Related papers: Cartesian trees and Lyndon trees
The intersection graph of a collection of trapezoids with corner points lying on two parallel lines is called a trapezoid graph. Using binary indexed tree data structure, we improve algorithms for calculating the size and the number of…
The $k^2$-tree is a successful compact representation of binary relations that exhibit sparseness and/or clustering properties. It can be extended to $d$ dimensions, where it is called a $k^d$-tree. The representation boils down to a long…
Generalized trees, we call them O-trees, are defined as hierarchical partial orders, i.e., such that the elements larger than any one are linearly ordered. Quasi-trees are, roughly speaking, undirected O-trees. For O-trees and quasi-trees,…
Relational structures are emerging as ubiquitous mathematical machinery in the semantics of open systems of various kinds. Cartesian bicategories are a well-known categorical algebra of relations that has proved especially useful in recent…
A data structure, called a biased range tree, is presented that preprocesses a set S of n points in R^2 and a query distribution D for 2-sided orthogonal range counting queries. The expected query time for this data structure, when queries…
We present a new characterization of $k$-trees based on their reduced clique graphs and $(k+1)$-line graphs, which are block graphs. We explore structural properties of these two classes, showing that the number of clique-trees of a…
In this article, we propose the approach to structural optimization of neural networks, based on the braid theory. The paper describes the basics of braid theory as applied to the description of graph structures of neural networks. It is…
This paper proves that two differently defined rooted binary trees are isomorphic. The first tree is one associated to a version of Farey sequences where the vertices correspond to the open intervals formed by two successive terms in the…
Dual-tree algorithms are a widely used class of branch-and-bound algorithms. Unfortunately, developing dual-tree algorithms for use with different trees and problems is often complex and burdensome. We introduce a four-part logical split:…
It has been hypothesized that the rather small number of crossings in real syntactic dependency trees is a side-effect of pressure for dependency length minimization. Here we answer a related important research question: what would be the…
Random Forests are one of the most popular classifiers in machine learning. The larger they are, the more precise is the outcome of their predictions. However, this comes at a cost: their running time for classification grows linearly with…
Rooted phylogenetic networks allow biologists to represent evolutionary relationships between present-day species by revealing ancestral speciation and hybridization events. A convenient and well-studied class of such networks are…
We develop a purely set-theoretic formalism for binary trees and binary graphs. We define a category of binary automata, and display it as a fibred category over the category of binary graphs. We also relate the notion of binary graphs to…
Working with tree graphs is always easier than with loopy ones and spanning trees are the closest tree-like structures to a given graph. We find a correspondence between the solutions of random K-satisfiability problem and those of spanning…
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…
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,…
A widely used method for determining the similarity of two labeled trees is to compute a maximum agreement subtree of the two trees. Previous work on this similarity measure is only concerned with the comparison of labeled trees of two…
Conversational recommender systems (CRS) aim to timely trace the dynamic interests of users through dialogues and generate relevant responses for item recommendations. Recently, various external knowledge bases (especially knowledge graphs)…
We discuss a notion of convergence for binary trees that is based on subtree sizes. In analogy to recent developments in the theory of graphs, posets and permutations we investigate some general aspects of the topology, such as a…
The sorting problem is one of the most relevant problems in computer science. Within the scope of modern computer science it has been studied for more than 70 years. In spite of these facts, new sorting algorithms have been developed in…