On Fast Pattern Formation by Autonomous Robots

Abstract

We consider the fundamental problem of arranging a set of n autonomous robots (points) on a plane according to a given pattern. Each robot operates in a, largely oblivious, look-compute-move step. In this paper, we present a framework for the pattern formation problem. Leader election is key to this framework. For a given leader election time of $T_{ LE}$ (that could be deterministic or randomized), we show that the pattern formation problem can be solved in $O(T_{ LE})$ time on the semi-synchronous model using robots that either are transparent (i.e., the classical oblivious robots model where complete visibility is guaranteed at all times) or have lights with a constant number of colors (i.e., the robots with lights model where robots are not transparent but the colors of the lights are persistent between steps). We also prove that, for some cases, the $O(T_{ LE})$ time is optimal for pattern formation on the semi-synchronous model. These are the first sublinear-time results on pattern formation by autonomous robots in the look-compute-move framework. Furthermore, our results on the semi-synchronous model indicate that transparency and lights compensate for each other in the pattern formation problem. The proposed method also runs in $O(T_{LE}+\log n)$ time on the asynchronous model of robots with lights.