Efficient Construction of Long Orientable Sequences

Authors Daniel Gabrić , Joe Sawada

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Author Details

Daniel Gabrić
  • University of Guelph, Canada
Joe Sawada
  • University of Guelph, Canada

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Daniel Gabrić and Joe Sawada. Efficient Construction of Long Orientable Sequences. In 35th Annual Symposium on Combinatorial Pattern Matching (CPM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 296, pp. 15:1-15:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


An orientable sequence of order n is a cyclic binary sequence such that each length-n substring appears at most once in either direction. Maximal length orientable sequences are known only for n ≤ 7, and a trivial upper bound on their length is 2^{n-1} - 2^{⌊(n-1)/2⌋}. This paper presents the first efficient algorithm to construct orientable sequences with asymptotically optimal length; more specifically, our algorithm constructs orientable sequences via cycle-joining and a successor-rule approach requiring O(n) time per bit and O(n) space. This answers a longstanding open question from Dai, Martin, Robshaw, Wild [Cryptography and Coding III (1993)]. Our sequences are applied to find new longest-known orientable sequences for n ≤ 20.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatorial algorithms
  • orientable sequence
  • de Bruijn sequence
  • concatenation tree
  • cycle-joining
  • universal cycle


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