Dirichlet random samplers for multiplicative structures

Olivier Bodini, Jérémie Lumbroso

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

In 2001, Duchon, Flajolet, Louchard and Schaeffer introduced Boltzmann samplers, a radically novel way to efficiently generate huge random combinatorial objects without any preprocessing; the insight was that the probabilities can be obtained directly by evaluating the generating functions of combinatorials classes. Over the following decade, a vast array of papers has increased the formal expressiveness of these random samplers. Our paper introduces a new kind of sampler which generates multiplicative combinatorial structures, which enumerated by Dirichlet generating functions. Such classes, which are significantly harder to analyze than their additive counterparts, are at the intersection of combinatorics and analytic number theory. Indeed, one example we fully discuss is that of ordered factorizations. While we recycle many of the concepts of Boltzmann random sampling, our samplers no longer obey a Boltzmann distribution; we thus have coined a new name for them: Dirichlet samplers. These are very efficient as they can generate objects of size n in O((log n)2) worst-case time complexity. By providing a means by which to generate very large random multiplicative objects, our Dirichlet samplers can facilitate the investigation of these interesting, yet little studied structures. We also hope to illustrate some of our general ideas regarding the future direction for random sampling.

Original languageEnglish (US)
Title of host publication9th Meeting on Analytic Algorithmics and Combinatorics 2012, ANALCO 2012
PublisherSociety for Industrial and Applied Mathematics Publications
Pages83-97
Number of pages15
ISBN (Electronic)9781618396235
StatePublished - Jan 1 2012
Externally publishedYes
Event9th Meeting on Analytic Algorithmics and Combinatorics, ANALCO 2012 - Kyoto, Japan
Duration: Jan 16 2012 → …

Publication series

Name9th Meeting on Analytic Algorithmics and Combinatorics 2012, ANALCO 2012

Conference

Conference9th Meeting on Analytic Algorithmics and Combinatorics, ANALCO 2012
CountryJapan
CityKyoto
Period1/16/12 → …

All Science Journal Classification (ASJC) codes

  • Applied Mathematics
  • Materials Chemistry
  • Discrete Mathematics and Combinatorics

Fingerprint Dive into the research topics of 'Dirichlet random samplers for multiplicative structures'. Together they form a unique fingerprint.

  • Cite this

    Bodini, O., & Lumbroso, J. (2012). Dirichlet random samplers for multiplicative structures. In 9th Meeting on Analytic Algorithmics and Combinatorics 2012, ANALCO 2012 (pp. 83-97). (9th Meeting on Analytic Algorithmics and Combinatorics 2012, ANALCO 2012). Society for Industrial and Applied Mathematics Publications.