Universal Dancing by Luminous Robots under Sequential Schedulers

TL;DR

This paper demonstrates that with luminous robots and sequential scheduling, universal choreographies can be achieved without prior restrictions, using an algorithm that leverages distributed counters and bounded choreography lengths.

Contribution

It introduces the concept of Universal Dancing problems under the LUMI model with sequential schedulers, removing previous restrictions and providing a new algorithmic solution.

Findings

Universal choreographies are feasible with luminous robots and sequential scheduling.

The choreography length is bounded by the number of robot colors and compositions of n.

An algorithm using distributed counters ensures choreography even with non-rigid movements.

Abstract

The Dancing problem requires a swarm of $n$ autonomous mobile robots to form a sequence of patterns, aka perform a choreography. Existing work has proven that some crucial restrictions on choreographies and initial configurations (e.g., on repetitions of patterns, periodicity, symmetries, contractions/expansions) must hold so that the Dancing problem can be solved under certain robot models. Here, we prove that these necessary constraints can be dropped by considering the LUMI model (i.e., where robots are endowed with a light whose color can be chosen from a constant-size palette) under the quite unexplored sequential scheduler. We formalize the class of Universal Dancing problems which require a swarm of $n$ robots starting from any initial configuration to perform a (periodic or finite) sequence of arbitrary patterns, only provided that each pattern consists of $n$ vertices (including multiplicities). However, we prove that, to be solvable under LUMI, the length of the feasible choreographies is bounded by the compositions of $n$ into the number of colors available to the robots. We provide an algorithm solving the Universal Dancing problem by exploiting the peculiar capability of sequential robots to implement a distributed counter mechanism. Even assuming non-rigid movements, our algorithm ensures spatial homogeneity of the performed choreography.