Searching for smallest grammars on large sequences and application to DNA

Motivated by the inference of the structure of genomic sequences, we address here the smallest grammar problem. In previous work, we introduced a new perspective on this problem, splitting the task into two different optimization problems: choosing which words will be considered constituents of the final grammar and finding a minimal parsing with these constituents. Here we focus on making these ideas applicable on large sequences. First, we improve the complexity of existing algorithms by using the concept of maximal repeats when choosing which substrings will be the constituents of the grammar. Then, we improve the size of the grammars by cautiously adding a minimal parsing optimization step. Together, these approaches enable us to propose new practical algorithms that return smaller grammars (up to 10%) in approximately the same amount of time than their competitors on a classical set of genomic sequences and on whole genomes of model organisms.