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DOI: 10.4230/LIPIcs.ICALP.2019.139
URN: urn:nbn:de:0030-drops-107153
URL: http://drops.dagstuhl.de/opus/volltexte/2019/10715/
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Dobrev, Stefan ; Narayanan, Lata ; Opatrny, Jaroslav ; Pankratov, Denis

Exploration of High-Dimensional Grids by Finite Automata

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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}
}

Keywords: Multi-agent systems, finite state machines, high-dimensional grids, robot exploration, randomized agents, semi-synchronous and synchronous agents
Seminar: 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)
Issue Date: 2019
Date of publication: 08.07.2019


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