Pattern Formation for Fat Robots with Lights

TL;DR

This paper presents improved algorithms for pattern formation with fat robots that use fewer colors and require less agreement, operating efficiently in the fully synchronous model with probabilistic guarantees.

Contribution

It reduces the number of colors needed and removes axis-agreement requirements for pattern formation with fat robots in the synchronous model.

Findings

Algorithm requiring 7 colors with pattern scaling.

Algorithm requiring 8 colors without pattern scaling.

Algorithms run in O(n) + O(q log n) rounds with high probability.

Abstract

Given a set of $n\geq 1$ unit disk robots in the Euclidean plane, we consider the Pattern Formation problem, i.e., the robots must reposition themselves to form a given target pattern. This problem arises under obstructed visibility, where a robot cannot see another robot if there is a third robot on the straight line segment between the two robots. Recently, this problem was solved in the asynchonous model for fat robots that agree on at least one axis in the robots with lights model where each robot is equipped with an externally visible persistent light that can assume colors from a fixed set of colors [K. Bose, R. Adhikary, M. K. Kundu, and B. Sau. Arbitrary pattern formation by opaque fat robots with lights. CALDAM, pages 347-359, 2020]. In this work, we reduce the number of colors needed and remove the axis-agreement requirement in the fully synchronous model. In particular, we present an algorithm requiring 7 colors when scaling the target pattern is allowed and an 8-color algorithm if scaling is not allowed. Our algorithms run in $O(n) + O(q \log n)$ rounds with probability at least $1 - n^{-q}$.