Broadcasting Automata and Patterns on Z^2

TL;DR

This paper studies the geometric shapes and metric approximation capabilities of broadcasting automata on Z^2, revealing how wave patterns can be used for digital shape encoding and approximating complex metrics like astroids.

Contribution

It introduces a novel analysis of wave-generated patterns in broadcasting automata and demonstrates their use in approximating discrete metrics, including complex concave shapes.

Findings

Categorization of digital discs and their patterns

Approximation of discrete metrics like astroids

Connection between wave patterns and metric approximation

Abstract

The Broadcasting Automata model draws inspiration from a variety of sources such as Ad-Hoc radio networks, cellular automata, neighbourhood se- quences and nature, employing many of the same pattern forming methods that can be seen in the superposition of waves and resonance. Algorithms for broad- casting automata model are in the same vain as those encountered in distributed algorithms using a simple notion of waves, messages passed from automata to au- tomata throughout the topology, to construct computations. The waves generated by activating processes in a digital environment can be used for designing a vari- ety of wave algorithms. In this chapter we aim to study the geometrical shapes of informational waves on integer grid generated in broadcasting automata model as well as their potential use for metric approximation in a discrete space. An explo- ration of the ability to vary the broadcasting radius of each node leads to results of categorisations of digital discs, their form, composition, encodings and gener- ation. Results pertaining to the nodal patterns generated by arbitrary transmission radii on the plane are explored with a connection to broadcasting sequences and ap- proximation of discrete metrics of which results are given for the approximation of astroids, a previously unachievable concave metric, through a novel application of the aggregation of waves via a number of explored functions.