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Tight Bounds for Deterministic High-Dimensional Grid Exploration

Authors Sebastian Brandt , Julian Portmann , Jara Uitto

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  • Computing methodologies → Mobile agents
Keywords
  • Mobile agents
  • finite automata
  • grid search

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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).

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Sebastian Brandt, Julian Portmann, and Jara Uitto. Tight Bounds for Deterministic High-Dimensional Grid Exploration. In 34th International Symposium on Distributed Computing (DISC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 179, pp. 13:1-13:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.DISC.2020.13

Author Details

Sebastian Brandt
  • ETH Zürich, Switzerland
Julian Portmann
  • ETH Zürich, Switzerland
Jara Uitto
  • Aalto University, Finland

Funding

  • Portmann, Julian: Supported by the Swiss National Foundation, under project number200021_184735.

References

  1. Susanne Albers and Monika Henzinger. Exploring Unknown Environments. SIAM Journal on Computing, 29:1164-1188, 2000. Google Scholar
  2. Romas Aleliunas, Richard M. Karp, Richard J. Lipton, Laszlo Lovasz, and Charles Rackoff. Random Walks, Universal Traversal Sequences, and the Complexity of Maze Problems. In Foundations of Computer Science (FOCS), pages 218-223, 1979. Google Scholar
  3. Ricardo A. Baeza-Yates, Joseph C. Culberson, and Gregory J. E. Rawlins. Searching in the Plane. Information and Computation, 106:234-252, 1993. Google Scholar
  4. Anatole Beck. On the Linear Search Problem. Israel Journal of Mathematics, 1964. Google Scholar
  5. M. Blum and W. J. Sakoda. On the Capability of Finite Automata in 2 and 3 Dimensional Space. In Foundations of Computer Science (FOCS), pages 147-161, 1977. Google Scholar
  6. Manuel Blum and Dexter Kozen. On the Power of the Compass (or, Why Mazes Are Easier to Search Than Graphs). In Foundations of Computer Science (FOCS), pages 132-142, 1978. Google Scholar
  7. Sebastian Brandt, Jara Uitto, and Roger Wattenhofer. A Tight Lower Bound for Semi-Synchronous Collaborative Grid Exploration. In International Symposium on Distributed Computing (DISC), pages 13:1-13:17, 2018. Google Scholar
  8. Lothar Budach. Automata and Labyrinths. Mathematische Nachrichten, 86(1):195-282, 1978. Google Scholar
  9. Lihi Cohen, Yuval Emek, Oren Louidor, and Jara Uitto. Exploring an Infinite Space with Finite Memory Scouts. In Symposium on Discrete Algorithms (SODA), pages 207-224, 2017. Google Scholar
  10. Xiaotie Deng and Christos Papadimitriou. Exploring an Unknown Graph. Journal of Graph Theory, 32:265-297, 1999. Google Scholar
  11. Krzysztof Diks, Pierre Fraigniaud, Evangelos Kranakis, and Andrzej Pelc. Tree Exploration with Little Memory. Journal of Algorithms, 51:38-63, 2004. Google Scholar
  12. Yann Disser, Jan Hackfeld, and Max Klimm. Undirected Graph Exploration with Θ(log log n) Pebbles. In Symposium on Discrete Algorithms (SODA), pages 25-39, 2016. Google Scholar
  13. Stefan Dobrev, Lata Narayanan, Jaroslav Opatrny, and Denis Pankratov. Exploration of High-Dimensional Grids by Finite Automata. In International Colloquium on Automata, Languages, and Programming (ICALP), pages 139:1-139:16, 2019. Google Scholar
  14. Stefan Dobrev, Lata Narayanan, Jaroslav Opatrny, and Denis Pankratov. Exploration of high-dimensional grids by finite state machines. CoRR, abs/1902.03693, 2019. URL: http://arxiv.org/abs/1902.03693.
  15. Yuval Emek, Tobias Langner, David Stolz, Jara Uitto, and Roger Wattenhofer. How Many Ants Does it Take to Find the Food? Theor. Comput. Sci., 608:255-267, 2015. Google Scholar
  16. Yuval Emek, Tobias Langner, Jara Uitto, and Roger Wattenhofer. Solving the ANTS Problem with Asynchronous Finite State Machines. In International Colloquium on Automata, Languages and Programming (ICALP), pages 471-482, 2014. Google Scholar
  17. Ofer Feinerman, Amos Korman, Zvi Lotker, and Jean-Sebastien Sereni. Collaborative Search on the Plane Without Communication. In Principles of Distributed Computing (PODC), pages 77-86, 2012. Google Scholar
  18. Pierre Fraigniaud, David Ilcinkas, Guy Peer, Andrzej Pelc, and David Peleg. Graph Exploration by a Finite Automaton. Theoretical Computer Science, 345(2-3):331-344, 2005. Google Scholar
  19. Frank Hoffmann. One Pebble Does Not Suffice to Search Plane Labyrinths. In FCT, pages 433-444, 1981. Google Scholar
  20. Alejandro López-Ortiz and Graeme Sweet. Parallel Searching on a Lattice. In CCCG, pages 125-128, 2001. Google Scholar
  21. Petrişor Panaite and Andrzej Pelc. Exploring Unknown Undirected Graphs. In Symposium on Discrete Algorithms (SODA), pages 316-322, 1998. Google Scholar
  22. H. A. Rollik. Automaten in Planaren Graphen, pages 266-275. Springer Berlin Heidelberg, Berlin, Heidelberg, 1979. Google Scholar
  23. Kazuo Sugihara and Ichiro Suzuki. Distributed Algorithms for Formation of Geometric Patterns with many Mobile Robots. Journal of Robotic Systems, 13(3):127-139, 1996. Google Scholar
  24. Ichiro Suzuki and Masafumi Yamashita. Distributed Anonymous Mobile Robots: Formation of Geometric Patterns. SIAM Journal on Computing, 28(4):1347-1363, 1999. Google Scholar
  25. Ichiro Suzuki and Masafurni Yamashita. Distributed Anonymous Mobile Robots - Formation and Agreement Problems. In Structural Information and Communication Complexity (SIROCCO), pages 1347-1363, 1996. Google Scholar
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