Various Solutions to the Firing Squad Synchronization Problems

TL;DR

This paper explores various algorithms for the Firing Squad Synchronization Problem across different network topologies and communication constraints, providing solutions with diverse time complexities and modular signal-based methods.

Contribution

It introduces multiple classes of solutions for the Firing Squad Synchronization Problem on various network shapes and communication modes, with algorithms tailored to different time functions.

Findings

Solutions for line, ring, and square networks.

Algorithms with time complexities: n^2, n log n, n√n, 2^n.

Signal-based building blocks for synchronization algorithms.

Abstract

We present different classes of solutions to the Firing Squad Synchronization Problem on networks of different shapes. The nodes are finite state processors that work at unison discrete steps. The networks considered are the line, the ring and the square. For all of these models we have considered one and two-way communication modes and also constrained the quantity of information that adjacent processors can exchange each step. We are given a particular time expressed as a function of the number of nodes of the network, $f(n)$ and present synchronization algorithms in time $n^2$, $n \log n$, $n\sqrt n$, $2^n$. The solutions are presented as {\em signals} that are used as building blocks to compose new solutions for all times expressed by polynomials with nonnegative coefficients.