A local strategy for cleaning expanding cellular domains by simple robots

TL;DR

This paper introduces a local strategy for simple robots to contain spreading contamination in cellular environments, demonstrating that a certain speed threshold ensures successful cleaning while maintaining connected contamination regions.

Contribution

The paper proposes a new local strategy SEP for finite state robots that guarantees containment of spreading contamination with a specific speed requirement, improving upon previous bounds.

Findings

Strategy SEP succeeds if robot speed d ≥ 3(h+w).

Contamination remains connected under strategy SEP.

Greedy strategies require higher speed d ≥ 4(h+w).

Abstract

We present a strategy SEP for finite state machines tasked with cleaning a cellular environment in which a contamination spreads. Initially, the contaminated area is of height $h$ and width $w$. It may be bounded by four monotonic chains, and contain rectangular holes. The robot does not know the initial contamination, sensing only the eight cells in its neighborhood. It moves from cell to cell, $d$ times faster than the contamination spreads, and is able to clean its current cell. A speed of $d<\sqrt{2}(h+w)$ is in general not sufficient to contain the contamination. Our strategy SEP succeeds if $d \geq 3(h+w)$ holds. It ensures that the contaminated cells stay connected. Greedy strategies violating this principle need speed at least $d \geq 4(h+w)$; all bounds are up to small additive constants.