Exploration of High-Dimensional Grids by Finite State Machines

We consider the problem of finding a treasure at an unknown point of an $n$-dimensional infinite grid, $n\geq 3$, by initially collocated finite state agents (scouts/robots). Recently, the problem has been well characterized for 2 dimensions for deterministic as well as randomized agents, both in synchronous and semi-synchronous models. It has been conjectured that $n+1$ randomized agents are necessary to solve this problem in the $n$-dimensional grid. In this paper we disprove the conjecture in a strong sense: we show that three randomized synchronous agents suffice to explore an $n$-dimensional grid for any $n$. Our algorithm is optimal in terms of the number of the agents. Our key insight is that a constant number of finite state machine agents can, by their positions and movements, implement a stack, which can store the path being explored. We also show how to implement our algorithm using: four randomized semi-synchronous agents; four deterministic synchronous agents; or five deterministic semi-synchronous agents. We give a different algorithm that uses $4$ deterministic semi-synchronous agents for the $3$-dimensional grid. This is provably optimal, and surprisingly, matches the result for $2$ dimensions. For $n\geq 4$, the time complexity of the solutions mentioned above is exponential in distance $D$ of the treasure from the starting point of the agents. We show that in the deterministic case, one additional agent brings the time down to a polynomial. Finally, we focus on algorithms that never venture much beyond the distance $D$. We describe an algorithm that uses $O(\sqrt{n})$ semi-synchronous deterministic agents that never go beyond $2D$, as well as show that any algorithm using $3$ synchronous deterministic agents in $3$ dimensions must travel beyond $\Omega(D^{3/2})$ from the origin.