A local strategy for cleaning expanding cellular domains by simple robots

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.