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Given a tree of weighted vertices, it is sometimes possible to break the tree into two equally-weighted subtrees within an allowable error. We give a fast algorithm that finds an edge which breaks the tree into equal-weight components or…

An important problem in geometric computing is defining and computing similarity between two geometric shapes, e.g. point sets, curves and surfaces, etc. Important geometric and topological information of many shapes can be captured by…

The reconstruction of large phylogenetic trees from data that violates clocklike evolution (or as a supertree constructed from any m input trees) raises a difficult question for biologists - how can one assign relative dates to the vertices…

Reconciling gene trees with a species tree is a fundamental problem to understand the evolution of gene families. Many existing approaches reconcile each gene tree independently. However, it is well-known that the evolution of gene families…

Considering a pair of genomes, the goal of rearrangement distance problems is to estimate how distant these genomes are from each other based on genome rearrangements. Seminal works in genome rearrangements assumed that both genomes being…

This paper presents a unified computational framework for the estimation of distances, geodesics and barycenters of merge trees. We extend recent work on the edit distance [106] and introduce a new metric, called the Wasserstein distance…

Early literature on genome rearrangement modelling views the problem of computing evolutionary distances as an inherently combinatorial one. In particular, attention was given to estimating distances using the minimum number of events…

This paper addresses the problem of finding a representation of a subtree distance, which is an extension of the tree metric. We show that a minimal representation is uniquely determined by a given subtree distance, and give a linear time…

The Genome Median Problem is an important problem in phylogenetic reconstruction under rearrangement models. It can be stated as follows: given three genomes, find a fourth that minimizes the sum of the pairwise rearrangement distances…

This paper give a simple linear-time algorithm that, given a weighted digraph, finds a spanning tree that simultaneously approximates a shortest-path tree and a minimum spanning tree. The algorithm provides a continuous trade-off: given the…

In a recent article [5], the authors claim that the distance between the b-chromatic index of a tree and a known upper bound is at most 1. At the same time, in [7] the authors claim to be able to construct a tree where this difference is…

We address the problem of computing distances between rankings that take into account similarities between candidates. The need for evaluating such distances is governed by applications as diverse as rank aggregation, bioinformatics, social…

An important application of distance geometry to biochemistry studies the embeddings of the vertices of a weighted graph in the three-dimensional Euclidean space such that the edge weights are equal to the Euclidean distances between…

Phylogenetic trees are a central tool in understanding evolution. They are typically inferred from sequence data, and capture evolutionary relationships through time. It is essential to be able to compare trees from different data sources…

The Hausdorff distance is a relatively new measure of similarity of graphs. The notion of the Hausdorff distance considers a special kind of a common subgraph of the compared graphs and depends on the structural properties outside of the…

Geometric graphs appear in many real-world data sets, such as road networks, sensor networks, and molecules. We investigate the notion of distance between embedded graphs and present a metric to measure the distance between two geometric…

Rotation distance between rooted binary trees measures the number of simple operations it takes to transform one tree into another. There are no known polynomial-time algorithms for computing rotation distance. We give an efficient,…

Many popular algorithms for searching the space of leaf-labelled trees are based on tree rearrangement operations. Under any such operation, the problem is reduced to searching a graph where vertices are trees and (undirected) edges are…

We describe a kernel of size 9k-8 for the NP-hard problem of computing the Tree Bisection and Reconnect (TBR) distance k between two unrooted binary phylogenetic trees. We achieve this by extending the existing portfolio of reduction rules…

In phylogenetic networks, it is desirable to estimate edge lengths in substitutions per site or calendar time. Yet, there is a lack of scalable methods that provide such estimates. Here we consider the problem of obtaining edge length…