A Space-Saving Approximation Algorithm for Grammar-Based Compression

A space-efficient approximation algorithm for the grammar-based compression problem, which requests for a given string to find a smallest context-free grammar deriving the string, is presented. For the input length n and an optimum CFG size g, the algorithm consumes only O(g log g) space and O(n log∗n) time to achieve O((log∗n) logn) approximation ratio to the optimum compression, where log∗n is the maximum number of logarithms satisfying log log · · · logn > 1. This ratio is thus regarded to almost O(log n), which is the currently best approximation ratio. While g depends on the string, it is known that g = Ω(log n) and g = O ( n logkn ) for strings from k-letter alphabet [12].