A Sparsified Four-Russian Algorithm for RNA Folding

Abstract

The basic RNA Secondary Structure Prediction problem or Single Sequence Folding Problem (SSF) was solved thirty-five years ago by a now well-known $O(n^3)$-time dynamic programming method.

Recently three methodologies - Valiant, Four-Russians, and Sparsification - have been applied to speedup RNA Secondary Structure prediction.

In this paper we combine the previously independent speedups of Sparsification and Four-Russians.

The Sparsification method exploits two properties of the input: the number of subsequence Z with the endpoints belonging to the optimal folding set and the maximum number base-pairs L. These sparsity properties satisfy $0 \le L \le n / 2$ and $n \le Z \le n^2 / 2$, and the method reduces the algorithmic running time to O(LZ). In this paper, we first reformulate the SSF Four-Russians $\varTheta (\frac{n^3}{\log ^2 n})$-time algorithm, implied by Pinhas et al. [24], to utilize an on-demand lookup table. This formulation not only removes all extraneous computation and allows us to incorporate more realistic scoring schemes, but leads us to take advantage of the sparsity properties.

Our main result is a framework that combines the fastest Sparsification and fastest Four-Russians Methods. For SSF, this combined method has worst-case running time of $O(\tilde{L}\tilde{Z})$, where $\frac{{L}}{\log n} \le \tilde{L}\le min({L},\frac{n}{\log n})$ and $\frac{{Z}}{\log n}\le \tilde{Z} \le min({Z},\frac{n^2}{\log n})$.

Through asymptotic analysis and empirical testing on the base-pair maximization variant, we show that this framework is able to achieve a speedup on every problem instance, that is asymptotically never worse, and empirically better than achieved by the minimum of the two methods alone.