5,248 research outputs found
Reconciling taxonomy and phylogenetic inference: formalism and algorithms for describing discord and inferring taxonomic roots
Although taxonomy is often used informally to evaluate the results of
phylogenetic inference and find the root of phylogenetic trees, algorithmic
methods to do so are lacking. In this paper we formalize these procedures and
develop algorithms to solve the relevant problems. In particular, we introduce
a new algorithm that solves a "subcoloring" problem for expressing the
difference between the taxonomy and phylogeny at a given rank. This algorithm
improves upon the current best algorithm in terms of asymptotic complexity for
the parameter regime of interest; we also describe a branch-and-bound algorithm
that saves orders of magnitude in computation on real data sets. We also
develop a formalism and an algorithm for rooting phylogenetic trees according
to a taxonomy. All of these algorithms are implemented in freely-available
software.Comment: Version submitted to Algorithms for Molecular Biology. A number of
fixes from previous versio
Product Dimension of Forests and Bounded Treewidth Graphs
The product dimension of a graph G is defined as the minimum natural number l
such that G is an induced subgraph of a direct product of l complete graphs. In
this paper we study the product dimension of forests, bounded treewidth graphs
and k-degenerate graphs. We show that every forest on n vertices has a product
dimension at most 1.441logn+3. This improves the best known upper bound of
3logn for the same due to Poljak and Pultr. The technique used in arriving at
the above bound is extended and combined with a result on existence of
orthogonal Latin squares to show that every graph on n vertices with a
treewidth at most t has a product dimension at most (t+2)(logn+1). We also show
that every k-degenerate graph on n vertices has a product dimension at most
\ceil{8.317klogn}+1. This improves the upper bound of 32klogn for the same by
Eaton and Rodl.Comment: 12 pages, 3 figure
Constrained Ramsey Numbers
For two graphs S and T, the constrained Ramsey number f(S, T) is the minimum
n such that every edge coloring of the complete graph on n vertices, with any
number of colors, has a monochromatic subgraph isomorphic to S or a rainbow
(all edges differently colored) subgraph isomorphic to T. The Erdos-Rado
Canonical Ramsey Theorem implies that f(S, T) exists if and only if S is a star
or T is acyclic, and much work has been done to determine the rate of growth of
f(S, T) for various types of parameters. When S and T are both trees having s
and t edges respectively, Jamison, Jiang, and Ling showed that f(S, T) <=
O(st^2) and conjectured that it is always at most O(st). They also mentioned
that one of the most interesting open special cases is when T is a path. In
this work, we study this case and show that f(S, P_t) = O(st log t), which
differs only by a logarithmic factor from the conjecture. This substantially
improves the previous bounds for most values of s and t.Comment: 12 pages; minor revision
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