Physical phenomena in science and engineering are frequently modeled using scalar fields. In scalar field topology, graph-based topological descriptors such as merge trees, contour trees, and Reeb graphs are commonly used to characterize topological changes in the (sub)level sets of scalar fields. One of the biggest challenges and opportunities to advance topology-based visualization is to understand and incorporate uncertainty into such topological descriptors to effectively reason about their underlying data. In this paper, we study a structural average of a set of labeled merge trees and use it to encode uncertainty in data. Specifically, we compute a 1-center tree that minimizes its maximum distance to any other tree in the set under a well-defined metric called the interleaving distance. We provide heuristic strategies that compute structural averages of merge trees whose labels do not fully agree. We further provide an interactive visualization system that resembles a numerical calculator that takes as input a set of merge trees and outputs a tree as their structural average. We also highlight structural similarities between the input and the average and incorporate uncertainty information for visual exploration. We develop a novel measure of uncertainty, referred to as consistency, via a metric-space view of the input trees. Finally, we demonstrate an application of our framework through merge trees that arise from ensembles of scalar fields. Our work is the first to employ interleaving distances and consistency to study a global, mathematically rigorous, structural average of merge trees in the context of uncertainty visualization.
The application of the interleaving distance to labeled merge trees can also be viewed as an interpretation of a metric for phylogenetic trees @cite_68 . Our framework differs from the previous work as it relies on a clean and simple metric-space view of input trees that is equipped with geodesics, as well as easy-to-implement algorithms.
@cite_68
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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 l∞-cophenetic metric introduced by This metric works by representing a phylogenetic tree with n labeled leaves as a point in ( R ^ n(n+1) 2 ) known as the cophenetic vector, then comparing the two resulting Euclidean points using the l∞ distance. Meanwhile, the interleaving distance is a formal categorical construction generalized from the definition of , originally introduced to compare persistence modules arising from the field of topological data analysis. We show that the l∞-cophenetic metric is an example of an interleaving distance. To do this, we define phylogenetic trees as a category of merge trees with some additional structure, namely, labelings on the leaves plus a requirement that morphisms respect these labels. Then we can use the definition of a flow on this category to give an interleaving distance. Finally, we show that, because of the additional structure given by the categories defined, the map sending a labeled merge tree to the cophenetic vector is, in fact, an isometric embedding, thus proving that the l∞-cophenetic metric is an interleaving distance. Abstract A metric on general phylogenetic trees is presented. This extends the work of most previous authors, who constructed metrics for binary trees. The metric presented in this paper makes possible the comparison of the many nonbinary phylogenetic trees appearing in the literature. This provides an objective procedure for comparing the different methods for constructing phylogenetic trees. The metric is based on elementary operations which transform one tree into another. Various results obtained in applying these operations are given. They enable the distance between any pair of trees to be calculated efficiently. This generalizes previous work by Bourque to the case where interior vertices can be labeled, and labels may contain more than one element or may be empty. We study distorted metrics on binary trees in the context of phylogenetic reconstruction. Given a binary tree T on n leaves with a path metric d, consider the pairwise distances d(u,v) between leaves. It is well known that these determine the tree and the d length of all edges. Here, we consider distortions @math of d such that, for all leaves u and v, it holds that @math if either d(u,v)< M + f 2 or @math , where d satisfies f ≤ d(e) ≤ g for all edges e. Given such distortions, we show how to reconstruct in polynomial time a forest T1, ... ,Tα such that the true tree T may be obtained from that forest by adding α-1 edges and α-1 ≤ 2-Ω(M g) n. Our distorted metric result implies a reconstruction algorithm of phylogenetic forests with a small number of trees from sequences of length logarithmic in the number of species. The reconstruction algorithm is applicable for the general Markov model. Both the distorted metric result and its applications to phylogeny are almost tight. Comparing and computing distances between phylogenetic trees are important biological problems, especially for models where edge lengths play an important role. The geodesic distance measure between two phylogenetic trees with edge lengths is the length of the shortest path between them in the continuous tree space introduced by Billera, Holmes, and Vogtmann. This tree space provides a powerful tool for studying and comparing phylogenetic trees, both in exhibiting a natural distance measure and in providing a euclidean-like structure for solving optimization problems on trees. An important open problem is to find a polynomial time algorithm for finding geodesics in tree space. This paper gives such an algorithm, which starts with a simple initial path and moves through a series of successively shorter paths until the geodesic is attained.