A parallel wakeup problem and multi-room light switch strategies

TL;DR

This paper investigates the minimum switch states and conditions needed for distributed processors or prisoners to coordinate actions in shared environments, addressing open questions in symmetric and parallel wakeup problems.

Contribution

It provides new bounds and exact conditions for the minimum number of switch states required for solutions in symmetric wakeup scenarios with multiple rooms.

Findings

Determined the minimum switch states needed for certain wakeup problems.

Established exact conditions for the feasibility of solutions based on processor and register counts.

Answered open questions from prior research on parallel wakeup problems.

Abstract

The wakeup problem in distributed computing asks for a symmetric protocol that enables one of several processors to eventually guarantee that all (or, in a more general setting, enough) other processors have acted, using a shared register but no global clock. Dropping the symmetry requirement gives a well-known exercise often phrased in terms of prisoners entering, in an unknown sequence, a room equipped with a single binary switch, and using it to communicate. Kane and Kominers recently analysed a more general version of the latter with multiple parallel and indistinguishable rooms. We answer some open questions of Kane and Kominers regarding the minimum number of switch states needed for the prisoners to solve the problem. We also consider the symmetric ``wakeup'' version of this scenario, and establish exactly for which numbers of processors and registers a solution is possible.