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.DISC.2020.13
URN: urn:nbn:de:0030-drops-130911
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2020/13091/
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Brandt, Sebastian ; Portmann, Julian ; Uitto, Jara

Tight Bounds for Deterministic High-Dimensional Grid Exploration

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LIPIcs-DISC-2020-13.pdf (0.5 MB)

Abstract

We study the problem of exploring an oriented grid with autonomous agents governed by finite automata. In the case of a 2-dimensional grid, the question how many agents are required to explore the grid, or equivalently, find a hidden treasure in the grid, is fully understood in both the synchronous and the semi-synchronous setting. For higher dimensions, Dobrev, Narayanan, Opatrny, and Pankratov [ICALP'19] showed very recently that, surprisingly, a (small) constant number of agents suffices to find the treasure, independent of the number of dimensions, thereby disproving a conjecture by Cohen, Emek, Louidor, and Uitto [SODA'17]. Dobrev et al. left as an open question whether their bounds on the number of agents can be improved. We answer this question in the affirmative for deterministic finite automata: we show that 3 synchronous and 4 semi-synchronous agents suffice to explore an n-dimensional grid for any constant n. The bounds are optimal and notably, the matching lower bounds already hold in the 2-dimensional case.
Our techniques can also be used to make progress on other open questions asked by Dobrev et al.: we prove that 4 synchronous and 5 semi-synchronous agents suffice for polynomial-time exploration, and we show that, under a natural assumption, 3 synchronous and 4 semi-synchronous agents suffice to explore unoriented grids of arbitrary dimension (which, again, is tight).

BibTeX - Entry

@InProceedings{brandt_et_al:LIPIcs:2020:13091,
  author =	{Sebastian Brandt and Julian Portmann and Jara Uitto},
  title =	{{Tight Bounds for Deterministic High-Dimensional Grid Exploration}},
  booktitle =	{34th International Symposium on Distributed Computing (DISC 2020)},
  pages =	{13:1--13:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-168-9},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{179},
  editor =	{Hagit Attiya},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2020/13091},
  URN =		{urn:nbn:de:0030-drops-130911},
  doi =		{10.4230/LIPIcs.DISC.2020.13},
  annote =	{Keywords: Mobile agents, finite automata, grid search}
}

Keywords: Mobile agents, finite automata, grid search
Collection: 34th International Symposium on Distributed Computing (DISC 2020)
Issue Date: 2020
Date of publication: 07.10.2020

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