Related papers: Covering Metric Spaces by Few Trees

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…

LRM-Trees are an elegant way to partition a sequence of values into sorted consecutive blocks, and to express the relative position of the first element of each block within a previous block. They were used to encode ordinal trees and to…

Based on decision trees, many fields have arguably made tremendous progress in recent years. In simple words, decision trees use the strategy of "divide-and-conquer" to divide the complex problem on the dependency between input features and…

Spanners for metric spaces have been extensively studied, both in general metrics and in restricted classes, perhaps most notably in low-dimensional Euclidean spaces -- due to their numerous applications. Euclidean spanners can be viewed as…

We propose a flexible and multi-scale method for organizing, visualizing, and understanding datasets sampled from or near stratified spaces. The first part of the algorithm produces a cover tree using adaptive thresholds based on a…

We define and study analogs of probabilistic tree embedding and tree cover for directed graphs. We define the notion of a DAG cover of a general directed graph $G$: a small collection $D_1,\dots D_g$ of DAGs so that for all pairs of…

In this paper, in the context of the ``Dessins d'enfants'' theory, we give a combinatorial criterion for a plane tree to cover a tree from the classes of "chains" or "stars''. Besides, we discuss some applications of this result which are…

Measuring the complexity of tree structures can be beneficial in areas that use tree data structures for storage, communication, and processing purposes. This complexity can then be used to compress tree data structures to their…

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…

Robinson spaces are structures equipped with a total order that encodes comparative dissimilarity relationships. We study the problem of representing Robinson dissimilarity spaces into low-dimensional metric spaces. These representations…

This article discusses some recent trends in Ramsey theory on infinite structures. Trees and their Ramsey theory have been vital to these investigations. The main ideas behind the author's recent method of trees with coding nodes are…

Tree ensembles are flexible predictive models that can capture relevant variables and to some extent their interactions in a compact and interpretable manner. Most algorithms for obtaining tree ensembles are based on versions of boosting or…

We propose an extension of tree-based space-partitioning indexing structures for data with low intrinsic dimensionality embedded in a high dimensional space. We call this extension an Angle Tree. Our extension can be applied to both…

Cartesian tree matching is the problem of finding all substrings of a given text which have the same Cartesian trees as that of a given pattern. So far there is one linear-time solution for Cartesian tree matching, which is based on the KMP…

Connected acyclic graphs (trees) are data objects that hierarchically organize categories. Collections of trees arise in a diverse variety of fields, including evolutionary biology, public health, machine learning, social sciences and…

Distances on merge trees facilitate visual comparison of collections of scalar fields. Two desirable properties for these distances to exhibit are 1) the ability to discern between scalar fields which other, less complex topological…

Large tree structures are ubiquitous and real-world relational datasets often have information associated with nodes (e.g., labels or other attributes) and edges (e.g., weights or distances) that need to be communicated to the viewers. Yet,…

Using isometric embedding of metric trees into Banach spaces, this paper will investigate barycenters, type and cotype, and various measures of compactness of metric trees. A metric tree ($T$, $d$) is a metric space such that between any…

We investigate the interrelations between labeled trees and ultrametric spaces generated by these trees. The labeled trees, which generate complete ultrametrics, totally bounded ultrametrics, and discrete ones, are characterized up to…

We consider the probability that a spanning tree chosen uniformly at random from a graph can be partitioned into a fixed number $k$ of trees of equal size by removing $k-1$ edges. In that case, the spanning tree is called {\em splittable}.…