License: 
Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ICALP.2019.139
URN: urn:nbn:de:0030-drops-107153
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2019/10715/
Dobrev, Stefan ;
Narayanan, Lata ;
Opatrny, Jaroslav ;
Pankratov, Denis
Exploration of High-Dimensional Grids by Finite Automata
Abstract
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.
BibTeX - Entry
@InProceedings{dobrev_et_al:LIPIcs:2019:10715,
author = {Stefan Dobrev and Lata Narayanan and Jaroslav Opatrny and Denis Pankratov},
title = {{Exploration of High-Dimensional Grids by Finite Automata}},
booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)},
pages = {139:1--139:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-109-2},
ISSN = {1868-8969},
year = {2019},
volume = {132},
editor = {Christel Baier and Ioannis Chatzigiannakis and Paola Flocchini and Stefano Leonardi},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2019/10715},
URN = {urn:nbn:de:0030-drops-107153},
doi = {10.4230/LIPIcs.ICALP.2019.139},
annote = {Keywords: Multi-agent systems, finite state machines, high-dimensional grids, robot exploration, randomized agents, semi-synchronous and synchronous agents}
}
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Keywords: |
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Multi-agent systems, finite state machines, high-dimensional grids, robot exploration, randomized agents, semi-synchronous and synchronous agents |
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Collection: |
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46th International Colloquium on Automata, Languages, and Programming (ICALP 2019) |
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Issue Date: |
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2019 |
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Date of publication: |
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04.07.2019 |
