Beyond Series-Parallel Concurrent Systems: The Case of Arch Processes

Authors Olivier Bodini, Matthieu Dien, Antoine Genitrini, Alfredo Viola

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Olivier Bodini
  • Laboratoire d'Informatique de Paris-Nord, CNRS UMR 7030 - Institut Galilée - Université Paris-Nord, 99, avenue Jean-Baptiste Clément, 93430 Villetaneuse, France.
Matthieu Dien
  • Institute of Statistical Science, Academia Sinica, Taipei 115, Taiwan.
Antoine Genitrini
  • Sorbonne Université, CNRS, Laboratoire d'Informatique de Paris 6 -LIP6- UMR 7606, F-75005 Paris, France.
Alfredo Viola
  • Universidad de la República, Montevideo, Uruguay.

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Olivier Bodini, Matthieu Dien, Antoine Genitrini, and Alfredo Viola. Beyond Series-Parallel Concurrent Systems: The Case of Arch Processes. In 29th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 110, pp. 14:1-14:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


In this paper we focus on concurrent processes built on synchronization by means of futures. This concept is an abstraction for processes based on a main execution thread but allowing to delay some computations. The structure of a general concurrent process is a directed acyclic graph (DAG). Since the quantitative study of increasingly labeled DAG (directly related to processes) seems out of reach (this is a #P-complete problem), we restrict ourselves to the study of arch processes, a simplistic model of processes with futures. They are based on two parameters related to their sizes and their numbers of arches. The increasingly labeled structures seems not to be specifiable in the classical sense of Analytic Combinatorics, but we manage to derive a recurrence equation for the enumeration. For this model we first exhibit an exact and an asymptotic formula for the number of runs of a given process. The second main contribution is composed of a uniform random sampler algorithm and an unranking one that allow efficient generation and exhaustive enumeration of the runs of a given arch process.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Generating functions
  • Theory of computation → Generating random combinatorial structures
  • Theory of computation → Concurrency
  • Concurrency Theory
  • Future
  • Uniform Random Sampling
  • Unranking
  • Analytic Combinatorics


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