733 research outputs found
Finding and counting vertex-colored subtrees
The problems studied in this article originate from the Graph Motif problem
introduced by Lacroix et al. in the context of biological networks. The problem
is to decide if a vertex-colored graph has a connected subgraph whose colors
equal a given multiset of colors . It is a graph pattern-matching problem
variant, where the structure of the occurrence of the pattern is not of
interest but the only requirement is the connectedness. Using an algebraic
framework recently introduced by Koutis et al., we obtain new FPT algorithms
for Graph Motif and variants, with improved running times. We also obtain
results on the counting versions of this problem, proving that the counting
problem is FPT if M is a set, but becomes W[1]-hard if M is a multiset with two
colors. Finally, we present an experimental evaluation of this approach on real
datasets, showing that its performance compares favorably with existing
software.Comment: Conference version in International Symposium on Mathematical
Foundations of Computer Science (MFCS), Brno : Czech Republic (2010) Journal
Version in Algorithmic
Improved Algorithms for the Point-Set Embeddability problem for Plane 3-Trees
In the point set embeddability problem, we are given a plane graph with
vertices and a point set with points. Now the goal is to answer the
question whether there exists a straight-line drawing of such that each
vertex is represented as a distinct point of as well as to provide an
embedding if one does exist. Recently, in \cite{DBLP:conf/gd/NishatMR10}, a
complete characterization for this problem on a special class of graphs known
as the plane 3-trees was presented along with an efficient algorithm to solve
the problem. In this paper, we use the same characterization to devise an
improved algorithm for the same problem. Much of the efficiency we achieve
comes from clever uses of the triangular range search technique. We also study
a generalized version of the problem and present improved algorithms for this
version of the problem as well
Phase transition in the sample complexity of likelihood-based phylogeny inference
Reconstructing evolutionary trees from molecular sequence data is a
fundamental problem in computational biology. Stochastic models of sequence
evolution are closely related to spin systems that have been extensively
studied in statistical physics and that connection has led to important
insights on the theoretical properties of phylogenetic reconstruction
algorithms as well as the development of new inference methods. Here, we study
maximum likelihood, a classical statistical technique which is perhaps the most
widely used in phylogenetic practice because of its superior empirical
accuracy.
At the theoretical level, except for its consistency, that is, the guarantee
of eventual correct reconstruction as the size of the input data grows, much
remains to be understood about the statistical properties of maximum likelihood
in this context. In particular, the best bounds on the sample complexity or
sequence-length requirement of maximum likelihood, that is, the amount of data
required for correct reconstruction, are exponential in the number, , of
tips---far from known lower bounds based on information-theoretic arguments.
Here we close the gap by proving a new upper bound on the sequence-length
requirement of maximum likelihood that matches up to constants the known lower
bound for some standard models of evolution.
More specifically, for the -state symmetric model of sequence evolution on
a binary phylogeny with bounded edge lengths, we show that the sequence-length
requirement behaves logarithmically in when the expected amount of mutation
per edge is below what is known as the Kesten-Stigum threshold. In general, the
sequence-length requirement is polynomial in . Our results imply moreover
that the maximum likelihood estimator can be computed efficiently on randomly
generated data provided sequences are as above.Comment: To appear in Probability Theory and Related Field
Convex Hull Realizations of the Multiplihedra
We present a simple algorithm for determining the extremal points in
Euclidean space whose convex hull is the nth polytope in the sequence known as
the multiplihedra. This answers the open question of whether the multiplihedra
could be realized as convex polytopes. We use this realization to unite the
approach to A_n-maps of Iwase and Mimura to that of Boardman and Vogt. We
include a review of the appearance of the nth multiplihedron for various n in
the studies of higher homotopy commutativity, (weak) n-categories,
A_infinity-categories, deformation theory, and moduli spaces. We also include
suggestions for the use of our realizations in some of these areas as well as
in related studies, including enriched category theory and the graph
associahedra.Comment: typos fixed, introduction revise
Connectivity Oracles for Graphs Subject to Vertex Failures
We introduce new data structures for answering connectivity queries in graphs
subject to batched vertex failures. A deterministic structure processes a batch
of failed vertices in time and thereafter
answers connectivity queries in time. It occupies space . We develop a randomized Monte Carlo version of our data structure
with update time , query time , and space
for any failure bound . This is the first connectivity oracle for
general graphs that can efficiently deal with an unbounded number of vertex
failures.
We also develop a more efficient Monte Carlo edge-failure connectivity
oracle. Using space , edge failures are processed in time and thereafter, connectivity queries are answered in
time, which are correct w.h.p.
Our data structures are based on a new decomposition theorem for an
undirected graph , which is of independent interest. It states that
for any terminal set we can remove a set of
vertices such that the remaining graph contains a Steiner forest for with
maximum degree
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