An O(n^5) algorithm for MFE prediction of kissing hairpins and 4-chains in nucleic acids

Efficient methods for prediction of minimum free energy (MFE) nucleic secondary structures are widely used, both to better understand structure and function of biological RNAs and to design novel nano-structures. Here, we present a new algorithm for MFE secondary structure prediction, which significantly expands the class of structures that can be handled in O(n^5) time. Our algorithm can handle H-type pseudoknotted structures, kissing hairpins, and chains of four overlapping stems, as well as nested substructures of these types

Alien Registration- Condon, Anne (Portland, Cumberland County)

https://digitalmaine.com/alien_docs/23838/thumbnail.jp

Alien Registration- Condon, Anne (Portland, Cumberland County)

https://digitalmaine.com/alien_docs/23838/thumbnail.jp

Constant Factor Approximation for Balanced Cut in the PIE model

We propose and study a new semi-random semi-adversarial model for Balanced Cut, a planted model with permutation-invariant random edges (PIE). Our model is much more general than planted models considered previously. Consider a set of vertices V partitioned into two clusters $L$ and $R$ of equal size. Let $G$ be an arbitrary graph on $V$ with no edges between $L$ and $R$. Let $E_{random}$ be a set of edges sampled from an arbitrary permutation-invariant distribution (a distribution that is invariant under permutation of vertices in $L$ and in $R$). Then we say that $G + E_{random}$ is a graph with permutation-invariant random edges. We present an approximation algorithm for the Balanced Cut problem that finds a balanced cut of cost $O(|E_{random}|) + n \text{polylog}(n)$ in this model. In the regime when $|E_{random}| = \Omega(n \text{polylog}(n))$, this is a constant factor approximation with respect to the cost of the planted cut.Comment: Full version of the paper at the 46th ACM Symposium on the Theory of Computing (STOC 2014). 32 page

A limit theorem for sets of stochastic matrices

AbstractThe following fact about (row) stochastic matrices is an easy consequence of well known results: for each positive integer n⩾1 there is a positive integer q=q(n) with the property that if A is any n×n stochastic matrix then the sequence of matrices Aq,A2q,A3q,… converges. We prove a generalization of this for sets of stochastic matrices under the Hausdorff metric. Let d be any metric inducing the standard topology on the set of n×n real matrices. For a matrix A and set of matrices B define d(A,B) to be the infimum of d(A,B) over all B∈B. For two sets of matrices A and B, define d+(A,B) to be the supremum of d(A,B) over all A∈A, and define d(A,B) to be the maximum of d+(A,B) and d+(B,A). This is the Hausdorff metric on the set of subsets of n×n stochastic matrices. If A is a set of stochastic matrices and k is a positive integer, define A(k) to be the set of all matrices expressible as a product of a sequence of k matrices from A. We prove: For each positive integer n there is a positive integer p=p(n) such that if A is any subset of n×n stochastic matrices then the sequence of subsets A(p),A(2p),A(3p),… converges with respect to the Hausdorff metric