Robot Positioning Using Torus Packing for Multisets

Authors Chung Shue Chen , Peter Keevash , Sean Kennedy, Élie de Panafieu , Adrian Vetta



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Author Details

Chung Shue Chen
  • Nokia Bell Labs, Nozay, France
Peter Keevash
  • Mathematical Institute, University of Oxford, UK
Sean Kennedy
  • Nokia Bell Labs, Ottawa, Canada
Élie de Panafieu
  • Nokia Bell Labs, Nozay, France
Adrian Vetta
  • McGill University, Montreal, Canada

Acknowledgements

Part of the work was carried out at the Laboratory for Information, Networking and Communication Sciences (https://www.lincs.fr). We thank Dr. Paolo Baracca, Siu-Wai Ho, Kenneth Shum and many colleagues for their great support and discussions.

Cite AsGet BibTex

Chung Shue Chen, Peter Keevash, Sean Kennedy, Élie de Panafieu, and Adrian Vetta. Robot Positioning Using Torus Packing for Multisets. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 43:1-43:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ICALP.2024.43

Abstract

We consider the design of a positioning system where a robot determines its position from local observations. This is a well-studied problem of considerable practical importance and mathematical interest. The dominant paradigm derives from the classical theory of de Bruijn sequences, where the robot has access to a window within a larger code and can determine its position if these windows are distinct. We propose an alternative model in which the robot has more limited observational powers, which we argue is more realistic in terms of engineering: the robot does not have access to the full pattern of colours (or letters) in the window, but only to the intensity of each colour (or the number of occurrences of each letter). This leads to a mathematically interesting problem with a different flavour to that arising in the classical paradigm, requiring new construction techniques. The parameters of our construction are optimal up to a constant factor, and computing the position requires only a constant number of arithmetic operations.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatorial algorithms
Keywords
  • Universal cycles
  • positioning systems
  • de Bruijn sequences

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References

  1. E. Aboufadel, T. Armstrong, and E. Smietana. Position coding. arXiv preprint, 2007. URL: https://arxiv.org/abs/0706.0869.
  2. R. Berkowitz and S. Kopparty. Robust positioning patterns. In Proceedings of the 27th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1937-1951, 2016. Google Scholar
  3. A. Blanca and A. Godbole. On universal cycles for new classes of combinatorial structures. SIAM Journal on Discrete Mathematics, 25(4):1832-1842, 2011. Publisher: SIAM. Google Scholar
  4. A. Bruckstein, T. Etzion, R. Giryes, N. Gordon, R. Holt, and D. Shuldiner. Simple and robust binary self-location patterns. IEEE Transactions on Information Theory, 58(7):4884-4889, 2012. Google Scholar
  5. J. Burns and C. Mitchell. Coding schemes for two-dimensional position sensing. Hewlett-Packard Laboratories, Technical Publications Department, 1992. Google Scholar
  6. Y. Chee, D. Dao, H. Kiah, S. Ling, and H. Wei. Binary robust positioning patterns with low redundancy and efficient locating algorithms. In Proceedings of the 30th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 2171-2184, 2019. Google Scholar
  7. C. S. Chen, P. Baracca, E. de Panafieu, and D. Michalopoulos. A positioning method exploiting ambient IoT deployments in controlled environments, Nokia Patent Application, 2023. Google Scholar
  8. Chung Shue Chen, Peter Keevash, Sean Kennedy, Élie de Panafieu, and Adrian Vetta. Robot positioning using torus packing for multisets, 2024. URL: https://arxiv.org/abs/2404.09981.
  9. F. Chung, P. Diaconis, and R. Graham. Universal cycles for combinatorial structures. Discrete Mathematics, 110(1-3):43-59, 1992. Publisher: North-Holland. Google Scholar
  10. D. Curtis, T. Hines, G. Hurlbert, and T. Moyer. Near-universal cycles for subsets exist. SIAM Journal on Discrete Mathematics, 23(3):1441-1449, 2009. Google Scholar
  11. M. Dȩbski and Z. Lonc. Universal cycle packings and coverings for k-subsets of an n-set. Graphs and Combinatorics, 32:2323-2337, 2016. Publisher: Springer. Google Scholar
  12. N. de Bruijn. Acknowledgement of priority to C. Flye Sainte-Marie on the counting of circular arrangements of 2ⁿ zeros and ones that show each n-letter word exactly once. EUT report. WSK, Dept. of Mathematics and Computing Science, 75-WSK-06, 1975. Google Scholar
  13. É. de Panafieu. Torus packing for multisets. Software, version 1.0., swhId: https://archive.softwareheritage.org/swh:1:dir:2eb591ca09fb8739e2135e17b5f595b7f25ed924;origin=https://gitlab.com/depanafieuelie/torus-packing-for-multisets;visit=swh:1:snp:977c7dfa75b83551f7238790b259989a998825a6;anchor=swh:1:rev:a59cfd49aabd0d6c7eab651836b2ebac621ccf31 (visited on 2024-06-17). URL: https://gitlab.com/depanafieuelie/torus-packing-for-multisets.
  14. S. Glock, F. Joos, D. Kühn, and D. Osthus. Euler tours in hypergraphs. arXiv preprint, 2020. URL: https://arxiv.org/abs/1808.07720.
  15. N. Hartsfield and G. Ringel. Pearls in graph theory: A comprehensive introduction. Courier Corporation, 2013. Google Scholar
  16. S.-W. Ho and C. S. Chen. Visible light communication based positioning using color sensor. In IEEE 8th Optoelectronics Global Conference (OGC), pages 1-5, 2023. Google Scholar
  17. G. Hurlbert and G. Isaak. On the de Bruijn Torus problem. Journal of Combinatorial Theory, Series A, 64(1):50-62, 1993. Publisher: Elsevier. Google Scholar
  18. G. Hurlbert and G. Isaak. New constructions for de Bruijn tori. Designs, Codes and Cryptography, 6(1):47-56, 1995. Google Scholar
  19. G. Hurlbert, T. Johnson, and J. Zahl. On universal cycles for multisets. Discrete Mathematics, 309(17):5321-5327, 2009. URL: https://doi.org/10.1016/j.disc.2008.04.050.
  20. G. Hurlbert, C. Mitchell, and K. Paterson. On the existence of de Bruijn tori with two by two windows. Journal of Combinatorial Theory, Series A, 76(2):213-230, 1996. Publisher: Elsevier. Google Scholar
  21. D. Knuth. Art of Computer Programming, Combinatorial Algorithms, volume 4A. Addison-Wesley Professional, 2011. Google Scholar
  22. H. Li, C. Zhou, S. Wang, Y. Lu, and X. Xiang. Two-dimensional gold matrix method for encoding two-dimensional optical arbitrary positions. Optics Express, 26(10):12742-12754, 2018. Google Scholar
  23. J. Luo, L. Fan, and H. Li. Indoor positioning systems based on visible light communication: State of the art. IEEE Commun. Surveys Tuts., 19(4):2871-2893, 2017. Google Scholar
  24. J. MacWilliams and N. Sloane. Pseudo-random sequences and arrays. Proceedings of the IEEE, 64(12):1715-1729, 1976. Google Scholar
  25. D. Makarov and A. Yashunsky. On a construction of easily decodable sub-de Bruijn arrays. Journal of Applied and Industrial Mathematics, 13:280-289, 2019. Publisher: Springer. Google Scholar
  26. M. Rahman, T. Li, and Y. Wang. Recent advances in indoor localization via visible lights: A survey. Sensors, 20(5), 2020. Google Scholar
  27. D. Schüsselbauer, A. Schmid, and R. Wimmer. Dothraki: Tracking tangibles atop tabletops through de-Bruijn tori. In Proceedings of the 15th International Conference on Tangible, Embedded, and Embodied Interaction, pages 1-10, 2021. Google Scholar
  28. F. Sinden. Sliding window codes. AT&T Bell Labs Technical Memorandum, 1985. Google Scholar
  29. I. Szentandrasi, M. Zachariáš, J. Havel, A. Herout, M. Dubska, and R. Kajan. Uniform marker fields: Camera localization by orientable de Bruijn tori. In IEEE International Symposium on Mixed and Augmented Reality (ISMAR), pages 319-320, 2012. Google Scholar
  30. Z. Yang, Z. Wang, J. Zhang, C. Huang, and Q. Zhang. Wearables can afford: Light-weight indoor positioning with visible light. In Proceedings of the 13th Annual International Conference on Mobile Systems, Applications, and Services, pages 317-330, 2015. Google Scholar
  31. F. Zafari, A. Gkelias, and K. Leung. A survey of indoor localization systems and technologies. IEEE Commun. Surveys Tuts., 2019. Google Scholar