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Balanced search trees are widely used in computer science to efficiently maintain dynamic ordered data. To support efficient set operations (e.g., union, intersection, difference) using trees, the join-based framework is widely studied.…
Efficient indexing is fundamental for multi-dimensional data management and analytics. An emerging tendency is to directly learn the storage layout of multi-dimensional data by simple machine learning models, yielding the concept of Learned…
Intersective polynomials are polynomials in $\Z[x]$ having roots every modulus. For example, $P_1(n)=n^2$ and $P_2(n)=n^2-1$ are intersective polynomials, but $P_3(n)=n^2+1$ is not. The purpose of this note is to deduce, using results of…
Arboreal networks are a generalization of rooted trees, defined by keeping the tree-like structure, but dropping the requirement for a single root. Just as the class of cographs is precisely the class of undirected graphs that can be…
Computational predictions of mass spectra from molecules have enabled the discovery of clinically relevant metabolites. However, such predictive tools are still limited as they occupy one of two extremes, either operating (a) by fragmenting…
For a graph G, the generating function of rooted forests, counted by the number of connected components, can be expressed in terms of the eigenvalues of the graph Laplacian. We generalize this result from graphs to cell complexes of…
Global optimization of decision trees is a long-standing challenge in combinatorial optimization, yet such models play an important role in interpretable machine learning. Although the problem has been investigated for several decades, only…
Variable trees are a new method for the exploration of discrete multivariate data. They display nested subsets and corresponding frequencies and percentages. Manual calculation of these quantities can be laborious, especially when there are…
Let $\mathcal{A}$ be a real line arrangement and $\mathcal{D}(\mathcal{A})$ the module of $\mathcal{A}$-derivations view as the set of polynomial vector fields which possess $\mathcal{A}$ as an invariant set. We first characterize…
A tree with $n$ vertices has at most $95^{n/13}$ minimal dominating sets. The growth constant $\lambda = \sqrt[13]{95} \approx 1.4194908$ is best possible. It is obtained in a semi-automatic way as a kind of "dominant eigenvalue" of a…
The paper is devoted to developing subdifferential theory for set-valued mappings taking values in ordered infinite-dimensional spaces. This study is motivated by applications to problems of vector and set optimization with various…
Decision trees are among the most popular machine learning models and are used routinely in applications ranging from revenue management and medicine to bioinformatics. In this paper, we consider the problem of learning optimal binary…
We introduce a semiparametric approach to neighbor-based classification. We build off the recently proposed Boundary Trees algorithm by Mathy et al.(2015) which enables fast neighbor-based classification, regression and retrieval in large…
In this paper we enumerate the cardinalities for the set of all vertices of outdegree $\ge k$ at level $\ge \ell$ among all rooted ordered $d$-trees with $n$ edges. Our results unite and generalize several previous works in the literature.
A weighted bicolored plane tree is a bicolored plane tree whose edges are endowed with positive integral weights. The degree of a vertex is defined as the sum of the weights of the edges incident to this vertex. Using the theory of dessins…
A tree $T$ on $2^n$ vertices is called set-sequential if the elements in $V(T)\cup E(T)$ can be labeled with distinct nonzero $(n+1)$-dimensional $01$-vectors such that the vector labeling each edge is the component-wise sum modulo $2$ of…
We tackle the problem of a combinatorial classification of finite metric spaces via their fundamental polytopes, as suggested by Vershik in 2010. In this paper we consider a hyperplane arrangement associated to every split pseudometric and,…
An alphabetic binary tree formulation applies to problems in which an outcome needs to be determined via alphabetically ordered search prior to the termination of some window of opportunity. Rather than finding a decision tree minimizing…
We define a bivariate polynomial for unlabeled rooted trees and show that the polynomial of an unlabeled rooted tree $T$ is the generating function of a class of subtrees of $T$. We prove that the polynomial is a complete isomorphism…
We divide the class of infinite computable trees into three types. For the first and second types, $0'$ computes a nontrivial self-embedding while for the third type $0''$ computes a nontrivial self-embedding. These results are optimal and…