Fast Collisionless Pattern Formation by Anonymous, Position-Aware Robots

Abstract

We consider a scenario of n identical autonomous robots on a 2D grid. They are memoryless and do not communicate. Their initial configuration does not have to be connected. Each robot r knows its position p _r  ∈ Z ². In addition, each robot knows the connected pattern F to be formed. F may be given by a set of n points in Z ², or may be only partially described, e.g., by ”form a connected pattern”, or ”build a connected formation with minimum diameter” (Collisonless Gathering). We employ the Look-Compute-Move (LCM) model, and assume that in a time step each robot is able to move to an unoccupied neighboring grid vertex, thus guaranteeing that two robots will never collide, i.e., occupy the same position. The decision where to move solely depends on the configuration of its 2-hop neighborhood in the grid Z ².

First we consider a helpful intermediate problem - we call it the Lemmings problem - where collision at one single point g, known to all robots, is allowed and the goal is that all robots gather at g. We present an algorithm solving this problem in 2n + D − 1 time steps, where D denotes the maximum initial distance from any robot to g. This time bound is easily shown to be optimal up to a constant factor.

Based on this strategy, forming a connected pattern can be done within the same time bound. Forming a connected pattern F needs additional considerations. We show how to do so in time O(n + D ^*), where D ^* denotes the diameter of the point set consisting of the initial configuration and F. For Collisionless gathering we obtain the same time bound, up to constant factors. This significantly improves upon the previous upper bound of O(nD) for this problem presented in [5].

Supported by the German Research Foundation (DFG) within the Collaborative Research Center ”On-The-Fly Computing” (SFB 901).