Tight Bounds for Deterministic High-Dimensional Grid Exploration

TL;DR

This paper establishes tight bounds on the number of deterministic finite automata agents needed for high-dimensional grid exploration, improving previous results and solving open questions in the field.

Contribution

It proves that 3 synchronous and 4 semi-synchronous agents are sufficient for exploring any constant-dimensional grid, with matching lower bounds, and extends these results to polynomial-time and unoriented grids.

Findings

3 synchronous agents suffice for high-dimensional exploration

4 semi-synchronous agents suffice for high-dimensional exploration

Bounds are proven to be optimal and tight

Abstract

We study the problem of exploring an oriented grid with autonomous agents governed by finite automata. In the case of a 2-dimensional grid, the question how many agents are required to explore the grid, or equivalently, find a hidden treasure in the grid, is fully understood in both the synchronous and the semi-synchronous setting. For higher dimensions, Dobrev, Narayanan, Opatrny, and Pankratov [ICALP'19] showed very recently that, surprisingly, a (small) constant number of agents suffices to find the treasure, independent of the number of dimensions, thereby disproving a conjecture by Cohen, Emek, Louidor, and Uitto [SODA'17]. Dobrev et al. left as an open question whether their bounds on the number of agents can be improved. We answer this question in the affirmative for deterministic finite automata: we show that 3 synchronous and 4 semi-synchronous agents suffice to explore an $n$-dimensional grid for any constant $n$. The bounds are optimal and notably, the matching lower bounds already hold in the 2-dimensional case. Our techniques can also be used to make progress on other open questions asked by Dobrev et al.: we prove that 4 synchronous and 5 semi-synchronous agents suffice for polynomial-time exploration, and we show that, under a natural assumption, 3 synchronous and 4 semi-synchronous agents suffice to explore unoriented grids of arbitrary dimension (which, again, is tight).