91 research outputs found
Combinatorial analysis of interacting RNA molecules
Recently several minimum free energy (MFE) folding algorithms for predicting
the joint structure of two interacting RNA molecules have been proposed. Their
folding targets are interaction structures, that can be represented as diagrams
with two backbones drawn horizontally on top of each other such that (1)
intramolecular and intermolecular bonds are noncrossing and (2) there is no
"zig-zag" configuration. This paper studies joint structures with arc-length at
least four in which both, interior and exterior stack-lengths are at least two
(no isolated arcs). The key idea in this paper is to consider a new type of
shape, based on which joint structures can be derived via symbolic enumeration.
Our results imply simple asymptotic formulas for the number of joint structures
with surprisingly small exponential growth rates. They are of interest in the
context of designing prediction algorithms for RNA-RNA interactions.Comment: 22 pages, 15 figure
Associative spectra of graph algebras II: Satisfaction of bracketing identities, spectrum dichotomy
Funding Information: This work is funded by National Funds through the FCT—Fundação para a Ciência e a Tecnologia, I.P., under the scope of the Project UIDB/00297/2020 (Center for Mathematics and Applications) and the Project PTDC/MAT-PUR/31174/2017. Funding Information: Research partially supported by the Hungarian Research, Development and Innovation Office Grant K115518, and by Grant TUDFO/47138-1/2019-ITM of the Ministry for Innovation and Technology, Hungary. Publisher Copyright: © 2021, The Author(s).A necessary and sufficient condition is presented for a graph algebra to satisfy a bracketing identity. The associative spectrum of an arbitrary graph algebra is shown to be either constant or exponentially growing.publishersversionepub_ahead_of_prin
Minimizing and Computing the Inverse Geodesic Length on Trees
For any fixed measure H that maps graphs to real numbers, the MinH problem is defined as follows: given a graph G, an integer k, and a target tau, is there a set S of k vertices that can be deleted, so that H(G - S) is at most tau? In this paper, we consider the MinH problem on trees.
We call H balanced on trees if, whenever G is a tree, there is an optimal choice of S such that the components of G - S have sizes bounded by a polynomial in n / k. We show that MinH on trees is Fixed-Parameter Tractable (FPT) for parameter n / k, and furthermore, can be solved in subexponential time, and polynomial space, whenever H is additive, balanced on trees, and computable in polynomial time.
A particular measure of interest is the Inverse Geodesic Length (IGL), which is used to gauge the efficiency and connectedness of a graph. It is defined as the sum of inverse distances between every two vertices: IGL(G) = sum_{{u,v} subseteq V} 1/d_G(u,v). While MinIGL is W[1]-hard for parameter treewidth, and cannot be solved in 2^{o(k + n + m)} time, even on bipartite graphs with n vertices and m edges, the complexity status of the problem remains open in the case where G is a tree. We show that IGL is balanced on trees, to give a 2^O((n log n)^(5/6)) time, polynomial space algorithm.
The distance distribution of G is the sequence {a_i} describing the number of vertex pairs distance i apart in G: a_i = |{{u, v}: d_G(u, v) = i}|. Given only the distance distribution, one can easily determine graph parameters such as diameter, Wiener index, and particularly, the IGL. We show that the distance distribution of a tree can be computed in O(n log^2 n) time by reduction to polynomial multiplication. We also extend the result to graphs with small treewidth by showing that the first p values of the distance distribution can be computed in 2^(O(tw(G))) n^(1 + epsilon) sqrt(p) time, and the entire distance distribution can be computed in 2^(O(tw(G))) n^{1 + epsilon} time, when the diameter of G is O(n^epsilon\u27) for every epsilon\u27 > 0
Smaller ACC0 Circuits for Symmetric Functions
What is the power of constant-depth circuits with gates, that can
count modulo ? Can they efficiently compute MAJORITY and other symmetric
functions? When is a constant prime power, the answer is well understood:
Razborov and Smolensky proved in the 1980s that MAJORITY and require
super-polynomial-size circuits, where is any prime power not
dividing . However, relatively little is known about the power of
circuits for non-prime-power . For example, it is still open whether every
problem in can be computed by depth- circuits of polynomial size and
only gates.
We shed some light on the difficulty of proving lower bounds for
circuits, by giving new upper bounds. We construct circuits computing
symmetric functions with non-prime power , with size-depth tradeoffs that
beat the longstanding lower bounds for circuits for prime power .
Our size-depth tradeoff circuits have essentially optimal dependence on and
in the exponent, under a natural circuit complexity hypothesis.
For example, we show for every that every symmetric
function can be computed with depth-3 circuits of
size, for a constant depending only on
. That is, depth- circuits can compute any symmetric
function in \emph{subexponential} size. This demonstrates a significant
difference in the power of depth- circuits, compared to other models:
for certain symmetric functions, depth- circuits require
size [H{\aa}stad 1986], and depth-
circuits (for fixed prime power ) require size
[Smolensky 1987]. Even for depth-two circuits,
lower bounds were known [Barrington Straubing Th\'erien 1990].Comment: 15 pages; abstract edited to fit arXiv requirement
Diagonalizations over polynomial time computable sets
AbstractA formal notion of diagonalization is developed which allows to enforce properties that are related to the class of polynomial time computable sets (the class of polynomial time computable functions respectively), like, e.g., p-immunity. It is shown that there are sets—called p-generic— which have all properties enforceable by such diagonalizations. We study the behaviour and the complexity of p-generic sets. In particular, we show that the existence of p-generic sets in NP is oracle dependent, even if we assume P ≠ NP
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