A simple, practical and complete O-time Algorithm for RNA folding using the Four-Russians Speedup

<p>Abstract</p> <p>Background</p> <p>The problem of computationally predicting the secondary structure (or folding) of RNA molecules was first introduced more than thirty years ago and yet continues to be an area of active research and development. The basic <it>RNA-folding problem </it>of finding a maximum cardinality, non-crossing, matching of complimentary nucleotides in an RNA sequence of length <it>n</it>, has an <it>O</it>(<it>n</it>³)-time dynamic programming solution that is widely applied. It is known that an <it>o</it>(<it>n</it>³) worst-case time solution is possible, but the published and suggested methods are complex and have not been established to be practical. Significant practical improvements to the original dynamic programming method have been introduced, but they retain the <it>O</it>(<it>n</it>³) worst-case time bound when <it>n </it>is the only problem-parameter used in the bound. Surprisingly, the most widely-used, general technique to achieve a worst-case (and often practical) speed up of dynamic programming, the <it>Four-Russians </it>technique, has not been previously applied to the RNA-folding problem. This is perhaps due to technical issues in adapting the technique to RNA-folding.</p> <p>Results</p> <p>In this paper, we give a simple, complete, and practical Four-Russians algorithm for the basic RNA-folding problem, achieving a worst-case time-bound of <it>O</it>(<it>n</it>³/log(<it>n</it>)).</p> <p>Conclusions</p> <p>We show that this time-bound can also be obtained for richer nucleotide matching scoring-schemes, and that the method achieves consistent speed-ups in practice. The contribution is both theoretical and practical, since the basic RNA-folding problem is often solved multiple times in the inner-loop of more complex algorithms, and for long RNA molecules in the study of RNA virus genomes.</p

Practicality and time complexity of a sparsified RNA folding algorithm

Commonly used RNA folding programs compute the minimum free energy structure of a sequence under the pseudoknot exclusion constraint. They are based on Zuker's algorithm which runs in time O(n^3). Recently, it has been claimed that RNA folding can be achieved in average time O(n^2) using a sparsification technique. A proof of quadratic time complexity was based on the assumption that computational RNA folding obeys the "polymer-zeta property". Several variants of sparse RNA folding algorithms were later developed. Here, we present our own version, which is readily applicable to existing RNA folding programs, as it is extremely simple and does not require any new data structure. We applied it to the widely used Vienna RNAfold program, to create sibRNAfold, the first public sparsified version of a standard RNA folding program. To gain a better understanding of the time complexity of sparsified RNA folding in general, we carried out a thorough run time analysis with synthetic random sequences, both in the context of energy minimization and base pairing maximization. Contrary to previous claims, the asymptotic time complexity of a sparsified RNA folding algorithm using standard energy parameters remains O(n^3) under a wide variety of conditions. Consistent with our run-time analysis, we found that RNA folding does not obey the "polymer-zeta property" as claimed previously. Yet, a basic version of a sparsified RNA folding algorithm provides 15- to 50-fold speed gain. Surprisingly, the same sparsification technique has a different effect when applied to base pairing optimization. There, its asymptotic running time complexity appears to be either quadratic or cubic depending on the base composition. The code used in this work is available at: http://sibRNAfold.sourceforge.net/

If the Current Clique Algorithms are Optimal, so is Valiant's Parser

The CFG recognition problem is: given a context-free grammar $\mathcal{G}$ and a string $w$ of length $n$, decide if $w$ can be obtained from $\mathcal{G}$. This is the most basic parsing question and is a core computer science problem. Valiant's parser from 1975 solves the problem in $O(n^{\omega})$ time, where $\omega<2.373$ is the matrix multiplication exponent. Dozens of parsing algorithms have been proposed over the years, yet Valiant's upper bound remains unbeaten. The best combinatorial algorithms have mildly subcubic $O(n^3/\log^3{n})$ complexity. Lee (JACM'01) provided evidence that fast matrix multiplication is needed for CFG parsing, and that very efficient and practical algorithms might be hard or even impossible to obtain. Lee showed that any algorithm for a more general parsing problem with running time $O(|\mathcal{G}|\cdot n^{3-\varepsilon})$ can be converted into a surprising subcubic algorithm for Boolean Matrix Multiplication. Unfortunately, Lee's hardness result required that the grammar size be $|\mathcal{G}|=\Omega(n^6)$. Nothing was known for the more relevant case of constant size grammars. In this work, we prove that any improvement on Valiant's algorithm, even for constant size grammars, either in terms of runtime or by avoiding the inefficiencies of fast matrix multiplication, would imply a breakthrough algorithm for the $k$-Clique problem: given a graph on $n$ nodes, decide if there are $k$ that form a clique. Besides classifying the complexity of a fundamental problem, our reduction has led us to similar lower bounds for more modern and well-studied cubic time problems for which faster algorithms are highly desirable in practice: RNA Folding, a central problem in computational biology, and Dyck Language Edit Distance, answering an open question of Saha (FOCS'14)