A Quasi-polynomial-time Algorithm for Sampling Words from a Context-free Language 1 Problem Speciication and History

A quasi-polynomial-time algorithm is presented for sampling almost uniformly at random from the n-slice of the language L(G) generated by an arbitrary context-free grammar G. (The n-slice of a language L over an alphabet is the subset L\ n of words of length exactly n.) The time complexity of the algorithm is " ?2 (n jGj) O(log n) , where the parameter " bounds the variation of the output distribution from uniform, and jGj is a natural measure of the size of grammar G. The algorithm applies to a class of language sampling problems that includes slices of context-free languages as a proper subclass. We address the problem of sampling (almost) uniformly at random a word of length n from a context-free language L, and the related problem of estimating the number of words of length n in L. Ideally, we would like to obtain a sampling procedure that runs in time polynomial in the length n and the size of the grammar used to specify L. This problem has been considered by many authors (for example, Mairson 7]), who have proposed eecient solutions based on dynamic programming, but always restricted to the special case of unambiguous grammars. No polynomial time algorithm has been proposed for general context-free grammars. Let G be a context-free grammar generating the language L = L(G) , and n a positive integer. The n-slice of L is just the subset L \ n containing all words in L of length n. The problem of determining the size of the n-slice of L(G) is #P-complete 1], and remains so even when the grammar G is restricted to be regular. (The latter claim can be established by reduction from #DNF 3].) This completeness result does not, however, rule out the possibility of eecient sampling from slices of context-free languages, nor eecient estimation of the size of slices. Kannan, Sweedyk and Mahaney 3] recently presented a \quasi-polynomial-time" (i.e., with running time exp(polylog(jGj; n))) algorithm for the case of a regular grammar G. In this article we extend their result to arbitrary context free grammars G. In fact, we operate within the more general setting of \ff; g-programs" which contains context free grammars as a special case. (See Theorem 1 for an exact statement of the main result.) The main techniques we use are Karp-Luby sampling from a union of sets 5] as extended by Kannan et al. 3], …