Related papers: Relating Interleaving and Fr\'echet Distances via …
Ancestral mixture model, proposed by Chen and Lindsay (2006), is an important model to build a hierarchical tree from high dimensional binary sequences. Mixture trees created from ancestral mixture models involve in the inferred…
In this paper, we study merge trees induced by a discrete Morse function on a tree. Given a discrete Morse function, we provide a method to constructing an induced merge tree and define a new notion of equivalence of discrete Morse…
We address the problem of building and maintaining distributed spanning trees in highly dynamic networks, in which topological events can occur at any time and any rate, and no stable periods can be assumed. In these harsh environments, we…
The mutational heterogeneity of tumours can be described with a tree representing the evolutionary history of the tumour. With noisy sequencing data there may be uncertainty in the inferred tree structure, while we may also wish to study…
In scientific visualization, scalar fields are often compared through edit distances between their merge trees. Typical tasks include ensemble analysis, feature tracking and symmetry or periodicity detection. Tree edit distances represent…
In this work we define a metric structure to compare functions defined on different merge trees. The metric introduced possesses some stability properties, which we illustrate within a standard topological data analysis (TDA) framework, and…
Comparative analysis of scalar fields is an important problem with various applications including feature-directed visualization and feature tracking in time-varying data. Comparing topological structures that are abstract and succinct…
In this work we define a novel edit distance for trees considered with some abstract weights on the edges. The metric is driven by the idea of considering trees as topological summaries in the context of persistence and topological data…
We give algorithms to compute the Fr\'echet distance of trees and graphs with bounded tree width. Our algorithms run in $O(n^2)$ time for trees of bounded degree, and $O(n^2\sqrt{n \log n})$ time for trees of arbitrary degree. For graphs of…
This is a survey article on trees, with a modest number of proofs to give a flavor of the way these topologies can be efficiently handled. Trees are defined in set-theorist fashion as partially ordered sets in which the elements below each…
The Fr\'echet distance is a popular similarity measure between curves. For some applications, it is desirable to match the curves under translation before computing the Fr\'echet distance between them. This variant is called the Translation…
The problem of comparing trees representing the evolutionary histories of cancerous tumors has turned out to be crucial, since there is a variety of different methods which typically infer multiple possible trees. A departure from the…
In a recent paper on 'Estimating Species Trees from Unrooted Gene Trees' Liu and Yu observe that the distance matrix on the underlying taxon set, which is built up from expected internode distances on gene trees under the multispecies…
We introduce new distance measures for comparing straight-line embedded graphs based on the Fr\'echet distance and the weak Fr\'echet distance. These graph distances are defined using continuous mappings and thus take the combinatorial…
There are many metrics available to compare phylogenetic trees since this is a fundamental task in computational biology. In this paper, we focus on one such metric, the $\ell^\infty$-cophenetic metric introduced by Cardona et al. This…
Given two binary trees on $N$ labeled leaves, the quartet distance between the trees is the number of disagreeing quartets. By permuting the leaves at random, the expected quartets distance between the two trees is…
Recently, $p$-presentation distances for $p\in [1,\infty]$ were introduced for merge trees and multiparameter persistence modules as more sensitive variations of the respective interleaving distances ($p=\infty)$. It is well-known that…
Metric graphs are meaningful objects for modeling complex structures that arise in many real-world applications, such as road networks, river systems, earthquake faults, blood vessels, and filamentary structures in galaxies. To study metric…
Agreement forests continue to play a central role in the comparison of phylogenetic trees since their introduction more than 25 years ago. More specifically, they are used to characterise several distances that are based on tree…
The reliability of a phylogenetic inference method from genomic sequence data is ensured by its statistical consistency. Bayesian inference methods produce a sample of phylogenetic trees from the posterior distribution given sequence data.…