Exploration of High-Dimensional Grids by Finite Automata

Authors Stefan Dobrev, Lata Narayanan, Jaroslav Opatrny, Denis Pankratov

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Stefan Dobrev
  • Institute of Mathematics, Slovak Academy of Sciences, Bratislava, Slovakia
Lata Narayanan
  • Department of CSSE, Concordia University, Montreal, Canada
Jaroslav Opatrny
  • Department of CSSE, Concordia University, Montreal, Canada
Denis Pankratov
  • Department of CSSE, Concordia University, Montreal, Canada

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Stefan Dobrev, Lata Narayanan, Jaroslav Opatrny, and Denis Pankratov. Exploration of High-Dimensional Grids by Finite Automata. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 139:1-139:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


We consider the problem of finding a treasure at an unknown point of an n-dimensional infinite grid, n >= 3, by initially collocated finite automaton 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 [S. Brandt et al., 2018; Y. Emek et al., 2015]. It has been conjectured that n+1 randomized agents are necessary to solve this problem in the n-dimensional grid [L. Cohen et al., 2017]. 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 automaton 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 >= 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, if it exists, must travel beyond Omega(D^{3/2}) from the origin.

Subject Classification

ACM Subject Classification
  • Theory of computation
  • Multi-agent systems
  • finite state machines
  • high-dimensional grids
  • robot exploration
  • randomized agents
  • semi-synchronous and synchronous agents


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