Abstract
BackgroundThe 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 RNA-folding problem of finding a maximum cardinality, non-crossing, matching of complimentary nucleotides in an RNA sequence of length n, has an O(n3)-time dynamic programming solution that is widely applied. It is known that an o(n3) 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 O(n3) worst-case time bound when n 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 Four-Russians 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.ResultsIn 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 O(n3/log(n)).ConclusionsWe 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.
Highlights
The problem of computationally predicting the secondary structure 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 O(n3)-time dynamic-programming solution to the RNA-folding problem remains a central tool in methods to predict RNA structure, and has been widely exposed in books and surveys on RNA folding, computational biology, and computer algorithms
In [5], Backofen et al present a compelling, practical reduction in space and time using the observations of [6] that yields a worst-case improvement when additional problem parameters are included in the time-bound i.e. O(nZ) where n ≤ Z ≤ n2
Summary
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 RNA-folding problem of finding a maximum cardinality, non-crossing, matching of complimentary nucleotides in an RNA sequence of length n, has an O(n3)-time dynamic programming solution that is widely applied. The most widely-used, general technique to achieve a worst-case (and often practical) speed up of dynamic programming, the Four-Russians technique, has not been previously applied to the RNA-folding problem. The basic RNA-folding problem of finding a maximum cardinality, non-crossing, matching of complementary nucleotides in an RNA sequence of length n, is at the heart of almost all methods to computationally predict RNA secondary structure, including more complex methods that incorporate more realistic folding models, such as allowing some crossovers (pseudoknots). If correct, would achieve a worst-case bound of
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