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We present a lattice algorithm specifically designed for some classical applications of lattice reduction. The applications are for lattice bases with a generalized knapsack-type structure, where the target vectors are boundably short. For…
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:…
Neural Networks and Decision Trees: two popular techniques for supervised learning that are seemingly disconnected in their formulation and optimization method, have recently been combined in a single construct. The connection pivots on…
Some top-down problem specifications, if executed directly, may compute sub-problems repeatedly. Instead, we may want a bottom-up algorithm that stores solutions of sub-problems in a table to be reused. It can be tricky, however, to figure…
Tree-child networks are a recently-described class of directed acyclic graphs that have risen to prominence in phylogenetics (the study of evolutionary trees and networks). Although these networks have a number of attractive mathematical…
The problem of determining whether a graph $G$ contains another graph $H$ as a minor, referred to as the minor containment problem, is a fundamental problem in the field of graph algorithms. While it is NP-complete when $G$ and $H$ are…
We introduce a new class of functions that can be minimized in polynomial time in the value oracle model. These are functions $f$ satisfying $f(x)+f(y)\ge f(x \sqcap y)+f(x \sqcup y)$ where the domain of each variable $x_i$ corresponds to…
The intrinsic structure of binary fields poses a challenging complexity problem from both hardware and software point of view. Motivated by applications to modern cryptography, we describe some simple techniques aimed at performing…
A spanning tree of a graph $G$ is a connected acyclic spanning subgraph of $G$. We consider enumeration of spanning trees when $G$ is a $2$-tree, meaning that $G$ is obtained from one edge by iteratively adding a vertex whose neighborhood…
This paper focuses on parallel hash functions based on tree modes of operation for an inner Variable-Input-Length function. This inner function can be either a single-block-length (SBL) and prefix-free MD hash function, or a sponge-based…
Submodular functions are an important class of functions in combinatorial optimization which satisfy the natural properties of decreasing marginal costs. The study of these functions has led to strong structural properties with applications…
We present a new formulation of the loop-tree duality theorem for higher loop diagrams valid both for massless and massive cases. $l$-loop integrals are expressed as weighted sum of trees obtained from cutting $l$ internal propagators of…
Tree-width and path-width are well-known graph parameters. Many NP-hard graph problems allow polynomial-time solutions, when restricted to graphs of bounded tree-width or bounded path-width. In this work, we study the behavior of tree-width…
In this paper, we address the scenario where nodes with sensor data are connected in a tree network, and every node wants to compute a given symmetric Boolean function of the sensor data. We first consider the problem of computing a…
A simple approach is presented to study the asymptotic behavior of some algorithms with an underlying tree structure. It is shown that some asymptotic oscillating behaviors can be precisely analyzed without resorting to complex analysis…
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…
We introduce in this paper a runtime-efficient tree hashing algorithm for the identification of isomorphic subtrees, with two important applications in genetic programming for symbolic regression: fast, online calculation of population…
Given a rooted tree and a ranking of its leaves, what is the minimum number of inversions of the leaves that can be attained by ordering the tree? This variation of the problem of counting inversions in arrays originated in mathematical…
These notes are a written version of my talk given at the CARMA workshop in June 2017, with some additional material. I presented a few concepts that have recently been used in the computation of tree-level scattering amplitudes (mostly…
Statistical mechanics is applied to lossy compression using multilayer perceptrons for unbiased Boolean messages. We utilize a tree-like committee machine (committee tree) and tree-like parity machine (parity tree) whose transfer functions…