Flat Embeddings of Genetic and Distance Data

The idea of displaying data in the plane is very attractive in many different fields of research. This thesis will focus on distance-based phylogenetics and multidimensional scaling (MDS). Both types of method can be viewed as a high-dimensional data reduction to pairwise distances and visualization of the data based on these distances. The difference between phylogenetics and multidimensional scaling is that the first one aims at finding a network or a tree structure that fits the distances, whereas MDS does not fix any structure and objects are simply placed in a low-dimensional space so that distances in the solution fit distances in the input as good as possible. Chapter 1 provides an introduction to the phylogenetics and multidimensional scaling. Chapter 2 focuses on the theoretical background of flat split systems (planar split networks). We prove equivalences between flat split systems, planar split networks and loop-free acyclic oriented matroids of rank three. The latter is a convenient mathematical structure that we used to design the algorithm for computing planar split networks that is described in Chapter 3. We base our approach on the well established agglomerative algorithms Neighbor-Joining and Neighbor-Net. In Chapter 4 we introduce multidimensional scaling and propose a new method for computing MDS plots that is based on the agglomerative approach and spring embeddings. Chapter 5 presents several case studies that we use to compare both of our methods and some classical agglomerative approaches in the distance-based phylogenetics

Constructing and Drawing Regular Planar Split Networks

Split networks are commonly used to visualize collections of bipartitions, also called splits, of a finite set. Such collections arise, for example, in evolutionary studies. Split networks can be viewed as a generalization of phylogenetic trees and may be generated using the SplitsTree package. Recently, the NeighborNet method for generating split networks has become rather popular, in part because it is guaranteed to always generate a circular split system, which can always be displayed by a planar split network. Even so, labels must be placed on the "outside" of the network, which might be problematic in some applications. To help circumvent this problem, it can be helpful to consider so-called flat split systems, which can be displayed by planar split networks where labels are allowed on the inside of the network too. Here we present a new algorithm that is guaranteed to compute a minimal planar split network displaying a flat split system in polynomial time, provided the split system is given in a certain format. We will also briefly discuss two heuristics that could be useful for analyzing phylogeographic data and that allow the computation of flat split systems in this format in polynomial time