2026-07-23T03:49:47Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:10715 2024-03-06T10:46:32Z ddc:004 open_access

Exploration of High-Dimensional Grids by Finite Automata Dobrev, Stefan Narayanan, Lata Opatrny, Jaroslav Pankratov, Denis Multi-agent systems finite state machines high-dimensional grids robot exploration randomized agents semi-synchronous and synchronous agents 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Stefan Dobrev and Lata Narayanan and Jaroslav Opatrny and Denis Pankratov 2019 Is Part Of LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ICALP.2019.139 urn:nbn:de:0030-drops-107153 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.139 eng https://creativecommons.org/licenses/by/3.0/legalcode