132 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

### Random induced subgraphs of Cayley graphs induced by transpositions

In this paper we study random induced subgraphs of Cayley graphs of the
symmetric group induced by an arbitrary minimal generating set of
transpositions. A random induced subgraph of this Cayley graph is obtained by
selecting permutations with independent probability, $\lambda_n$. Our main
result is that for any minimal generating set of transpositions, for
probabilities $\lambda_n=\frac{1+\epsilon_n}{n-1}$ where $n^{-{1/3}+\delta}\le
\epsilon_n0$, a random induced subgraph has a.s. a unique
largest component of size $\wp(\epsilon_n)\frac{1+\epsilon_n}{n-1}n!$, where
$\wp(\epsilon_n)$ is the survival probability of a specific branching process.Comment: 18 pages, 1 figur

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