A Simple Online Competitive Adaptation of Lempel-Ziv Compression with Efficient Random Access Support

Abstract

We present a simple adaptation of the Lempel Ziv 78' (LZ78) compression scheme that supports efficient random access to the input string. The compression algorithm is given as input a parameter ε > 0, and with very high probability increases the length of the compressed string by at most a factor of (1 + ε). The access time is O(log n + 1/ε <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ) in expectation, and O(log n/ε <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ) with high probability. The scheme relies on sparse transitive-closure spanners. Any (consecutive) substring of the input string can be retrieved at an additional additive cost in the running time of the length of the substring. The main benefit of the proposed scheme is that it preserves the online nature and simplicity of LZ78, and that for every input string, the length of the compressed string is only a small factor larger than that obtained by running LZ78.