Approximation and exact algorithms for RNA secondary structure prediction and recognition of stochastic context-free languages

Abstract

For a basic version (i.e., maximizing the number of base-pairs) of the RNA secondary structure prediction problem and the construction of a parse tree for a stochastic context-free language, O(n 3 ) time algorithms were known. For both problems, this paper shows slightly improved O(n 3 (log logn) 1/2 /(log n) 1/2 ) time exact algorithms, which are obtained by combining Valiant's algorithm for context-free recognition with fast funny matrix multiplication. Moreover, this paper shows an O(n 2.776 + (1/∈) O(1) ) time approximation algorithm for the former problem and an O(n 2.976 log n + (1/∈) O(1) ) time approximation algorithm for the latter problem, each of which has a guaranteed approximation ratio I - ∈ for any positive constant ∈, where the absolute value of the logarithm of the probability is considered as an objective value in the latter problem. The former algonthm is obtained from a non-trivial modification of the well-known O(n 3 ) time dynamic programming algorithm, and the latter algorithm is obtained by combining Valiant's algorithm with approximate funny matrix multiplication. Several related results are shown too.