On the combinatorics of sparsification

Background: We study the sparsification of dynamic programming folding algorithms of RNA structures. Sparsification applies to the mfe-folding of RNA structures and can lead to a significant reduction of time complexity. Results: We analyze the sparsification of a particular decomposition rule, $\Lambda^*$, that splits an interval for RNA secondary and pseudoknot structures of fixed topological genus. Essential for quantifying the sparsification is the size of its so called candidate set. We present a combinatorial framework which allows by means of probabilities of irreducible substructures to obtain the expected size of the set of $\Lambda^*$-candidates. We compute these expectations for arc-based energy models via energy-filtered generating functions (GF) for RNA secondary structures as well as RNA pseudoknot structures. For RNA secondary structures we also consider a simplified loop-energy model. This combinatorial analysis is then compared to the expected number of $\Lambda^*$-candidates obtained from folding mfe-structures. In case of the mfe-folding of RNA secondary structures with a simplified loop energy model our results imply that sparsification provides a reduction of time complexity by a constant factor of 91% (theory) versus a 96% reduction (experiment). For the "full" loop-energy model there is a reduction of 98% (experiment).Comment: 27 pages, 12 figure

Random 3-noncrossing partitions

In this paper, we introduce polynomial time algorithms that generate random 3-noncrossing partitions and 2-regular, 3-noncrossing partitions with uniform probability. A 3-noncrossing partition does not contain any three mutually crossing arcs in its canonical representation and is 2-regular if the latter does not contain arcs of the form $(i,i+1)$. Using a bijection of Chen {\it et al.} \cite{Chen,Reidys:08tan}, we interpret 3-noncrossing partitions and 2-regular, 3-noncrossing partitions as restricted generalized vacillating tableaux. Furthermore, we interpret the tableaux as sampling paths of Markov-processes over shapes and derive their transition probabilities.Comment: 17 pages, 7 figure

Shapes of interacting RNA complexes

Shapes of interacting RNA complexes are studied using a filtration via their topological genus. A shape of an RNA complex is obtained by (iteratively) collapsing stacks and eliminating hairpin loops. This shape-projection preserves the topological core of the RNA complex and for fixed topological genus there are only finitely many such shapes.Our main result is a new bijection that relates the shapes of RNA complexes with shapes of RNA structures.This allows to compute the shape polynomial of RNA complexes via the shape polynomial of RNA structures. We furthermore present a linear time uniform sampling algorithm for shapes of RNA complexes of fixed topological genus.Comment: 38 pages 24 figure

A bijection between unicellular and bicellular maps

In this paper we present a combinatorial proof of a relation between the generating functions of unicellular and bicellular maps. This relation is a consequence of the Schwinger-Dyson equation of matrix theory. Alternatively it can be proved using representation theory of the symmetric group. Here we give a bijective proof by rewiring unicellular maps of topological genus $(g+1)$ into bicellular maps of genus $g$ and pairs of unicellular maps of lower topological genera. Our result has immediate consequences for the folding of RNA interaction structures, since the time complexity of folding the transformed structure is $O((n+m)^5)$, where $n,m$ are the lengths of the respective backbones, while the folding of the original structure has $O(n^6)$ time complexity, where $n$ is the length of the longer sequence.Comment: 18 pages, 13 figure