, test if T ε L, the alphabet is unary,

• context-free language recognition: given LZ(T) and a language L described by a context-free grammar, test if T ε L, the alphabet is unary.

We show that the first recognition problem has a polynomial time algorithm and the other two problems are NP-hard.

We show also that the LZ encoding can be computed on-line in polynomial time delay and small space (i.e. proportional to the size of the compressed text). Also the compressed representation of a patternmatching automaton for the compressed pattern is computed in polynomial time.

On leave from Institute of Informatics, Warsaw University, ul. Banacha 2, 02097, Warszawa, Poland.

This research was partially supported by the DFG Grant KA 673/4-1, and by the ESPRIT BR Grant 7097 and the ECUS 030.

Supported partially by the grant KBN 8T11C01208.

Supported partially by the grant KBN 8T11C01208.