The Wake Up and Report Problem is Time-Equivalent to the Firing Squad Synchronization Problem

TL;DR

This paper demonstrates that the Wake Up and Report Problem and the Firing Squad Synchronization Problem are asymptotically time-equivalent in strongly-connected directed networks, unifying their complexity classes.

Contribution

It establishes the asymptotic time-equivalence of two fundamental synchronization problems using a novel algorithmic approach with the snake data structure.

Findings

The two problems are asymptotically time-equivalent up to a constant factor.

Several related synchronization problems are included in the same asymptotic time class.

The approach uses a complex application of the snake data structure.

Abstract

We consider several problems relating to strongly-connected directed networks of identical finite-state processors that work synchronously in discrete time steps. The conceptually simplest of these is the Wake Up and Report Problem; this is the problem of having a unique "root" processor send a signal to all other processors in the network and then enter a special "done" state only when all other processors have received the signal. The most difficult of the problems we consider is the classic Firing Squad Synchronization Problem; this is the much-studied problem of achieving macro-synchronization in a network given micro-synchronization. We show via a complex algorithmic application of the "snake" data structure first introduced in Even, Litman, and Winkler [ELW], that these two problems are asymptotically time-equivalent up to a constant factor. This result leads immediately to the inclusion of several other related problems into this new asymptotic time-class.