Interleaving Distance between Merge Trees
نویسندگان
چکیده
Merge trees are topological descriptors of scalar functions. They record how the subsets of the domain where the function value does not exceed a given threshold are connected. We define a distance between merge trees, called an interleaving distance, and prove the stability of these trees, with respect to this distance, to perturbations of the functions that define them. We show that the interleaving distance is never smaller than the bottleneck distance between persistence diagrams.
منابع مشابه
The $\ell^\infty$-Cophenetic Metric for Phylogenetic Trees as an Interleaving Distance
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 `∞-cophenetic metric introduced by Cardona et al. This metric works by representing a phylogenetic tree with n labeled leaves as a point in R known as the cophenetic vector, then comparing the two resulting Euclidean points usi...
متن کاملMeasuring the Distance Between Merge Trees
Merge trees represent the topology of scalar functions. To assess the topological similarity of functions, one can compare their merge trees. To do so, one needs a notion of a distance between merge trees, which we define. We provide examples of using our merge tree distance and compare this new measure to other ways used to characterize topological similarity (bottleneck distance for persisten...
متن کاملA generalization of ACP using Belnap's logic
ACP is combined with Belnap’s four-valued logic via conditional composition (if–then–else). We show that the operators of ACP can be seen as instances of more general, conditional operators. For example, both the choice operator + and δ (deadlock) can be seen as instances of conditional composition, and the axiom x + δ = x follows from this perspective. Parallel composition is generalized to th...
متن کاملThe Behavior of Admixed Populations in Neighbor-Joining Inference of Population Trees
Neighbor-joining is one of the most widely used methods for constructing evolutionary trees. This approach from phylogenetics is often employed in population genetics, where distance matrices obtained from allele frequencies are used to produce a representation of population relationships in the form of a tree. In phylogenetics, the utility of neighbor-joining derives partly from a result that ...
متن کاملOn the spectra of reduced distance matrix of the generalized Bethe trees
Let G be a simple connected graph and {v_1,v_2,..., v_k} be the set of pendent (vertices of degree one) vertices of G. The reduced distance matrix of G is a square matrix whose (i,j)-entry is the topological distance between v_i and v_j of G. In this paper, we compute the spectrum of the reduced distance matrix of the generalized Bethe trees.
متن کامل