Infinite Grid Exploration by Disoriented Robots

TL;DR

This paper investigates the problem of infinite grid exploration by autonomous, opaque robots with limited capabilities, providing impossibility results and three algorithms that optimize robot number and capabilities for solving the problem.

Contribution

It presents new impossibility proofs and three algorithms for grid exploration, including an optimal robot count solution and methods with different robot capabilities.

Findings

Two robots are insufficient for exploration.

Three or four robots are insufficient if coordinate systems are inconsistent.

Algorithms successfully explore the grid with 5, 6, or 7 robots under various conditions.

Abstract

We deal with a set of autonomous robots moving on an infinite grid. Those robots are opaque, have limited visibility capabilities, and run using synchronous Look-Compute-Move cycles. They all agree on a common chirality, but have no global compass. Finally, they may use lights of different colors, but except from that, robots have neither persistent memories, nor communication mean. We consider the infinite grid exploration (IGE) problem. For this problem we give two impossibility results and three algorithms, including one which is optimal in terms of number of robots. In more detail, we first show that two robots are not sufficient in our settings to solve the problem, even when robots have a common coordinate system. We then show that if the robots' coordinate systems are not self-consistent, three or four robots are not sufficient to solve the problem. Finally, we present three algorithms that solve the IGE problem in various settings. The first algorithm uses six robots with constant colors and a visibility range of one. The second one uses the minimum number of robots, i.e., five, as well as five modifiable colors, still under visibility one. The last algorithm requires seven oblivious anonymous robots, yet assuming visibility two. Notice that the two last algorithms also satisfy achieve exclusiveness.