Polyhedral computational geometry for averaging metric phylogenetic trees

This paper investigates the computational geometry relevant to calculations of the Frechet mean and variance for probability distributions on the phylogenetic tree space of Billera, Holmes and Vogtmann, using the theory of probability measures on spaces of nonpositive curvature developed by Sturm. We show that the combinatorics of geodesics with a specified fixed endpoint in tree space are determined by the location of the varying endpoint in a certain polyhedral subdivision of tree space. The variance function associated to a finite subset of tree space has a fixed $C^\infty$ algebraic formula within each cell of the corresponding subdivision, and is continuously differentiable in the interior of each orthant of tree space. We use this subdivision to establish two iterative methods for producing sequences that converge to the Frechet mean: one based on Sturm's Law of Large Numbers, and another based on descent algorithms for finding optima of smooth functions on convex polyhedra. We present properties and biological applications of Frechet means and extend our main results to more general globally nonpositively curved spaces composed of Euclidean orthants.Comment: 43 pages, 6 figures; v2: fixed typos, shortened Sections 1 and 5, added counter example for polyhedrality of vistal subdivision in general CAT(0) cubical complexes; v1: 43 pages, 5 figure

The Complexity of Reliability Computations in Planar and Acyclic Graphs

We show that the problem of computing source-sink reliability is NP-hard, in fact # P-complete, even for undirected and acyclic directed source-sink planar graphs having vertex degree at most three. Thus the source-sink reliability problem is unlikely to have an efficient algorithm, even when the graph can be laid out on a rectilinear grid

<i>Ceramium botryocarpum</i> and <i>C. secundatum</i> re-evaluated

In the British Isles the genus Ceramium is represented by 15 species, divided into to groups. The two groups without cortical spines are (1) fully corticated species and (2) those with ecorticated internodes. Group 1 species are very difficult to distinguish. In particular the key morphological features that discriminate between C. botryocarpum and C. secundatum include the number of periaxial cells and presence of adventitious branching (Maggs and Hommersand, 1993). However, these features may be influenced by the environment. By using various molecular markers, growing cultures in different conditions and crossing experiments we aim to clarify the relationship between the species of group 1.Analysis of the formalin preserved vouchers showed that C. botryocarpum and C. secundatum are morphologically almost identical. They only differ in the number of periaxial cells (6-7 for C. botryocarpum and 7-8 for C. secundatum) and by the more robust, larger thallus of C. secundatum. Culture studies showed that the morphology of Ceramium is highly influenced by the environment. There was crossing with formation of tetrasporophytes between C. botryocarpum and C. secundatum. The phylogenetic analysis with the chloroplast marker (tufA/rpl31) and the mitochondrial marker (cox2-3 spacer, Gabrielsen 2002) clearly demonstrate that C. botryocarpum and C. secundatum are not respectively monophyletic. Analysis of multiple samples and with different techniques confirmed that C. botryocapum Griffiths ex Harvey (1848) is a later synonym of C. secundatum Lyngbye (1819)

A new approach to solving three combinatorial enumeration problems on planar graphs

The purpose of this paper is to show how the technique of delta-wye graph reduction provides an alternative method for solving three enumerative function evaluation problems on planar graphs. In particular, it is shown how to compute the number of spanning trees and perfect matchings, and how to evaluate energy in the Ising spin glass model of statistical mechanics. These alternative algorithms require O(n2) arithmetic operations on an n-vertex planar graph, and are relatively easy to implement