239 research outputs found

### 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

### Large components in random induced subgraphs of n-cubes

In this paper we study random induced subgraphs of the binary $n$-cube,
$Q_2^n$. This random graph is obtained by selecting each $Q_2^n$-vertex with
independent probability $\lambda_n$. Using a novel construction of
subcomponents we study the largest component for
$\lambda_n=\frac{1+\chi_n}{n}$, where $\epsilon\ge \chi_n\ge n^{-{1/3}+
\delta}$, $\delta>0$. We prove that there exists a.s. a unique largest
component $C_n^{(1)}$. We furthermore show that $\chi_n=\epsilon$, $|
C_n^{(1)}|\sim \alpha(\epsilon) \frac{1+\chi_n}{n} 2^n$ and for $o(1)=\chi_n\ge
n^{-{1/3}+\delta}$, $| C_n^{(1)}| \sim 2 \chi_n \frac{1+\chi_n}{n} 2^n$ holds.
This improves the result of \cite{Bollobas:91} where constant $\chi_n=\chi$ is
considered. In particular, in case of $\lambda_n=\frac{1+\epsilon} {n}$, our
analysis implies that a.s. a unique giant component exists.Comment: 18 Page

### 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

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