Related papers: Abstract Huffman Coding and PIFO Tree Embeddings

We introduce the notions of preordered and heap-preordered forests, generalizing the construction of ordered and heap-ordered forests. We prove that the algebras of preordered and heap-preordered forests are Hopf for the cut coproduct, and…

Combinatorial Hopf algebras of trees exemplify the connections between operads and bialgebras. Painted trees were introduced recently as examples of how graded Hopf operads can bequeath Hopf structures upon compositions of coalgebras. We…

We study tree-to-tree transformations that can be defined in first-order logic or monadic second-order logic. We prove a decomposition theorem, which shows that every transformation can be obtained from prime transformations, such as…

This paper is concerned with the approximation of tensors using tree-based tensor formats, which are tensor networks whose graphs are dimension partition trees. We consider Hilbert tensor spaces of multivariate functions defined on a…

Description of cocommutative Hopf algebras associated with families of trees. Applications include Cayley's theorem on the number of rooted trees with n nodes, and Catalan's theorem on the number of rooted ordered trees with n nodes.

This paper presents a tree-to-tree transduction method for sentence compression. Our model is based on synchronous tree substitution grammar, a formalism that allows local distortion of the tree topology and can thus naturally capture…

Raising the order of the multipole expansion is a feasible approach for improving the accuracy of the treecode algorithm. However, a uniform order for the expansion would result in the inefficiency of the implementation, especially when the…

We prove that the Hopf algebra of parking functions and the Hopf algebra of ordered forests are isomorphic, using a rigidity theorem for a particular type of bialgebras.

In a recent series of communications we have shown that the reordering problem of bosons leads to certain combinatorial structures. These structures may be associated with a certain graphical description. In this paper, we show that there…

In this paper, we revisit the classical data compression problem for domain specific texts. It is well-known that classical Huffman algorithm is optimal with respect to prefix encoding and the compression is done at character level. Since…

The Grassmannian is an important object in Algebraic Geometry. One of the many techniques used to study the Grassmannian is to build a vector space from its points in the projective embedding and study the properties of the resulting linear…

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:…

We show that the genealogy of any self-similar fragmentation process can be encoded in a compact measured real tree. Under some Malthusian hypotheses, we compute the fractal Hausdorff dimension of this tree through the use of a natural…

A grove is a spanning forest of a planar graph in which every component tree contains at least one of a special subset of vertices on the outer face called nodes. For the natural probability measure on groves, we compute various connection…

In the context of Higman embeddings of recursive groups into finitely presented groups we suggest an algorithm which uses Higman operations to explicitly constructs the specific recursively enumerable sets of integer sequences arising…

Efficient optimal prefix coding has long been accomplished via the Huffman algorithm. However, there is still room for improvement and exploration regarding variants of the Huffman problem. Length-limited Huffman coding, useful for many…

Feature-mapping methods for topology optimization (FMTO) facilitate direct geometry extraction by leveraging high-level geometric descriptions of the designs. However, FMTO often relies solely on Boolean unions, which can restrict the…

Embeddings of graphs into distributions of trees that preserve distances in expectation are a cornerstone of many optimization algorithms. Unfortunately, online or dynamic algorithms which use these embeddings seem inherently randomized and…

This note presents an encoding and a decoding algorithms for a forest of (labelled) rooted uniform hypertrees and hypercycles in linear time, by using as few as $n - 2$ integers in the range $[1,n]$. It is a simple extension of the…

We investigate the structural relationship between prefix-free codes over the binary alphabet and a class of unlabeled rooted trees, which we call \emph{symmetric} trees. We establish a canonical correspondence between prefix-free codes and…