Probabilistic Asynchronous Arbitrary Pattern Formation

TL;DR

This paper introduces a probabilistic algorithm for oblivious mobile robots in the asynchronous model that forms arbitrary patterns without shared coordinate systems, using minimal randomness and enabling multiplicity points.

Contribution

It presents a novel probabilistic pattern formation algorithm that works asynchronously without shared coordinates and includes multiplicity points, advancing previous deterministic and probabilistic methods.

Findings

Operates fully asynchronously without shared coordinate assumptions.

Uses only one random bit per cycle per robot.

Enables formation of arbitrary patterns with multiplicity points.

Abstract

We propose a new probabilistic pattern formation algorithm for oblivious mobile robots that operates inthe ASYNC model. Unlike previous work, our algorithm makes no assumptions about the local coordinatesystems of robots (the robots do not share a common "North" nor a common "Right"), yet it preserves theability from any initial configuration that contains at least 5 robots to form any general pattern (and not justpatterns that satisfy symmetricity predicates). Our proposal also gets rid of the previous assumption (in thesame model) that robots do not pause while moving (so, our robots really are fully asynchronous), and theamount of randomness is kept low -- a single random bit per robot per Look-Compute-Move cycle is used.Our protocol consists in the combination of two phases, a probabilistic leader election phase, and a deterministicpattern formation one. As the deterministic phase does not use chirality, it may be of independentinterest in the deterministic context. A noteworthy feature of our algorithm is the ability to form patternswith multiplicity points (except the gathering case due to impossibility results), a new feature in the contextof pattern formation that we believe is an important asset of our approach.