Exploration of High-Dimensional Grids by Finite Automata
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.
Multi-agent systems
finite state machines
high-dimensional grids
robot exploration
randomized agents
semi-synchronous and synchronous agents
Theory of computation
139:1-139:16
Track C: Foundations of Networks and Multi-Agent Systems: Models, Algorithms and Information Management
Research supported by NSERC, Canada.
A full version of the paper is available at https://arXiv.org/abs/1902.03693.
Stefan
Dobrev
Stefan Dobrev
Institute of Mathematics, Slovak Academy of Sciences, Bratislava, Slovakia
http://www.ifi.savba.sk/~stefan/
Lata
Narayanan
Lata Narayanan
Department of CSSE, Concordia University, Montreal, Canada
https://users.encs.concordia.ca/~lata/
Jaroslav
Opatrny
Jaroslav Opatrny
Department of CSSE, Concordia University, Montreal, Canada
https://users.encs.concordia.ca/~opatrny/
Denis
Pankratov
Denis Pankratov
Department of CSSE, Concordia University, Montreal, Canada
https://users.encs.concordia.ca/~denisp/
10.4230/LIPIcs.ICALP.2019.139
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Stefan Dobrev, Lata Narayanan, Jaroslav Opatrny, and Denis Pankratov
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