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, \(n\ge 3\), by initially collocated finite automaton (FA) agents. Recently, the problem has been well characterized for 2 dimensions for deterministic as well as randomized FA agents, both in synchronous and semi-synchronous models (Brandt et al. in Proceedings of 32nd International Symposium on Distributed Computing (DISC) LIPCS 121:13:1–13:17, 2018; Emek et al. in Theor Comput Sci 608:255–267, 2015). It has been conjectured that \(n+1\) randomized FA agents are necessary to solve this problem in the n-dimensional grid (Cohen et al. in Proceedings of the 28th SODA, SODA ’17, pp 207–224, 2017). In this paper we disprove the conjecture in a strong sense: we show that three randomized synchronous FA 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 FA 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 FA agents; four deterministic synchronous FA agents; or five deterministic semi-synchronous FA agents. We give a different, no-stack algorithm that uses 4 deterministic semi-synchronous FA agents for the 3-dimensional grid. This is provably optimal in the number of agents and the exploration cost, and surprisingly, matches the result for 2 dimensions. For \(n\ge 4\), the time complexity of the stack-based algorithms 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 finite automaton agent brings the time down to a polynomial. We also show that any algorithm using 3 synchronous deterministic FA agents in 3 dimensions must travel beyond \(\Omega (D^{3/2})\) from the origin. Finally, we show that all the above algorithms can be generalized to unoriented grids. More specifically, six deterministic semi-synchronous FA agents are sufficient to locate the treasure in an unoriented n-dimensional grid.