We present two algorithms for computing the geodesic distance between phylogenetic trees in tree space, as introduced by Billera, Holmes, and Vogtmann (2001). We show that the possible combinatorial types of shortest paths between two trees can be compactly represented by a partially ordered set. We calculate the shortest distance along each candidate path by converting the problem into one of finding the shortest path through a certain region of Euclidean space. In particular, we show there is a linear time algorithm for finding the shortest path between a point in the all positive orthant and a point in the all negative orthant of R^k contained in the subspace of R^k consisting of all orthants with the first i coordinates non-positive and the remaining coordinates non-negative for 0 <= i <= k.
We study the geometry of metrics and convexity structures on the space of phylogenetic trees, which is here realized as the tropical linear space of all \ ultrametrics. The ${\rm CAT}(0)$-metric of Billera-Holmes-Vogtman arises from the theory of orthant spaces. While its geodesics can be computed by the Owen-Provan algorithm, geodesic triangles are complicated. We show that the dimension of such a triangle can be arbitrarily high. Tropical convexity and the tropical metric behave better. They exhibit properties desirable for geometric statistics, such as geodesics of small depth.
Geodesic distance, sometimes called shortest path length, has proven useful in a great variety of applications, such as information retrieval on networks including treelike networked models. Here, our goal is to analytically determine the exact solutions to geodesic distances on two different families of growth trees which are recursively created upon an arbitrary tree $\mathcal{T}$ using two types of well-known operations, first-order subdivision and ($1,m$)-star-fractal operation. Different from commonly-used methods, for instance, spectral techniques, for addressing such a problem on growth…
In this paper we exhibit two infinite families of trees $\{T^1_n\}_{n \geq 17}$ and $\{T^2_n\}_{n \geq 17}$ on $n$ vertices, such that $T^1_n$ and $T^2_n$ are non-isomorphic, co-spectral, and the right-angled Coxeter groups (RACGs) based on $T^1_n$ and $T^2_n$ have the same geodesic growth with respect to the standard generating set. We then show that the spectrum of a tree does is not sufficient to determine the geodesic growth of the RACG based on that tree, by providing two infinite families of trees $\{S^1_n\}_{n \geq 11}$ and $\{S^2_n\}_{n \geq 11}$, on $n$ vertices, such that $S^1_n$ and…
Geodesic distance, commonly called shortest path length, has proved useful in a great variety of disciplines. It has been playing a significant role in search engine at present and so attracted considerable attention at the last few decades, particularly, almost all data structures and corresponding algorithms suitable to searching information generated based on treelike models. Hence, we, in this paper, study in detail geodesic distance on some treelike models which can be generated by three different types of operations, including first-order subdivision, ($1,m$)-star-fractal operation and $…
Let $G$ be a connected graph and $\ell : E(G) \to \mathbb{R}^+$ a length-function on the edges of $G$. The Steiner distance $\mathrm{sd}_G(A)$ of $A \subseteq V(G)$ within $G$ is the minimum length of a connected subgraph of $G$ containing $A$, where the length of a subgraph is the sum of the lengths of its edges. It is clear that every subgraph $H \subseteq G$, with the induced length-function $\ell|_{E(H)}$, satisfies $\mathrm{sd}_H(A) \geq \mathrm{sd}_G(A)$ for every $A \subseteq V(H)$. We call $H \subseteq G$ $k$-geodesic in $G$ if equality is attained for every $A \subseteq V(H)$ with $|A…
The problem of how to estimate diffusion on a graph effectively is of importance both theoretically and practically. In this paper, we make use of two widely studied indices, geodesic distance and mean first-passage time ($MFPT$) for random walk, to consider such a problem on some treelike models of interest. To this end, we first introduce several types of operations, for instance, $m$th-order subdivision and ($1,m$)-star-fractal operation, to generate the potential candidate models. And then, we develop a class of novel techniques based on mapping for calculating the exact formulas of the bo…
Consider all geodesics between two given points on a polyhedron. On the regular tetrahedron, we describe all the geodesics from a vertex to a point, which could be another vertex. Using the Stern--Brocot tree to explore the recursive structure of geodesics between vertices on a cube, we prove, in some precise sense, that there are twice as many geodesics between certain pairs of vertices than other pairs. We also obtain the fact that there are no geodesics that start and end at the same vertex on the regular tetrahedron or the cube.
We study the behavior of phylogenetic tree shapes in the tropical geometric interpretation of tree space. Tree shapes are formally referred to as tree topologies; a tree topology can also be thought of as a tree combinatorial type, which is given by the tree's branching configuration and leaf labeling. We use the tropical line segment as a framework to define notions of variance as well as invariance of tree topologies: we provide a combinatorial search theorem that describes all tree topologies occurring along a tropical line segment, as well as a setting under which tree topologies do not ch…
There are no known efficient algorithms to calculate distance in the one-skeleta of associahedra, a problem that is equivalent to finding rotation distance between rooted binary trees or the flip distance between polygonal triangulations. One measure of the difference between trees is the number of conflicting edge pairs, and a natural way of trying to find short paths is to minimize successively this number of conflicting edge pairs using flip operations in the corresponding triangulations. We describe examples that show that the number of such conflicts does not always decrease along geodesi…
In this paper, we study Gromov hyperbolicity and related parameters, that represent how close (locally) a metric space is to a tree from a metric point of view. The study of Gromov hyperbolicity for geodesic metric spaces can be reduced to the study of graph hyperbolicity. The main contribution of this paper is a new characterization of the hyperbolicity of graphs. This characterization has algorithmic implications in the field of large-scale network analysis. A sharp estimate of graph hyperbolicity is useful, e.g., in embedding an undirected graph into hyperbolic space with minimum distortion…
We study the statistical properties of geodesics, i.e. paths of minimal length, in large random planar quadrangulations. We extend Schaeffer's well-labeled tree bijection to the case of quadrangulations with a marked geodesic, leading to the notion of "spine trees", amenable to a direct enumeration. We obtain the generating functions for quadrangulations with a marked geodesic of fixed length, as well as with a set of "confluent geodesics", i.e. a collection of non-intersecting minimal paths connecting two given points. In the limit of quadrangulations with a large area n, we find in particula…
The phylogenetic tree space introduced by Billera, Holmes, and Vogtmann (BHV tree space) is a CAT(0) continuous space that represents trees with edge weights with an intrinsic geodesic distance measure. The geodesic distance measure unique to BHV tree space is well known to be computable in polynomial time, which makes it a potentially powerful tool for optimization problems in phylogenetics and phylogenomics. Specifically, there is significant interest in comparing and combining phylogenetic trees. For example, BHV tree space has been shown to be potentially useful in tree summary and consens…
We study the statistics of edges and vertices in the vicinity of a reference vertex (origin) within random planar quadrangulations and Eulerian triangulations. Exact generating functions are obtained for theses graphs with fixed numbers of edges and vertices at given geodesic distances from the origin. Our analysis relies on bijections with labeled trees, in which the labels encode the information on the geodesic distance from the origin. In the case of infinitely large graphs, we give in particular explicit formulas for the probabilities that the origin have given numbers of neighboring edges…
In this paper we consider the directed path-width and directed tree-width of recursively defined digraphs. As an important combinatorial tool, we show how the directed path-width and the directed tree-width can be computed for the disjoint union, order composition, directed union, and series composition of two directed graphs. These results imply the equality of directed path-width and directed tree-width for all digraphs which can be defined by these four operations. This allows us to show a linear-time solution for computing the directed path-width and directed tree-width of all these digrap…
We discuss the enumeration of planar graphs using bijections with suitably decorated trees, which allow for keeping track of the geodesic distances between faces of the graph. The corresponding generating functions obey non-linear recursion relations on the geodesic distance. These are solved by use of stationary multi-soliton tau-functions of suitable reductions of the KP hierarchy. We obtain a unified formulation of the (multi-) critical continuum limit describing large graphs with marked points at large geodesic distances, and obtain integrable differential equations for the corresponding s…
A connected graph has tree-depth at most $k$ if it is a subgraph of the closure of a rooted tree whose height is at most $k$. We give an algorithm which for a given $n$-vertex graph $G$, in time $\mathcal{O}(1.9602^n)$ computes the tree-depth of $G$. Our algorithm is based on combinatorial results revealing the structure of minimal rooted trees whose closures contain $G$.
Here we present a new fixed parameter tractable algorithm to compute the hybridization number r of two rooted, not necessarily binary phylogenetic trees on taxon set X in time (6^r.r!).poly(n)$, where n=|X|. The novelty of this approach is its use of terminals, which are maximal elements of a natural partial order on X, and several insights from the softwired clusters literature. This yields a surprisingly simple and practical bounded-search algorithm and offers an alternative perspective on the underlying combinatorial structure of the hybridization number problem.
Within the field of phylogenetics there is great interest in distance measures to quantify the dissimilarity of two trees. Recently, a new distance measure has been proposed: the Maximum Parsimony (MP) distance. This is based on the difference of the parsimony scores of a single character on both trees under consideration, and the goal is to find the character which maximizes this difference. Here we show that computation of MP distance on two \emph{binary} phylogenetic trees is NP-hard. This is a highly nontrivial extension of an earlier NP-hardness proof for two multifurcating phylogenetic t…
Consider finitely many points in a geodesic space. If the distance of two points is less than a fixed threshold, then we regard these two points as "near". Connecting near points with edges, we obtain a simple graph on the points, which is called a unit ball graph. If the space is the real line, then it is known as a unit interval graph. Unit ball graphs on a geodesic space describe geometric characteristics of the space in terms of graphs. In this article, we show that every unit ball graph on a geodesic space is (strongly) chordal if and only if the space is an $ \mathbb{R} $-tree and that e…
We derive the exact generating function for planar maps (genus zero fatgraphs) with vertices of arbitrary even valence and with two marked points at a fixed geodesic distance. This is done in a purely combinatorial way based on a bijection with decorated trees, leading to a recursion relation on the geodesic distance. The latter is solved exactly in terms of discrete soliton-like expressions, suggesting an underlying integrable structure. We extract from this solution the fractal dimensions at the various (multi)-critical points, as well as the precise scaling forms of the continuum two-point …
The minimum rank of a graph is the minimum possible rank of a real matrix whose zero-nonzero pattern is described by the graph. The current algorithms can compute efficiently the minimum rank of undirected trees. This paper provides an algorithm to compute in polynomial time the minimum rank of directed trees allowing loops.
