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


Part of the work was carried out at the Laboratory for Information, Networking and Communication Sciences ( 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)


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
  • Universal cycles
  • positioning systems
  • de Bruijn sequences


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