# The Combinatorics of Barrier Synchronization

## TL;DR

This paper explores the combinatorial aspects of barrier synchronization, providing formulas and algorithms for counting and randomly generating process executions without explicit state-space construction.

## Contribution

It introduces a systematic combinatorial approach and efficient algorithms for counting and sampling process executions in barrier synchronization.

## Key findings

- Developed a symbolic integral formula for counting executions
- Created a generic algorithm for uniform random sampling
- Proposed efficient algorithms for specific process subclasses

## Abstract

In this paper we study the notion of synchronization from the point of view of combinatorics. As a first step, we address the quantitative problem of counting the number of executions of simple processes interacting with synchronization barriers. We elaborate a systematic decomposition of processes that produces a symbolic integral formula to solve the problem. Based on this procedure, we develop a generic algorithm to generate process executions uniformly at random. For some interesting sub-classes of processes we propose very efficient counting and random sampling algorithms. All these algorithms have one important characteristic in common: they work on the control graph of processes and thus do not require the explicit construction of the state-space.