Related papers: On cycles and merge trees
Every finite metric tree has generalized roundness strictly greater than one. On the other hand, some countable metric trees have generalized roundness precisely one. The purpose of this paper is to identify some large classes of countable…
A matching complex of a simple graph $G$ is a simplicial complex with faces given by the matchings of $G$. The topology of matching complexes is mysterious; there are few graphs for which the homotopy type is known. Marietti and Testa…
We study an abstract notion of tree structure which lies at the common core of various tree-like discrete structures commonly used in combinatorics: trees in graphs, order trees, nested subsets of a set, tree-decompositions of graphs and…
Much information about a graph can be obtained by studying its spanning trees. On the other hand, a graph can be regarded as a 1-dimensional cell complex, raising the question of developing a theory of trees in higher dimension. As observed…
Merge trees, a type of topological descriptor, serve to identify and summarize the topological characteristics associated with scalar fields. They present a great potential for the analysis and visualization of time-varying data. First,…
Quasi-trees generalize trees in that the unique "path" between two nodes may be infinite and have any countable order type. They are used to define the rank-width of a countable graph in such a way that it is equal to the least upper-bound…
The tree is an essential data structure in many applications. In a distributed application, such as a distributed file system, the tree is replicated.To improve performance and availability, different clients should be able to update their…
We consider the inference of the structure of an undirected graphical model in an exact Bayesian framework. More specifically we aim at achieving the inference with close-form posteriors, avoiding any sampling step. This task would be…
A new tree model is introduced based on ordered trees, by distinguishing exactly one child of each node that \emph{has} children. The basic enumeration leads to a cubic equation of the generating function. The extraction of its coefficients…
We introduce a family of branch merging operations on continuum trees and show that Ford CRTs are distributionally invariant. This operation is new even in the special case of the Brownian CRT, which we explore in more detail. The…
The output of a machine learning algorithm can usually be represented by one or more multivariate functions of its input variables. Knowing the global properties of such functions can help in understanding the system that produced the data…
A hypergraph can be obtained from a simplicial complex by deleting some non-maximal simplices. By [11], a hypergraph gives an associated simplicial complex. By [4], the embedded homology of a hypergraph is the homology of the infimum chain…
Work introduces a hierarchical binary tree-based reduction that replaces standard self-attention. The core idea is to use a recursive Gated Linear Unit merge operation, achieving O(n) total merge operations O(log n) parallel depth O(n d^2)…
Chessboard complexes and their generalizations, as objects, and Discrete Morse theory, as a tool, are presented as a unifying theme linking different areas of geometry, topology, algebra and combinatorics. Edmonds and Fulkerson bottleneck…
Optimal Morse matchings reveal essential structures of cell complexes which lead to powerful tools to study discrete geometrical objects, in particular discrete 3-manifolds. However, such matchings are known to be NP-hard to compute on…
In this paper we face the problem of representation of functional data with the tools of algebraic topology. We represent functions by means of merge trees, which, like the more commonly used persistence diagrams, are invariant under…
Recent work has proven the existence of extreme inbreeding in a European ancestry sample taken from the contemporary UK population \cite{nature_01}. This result brings our attention again to a math problem related to inbreeding family trees…
The use of machine learning algorithms in finance, medicine, and criminal justice can deeply impact human lives. As a consequence, research into interpretable machine learning has rapidly grown in an attempt to better control and fix…
Consider the generalized iterated wreath product $S_{r_1}\wr \ldots \wr S_{r_k}$ of symmetric groups. We give a complete description of the traversal for the generalized iterated wreath product. We also prove an existence of a bijection…
This work addresses the intrinsic relationship between trees and networks (i.e. graphs). A complete (invertible) mapping is presented which allows trees to be mapped into weighted graphs and then backmapped into the original tree without…