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Exploration of High-Dimensional Grids by Finite State Machines

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Abstract

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.

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Notes

  1. In some related literature [17, 21, 21] the same model was referred to as asynchronous. We follow the terminology of semi-synchronous of [12] and the vast literature on autonomous mobile robots to avoid confusion with a fully asynchronous model.

  2. This is similar to the simulation of PDAs by counter machines - see Chapter 8.5 in Hopcroft, Motwani, and Ullman text [30]; however, the details of our implementation are completely different.

  3. Relative to \(a_1\).

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Authors and Affiliations

  1. Institute of Mathematics, Slovak Academy of Sciences, Bratislava, Slovakia

    Stefan Dobrev

  2. Department of CSSE, Concordia University, Montreal, Canada

    Lata Narayanan, Jaroslav Opatrny & Denis Pankratov

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Dobrev, S., Narayanan, L., Opatrny, J. et al. Exploration of High-Dimensional Grids by Finite State Machines. Algorithmica 86, 1700–1729 (2024). https://doi.org/10.1007/s00453-024-01207-6

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