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#### A really Simple Approximation of Smallest Grammar

##### Externe Ressourcen

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##### Volltexte (frei zugänglich)

arXiv:1403.4445.pdf

(Preprint), 490KB

##### Ergänzendes Material (frei zugänglich)

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##### Zitation

Jeż, A. (2014). A really Simple Approximation of Smallest Grammar. Retrieved from http://arxiv.org/abs/1403.4445.

Zitierlink: http://hdl.handle.net/11858/00-001M-0000-0024-3CFA-7

##### Zusammenfassung

In this paper we present a really simple linear-time algorithm constructing a
context-free grammar of size 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 assuming that the alphabet Sigma of the input string can be
identified with numbers from 1,ldots, N^c for some constant c. Algorithms with
such an approximation guarantee and running time are known, however all of them
were non-trivial and their analyses were involved. The here presented algorithm
computes the LZ77 factorisation 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 O(l) letters, where l was the size of the
original LZ77 factorisation. In one phase in a greedy way (by a left-to-right
sweep and a help of the factorisation) 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 O(l)
many different pairs among them. Hence there are O(log N) phases, each of them
introduces O(l) nonterminals to a grammar. A more precise analysis yields a
bound O(l log(N/l)). As l \leq g, this yields the desired bound O(g log(N/g)).