Ju n 20 13 A Succinct Grammar Compression ⋆

We solve an open problem related to an optimal encoding of a straight line program (SLP), a canonical form of grammar compression deriving a single string deterministically. We show that an informationtheoretic lower bound for representing an SLP with n symbols requires at least 2n + log n! + o(n) bits. We then present a succinct representation of an SLP; this representation is asymptotically equivalent to the lower bound. The space is at most 2n log ρ(1 + o(1)) bits for ρ ≤ 2√n, while supporting random access to any production rule of an SLP in O(log log n) time. In addition, we present a novel dynamic data structure associating a digram with a unique symbol. Such a data structure is called a naming function and has been implemented using a hash table that has a space-time tradeoff. Thus, the memory space is mainly occupied by the hash table during the development of production rules. Alternatively, we build a dynamic data structure for the naming function by leveraging the idea behind the wavelet tree. The space is strictly bounded by 2n log n(1+ o(1)) bits, while supporting O(log n) query and update time.