On Two LZ78-style Grammars: Compression Bounds and Compressed-Space Computation

Abstract

We investigate two closely related LZ78-based compression schemes: LZMW (an old scheme by Miller and Wegman) and LZD (a recent variant by Goto et al.). Both LZD and LZMW naturally produce a grammar for a string of length n; we show that the size of this grammar can be larger than the size of the smallest grammar by a factor Ω(n 1 3 ) but is always within a factor O(( n log n ) 2 3 ). In addition, we show that the standard algorithms using Θ(z) working space to construct the LZD and LZMW parsings, where z is the size of the parsing, work in Ω(n 5 4 ) time in the worst case. We then describe a new Las Vegas LZD/LZMW parsing algorithm that uses O(z log n) space and O(n + z log 2 n) time with high probability.of the smallest grammar exists. This paper addresses that need. Moreover, we show that the standard algorithms for computing LZD and LZMW have undesirable worst case performance, and provide an alternative algorithm that runs in log-linear randomized time. In particular the contributions of this article are as follows:1. We show that the size of the grammar produced by LZD and LZMW can be larger than the size of the smallest grammar by a factor Ω(n 1 3 ) but is always within a factor O(( n log n ) 2 3 ). To our knowledge these are the first non-trivial bounds on compression performance known for these algorithms.2. Space usage during compression is often a concern. For both LZD and LZMW, parsing algorithms are known that use O(z) space, where z is the size of the final parsing. We describe strings for which these algorithms require Ω(n 5 4 ) time. (The only previous analysis is an O(n 2 / log n) upper bound [8].)3. We describe a Monte-Carlo parsing algorithm for LZD/LZMW that uses a z-fast trie [2] and an AVL-grammar [15] to achieve O(z log n) space and O(n + z log 2 n) time for inputs over the integer alphabet {0, 1, . . . , n O(1) }. This algorithm works in the streaming model and computes the parsing with high probability. Using the Monte-Carlo solution, we obtain a Las Vegas algorithm that, with high probability, works in the same space and time.In what follows we provide formal definitions and examples of LZD and LZMW parsings. Section 2 then establishes bounds for the approximation ratios for the sizes of the LZD/LZMW grammars. In Section 3 we consider the time efficiency of current space-efficient parsing schemes for LZD/LZMW. Section 4 provides an algorithm with significantly better (albeit randomized) performance. Conclusions and reflections are offered in Section 5.