A really Simple Approximation of Smallest Grammar

Abstract

We present a really simple linear-time algorithm constructing a context-free grammar of size $\mathcal{O}(g log (N/g))$ for the input string, where N is the size of the input string and g the size of the optimal grammar generating this string. The algorithm works for arbitrary size alphabets, but the running time is linear when the alphabet Σ of the input string can be identified with numbers from {1,…, N ^}. Algorithms with such an approximation guarantee and running time are known, however all of them were non-trivial and their analyses involved. The here presented algorithm computes the LZ77 factorisation (of size l) and transforms it in phases to a grammar. In each phase it maintains an LZ77-like factorisation of the word with at most l factors as well as additional $\mathcal{O}(l)$ letters. In one phase in a greedy way (by a left-to-right sweep) we choose a set of pairs of consecutive letters to be replaced with new symbols, i.e. nonterminals of the constructed grammar. We choose at least 2/3 of the letters in the word and there are $\mathcal{O}(l)$ many different pairs among them. Hence there are $\mathcal{O}(log N)$ phases, each introduces $\mathcal{O}(l)$ nonterminals. A more precise analysis yields a bound $\mathcal{O}(l log(N/l))$. As l ≤ g, this yields $\mathcal{O}(g log(N/g))$.