Robot Networks with Homonyms: The Case of Patterns Formation

TL;DR

This paper explores how sharing limited identifiers among oblivious mobile robots affects their ability to form geometric patterns, revealing conditions under which anonymity does not limit computational power.

Contribution

It introduces a new model with shared identifiers, provides necessary and sufficient conditions for pattern formation, and shows that full anonymity does not reduce computational power in certain cases.

Findings

Shared identifiers enable pattern formation under specific conditions.

Full anonymity does not limit computational power when all robots share identifiers.

The Weber point computation is key for pattern algorithms.

Abstract

In this paper, we consider the problem of formation of a series of geometric patterns [4] by a network of oblivious mobile robots that communicate only through vision. So far, the problem has been studied in models where robots are either assumed to have distinct identifiers or to be completely anonymous. To generalize these results and to better understand how anonymity affects the computational power of robots, we study the problem in a new model, introduced recently in [5], in which n robots may share up to 1 <= h <= n different identifiers. We present necessary and sufficient conditions, relating symmetricity and homonymy, that makes the problem solvable. We also show that in the case where h = n, making the identifiers of robots invisible does not limit their computational power. This contradicts a result of [4]. To present our algorithms, we use a function that computes the Weber point for many regular and symmetric configurations. This function is interesting in its own right, since the problem of finding Weber points has been solved up to now for only few other patterns.