Related papers: Random generation of group elements using combinat…
Let $G$ be a finite group generated by $k$ elements. The well-known product replacement algorithm provides an effective method for sampling generating sets of $G$. We study a refinement of this algorithm that is designed to output…
The Boltzmann model for the random generation of "decomposable" combinatorial structures is a set of techniques that allows for efficient random sampling algorithms for a large class of families of discrete objects. The usual requirement of…
Many groups possess highly symmetric generating sets that are naturally endowed with an underlying combinatorial structure. Such generating sets can prove to be extremely useful both theoretically in providing new existence proofs for…
Dixon's famous theorem states that the group generated by two random permutations of a finite set is generically either the whole symmetric group or the alternating group. In the context of random generation of finite groups this means that…
It is well known that every finite simple group can be generated by two elements and this leads to a wide range of problems that have been the focus of intensive research in recent years. In this survey article we discuss some of the…
Algorithms to generate various combinatorial structures find tremendous importance in computer science. In this paper, we begin by reviewing an algorithm proposed by Rohl that generates all unique permutations of a list of elements which…
We give an efficient algorithm to randomly generate finitely generated subgroups of a given size, in a finite rank free group. Here, the size of a subgroup is the number of vertices of its representation by a reduced graph such as can be…
In 1986 S. Sattolo introduced a simple algorithm for uniform random generation of cyclic permutations on a fixed number of symbols. This algorithm is very similar to the standard method for generating a random permutation, but is less well…
Various specifiable combinatorial structures, with d extensive parameters, can be exactly sampled both by the recursive method, with linear arithmetic complexity if a heavy preprocessing is performed, or by the Boltzmann method, with…
Group theory is a particularly fertile field for the design of practical algorithms. Algorithms have been developed across the various branches of the subject and they find wide application. Because of its relative maturity, computational…
In this work we present a general and versatile algorithmic framework for exhaustively generating a large variety of different combinatorial objects, based on encoding them as permutations. This approach provides a unified view on many…
We show how to generate random derangements efficiently by two different techniques: random restricted transpositions and sequential importance sampling. The algorithm employing restricted transpositions can also be used to generate random…
The generation of random graphs using edge swaps provides a reliable method to draw uniformly random samples of sets of graphs respecting some simple constraints, e.g. degree distributions. However, in general, it is not necessarily…
This article introduces a new approach to principled and practical random variate generation with formal guarantees. The key idea is to first specify the desired probability distribution in terms of a finite-precision numerical program that…
In this short review we introduce group field theory, a particular class of random tensor models, which represents nowadays one of the candidates for a fundamental theory of quantum gravity. We insist on the combinatorial richness of…
The rapid development of derandomization theory, which is a fundamental area in theoretical computer science, has recently led to many surprising applications outside its initial intention. We will review some recent such developments…
Let $G$ be a finite simple group. In this paper we consider the existence of small subsets $A$ of $G$ with the property that, if $y \in G$ is chosen uniformly at random, then with high probability $y$ invariably generates $G$ together with…
In this work we employ machine learning to understand structured mathematical data involving finite groups and derive a theorem about necessary properties of generators of finite simple groups. We create a database of all 2-generated…
Consider a class of decomposable combinatorial structures, using different types of atoms $\Atoms = \{\At_1,\ldots ,\At_{|{\Atoms}|}\}$. We address the random generation of such structures with respect to a size $n$ and a targeted…
This document presents a combinatorial framework for analyzing assembly systems using generating functions. We explore the theory through concrete examples, such as linear polymers, and develop recursive equations to characterize valid…