[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81488-en":3,"doc-seo-81488-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},81488,1099513958607,"Jiven","https://ap-avatar.wpscdn.com/avatar/100002390cf8733938c?x-image-process=image/resize,m_fixed,w_180,h_180&k=1778829742770036399",8,"Research & Report","Construction of Orientable Sequences in O(1)-Amortized Time Per Bit","Orientable sequences of order n are cyclic binary sequences whose length-n substrings occur at most once in either direction, excluding palindromic substrings. Maximal sequences are only known for n ≤ 7, with a general upper bound 2^{n-1} − 2⌊(n−1)/2⌋. This work gives an efficient algorithm achieving this asymptotically optimal bound, constructing sequences via cycle-joining and successor rules in O(n) time per bit and O(n) space, and using concatenation trees to obtain O(1)-amortized time per bit with O(n^2) space.","Construction of orientable sequences in O(1)-amortized time per bit  \nDaniel Gabri  \nUniversity of Guelph, Canada  \nJoe Sawada  \nUniversity of Guelph, Canada  \n~~ Abstract ~~  \nAn 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 2n−1 − 2 ⌊ (n−1)/2⌋ . This paper presents the first efficient algorithm to construct orientable sequences that reach this upper bound, asymptotically; 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)] . Applying a recent concatenation-tree framework, the same sequences can be generated in O(1)-amortized time per bit using O (n2 ) space. Our sequences are applied to find new longest-known (aperiodic) orientable sequences for n ≤ 20.  \narXiv :2401 . 1434 1v4 [ cs .DS] 10 Jul 2026  \n© D. Gabri, J. Sawada;  \nlicensed under Creative Commons License CC-BY  \nLeibniz International Proceedings in Informatics  \nSchloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany  \n2 Orientable sequences  \n 1  Introduction  \nOrientable sequences were introduced by Burns and Mitchell [1] and studied by Dai, Martin, Robshaw, and Wild [2] with applications related to robotic position sensing. In particular, consider an autonomous robot with limited sensors. To determine its location on a cyclic track labeled with black and white squares, the robot scans a window of n squares directly beneath it. For the position and orientation to be uniquely determined, the track must be designed with the property that each length n window can appear at most once in either direction. A cyclic binary sequence (track) with such a property is called an orientable sequence of order n (an OS (n)) . By this definition, an orientable sequence does not contain a length-n substring that is a palindrome.  \nExample 1 Consider S = 001011 . In the forward direction, including the wraparound, S contains the six 5-tuples 00101, 01011, 10110, 01100, 11001, and 10010 ; in the reverse direction S contains 11010, 10100, 01001, 10011, 00110, and 01101. Since each substring is unique, S is an OS(5) with length (period) six.  \nOrientable sequences do not exist for n = 1, and somewhat surprisingly, the maximum length Mn of an OS (n) is known only for 1 \u003C n ≤ 7. Since the number of palindromes of length n is 2 ⌊ (n+1)/2⌋, a trivial upper bound on Mn is (2n − 2⌊ (n+1)/2⌋ )/2 = 2n−1 − 2 ⌊ (n−1)/2⌋ .  \nIn addition to providing a tighter upper bound, Dai, Martin, Robshaw, and Wild [2] provide a lower bound Ln on Mn by demonstrating the existence of OS (n)s via cycle-joining with length Ln asymptotic to their upper bound. They conclude by stating the following open problem relating to orientable sequences whose lengths (periods) attain the lower bound. See Section 2.1 for the explicit upper and lower bounds.  \nWe note that the lower bound on the maximum period was obtained using an existence construction . . . It is an open problem whether a more practical procedure exists for the construction of orientable sequences that have this asymptotically optimal period.  \nRecently, some progress was made in this direction by Mitchell and Wild [3] . They apply Lempel’s lift [4] to obtain an OS(n) recursively from an OS (n−1) . This construction can generate orientable sequences in O(1)-amortized time per bit; however, it requires exponential space, and there is an exponential time delay before the first bit can be output. Furthermore, they state that their work “only partially answers the question, since the periods/lengths of the sequences produced are not asymptotically optimal.”  \nMain result: By developing a parent rule to define a cycle-joining tre","cbCaimv3jaar23ch","https://ap.wps.com/l/cbCaimv3jaar23ch","pdf",646438,3,1,16,"English","en",105,"# Introduction\n## Related work\n# Orientable sequences background\n## Bounds and cycle-joining technique\n# Cycle-joining tree construction\n## Parent rule for reverse-disjoint cycles\n# Properties of periodic nodes and child computation\n# Conversion to concatenation trees\n# Applications for n ≤ 20\n## Extending to longer sequences\n## Longest-known acyclic orientable sequences\n# Conclusion and future research","[{\"question\":\"What makes a cyclic binary sequence an orientable sequence of order n (OS(n))?\",\"answer\":\"An OS(n) is cyclic such that each length-n substring appears at most once in either direction, and by this definition no length-n substring is a palindrome.\"},{\"question\":\"What prior open problem does the paper address?\",\"answer\":\"It addresses the longstanding question of whether orientable sequences with asymptotically optimal period can be constructed by a more practical procedure than known existence constructions.\"},{\"question\":\"How does the proposed algorithm achieve O(1)-amortized time per bit?\",\"answer\":\"It first builds the desired sequences with a cycle-joining tree using O(n) time per bit and O(n) space, then applies a concatenation-tree framework to convert the construction to O(1)-amortized time per bit using O(n^2) 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makes a cyclic binary sequence an orientable sequence of order n (OS(n))?","Question",{"text":75,"@type":76},"An OS(n) is cyclic such that each length-n substring appears at most once in either direction, and by this definition no length-n substring is a palindrome.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What prior open problem does the paper address?",{"text":80,"@type":76},"It addresses the longstanding question of whether orientable sequences with asymptotically optimal period can be constructed by a more practical procedure than known existence constructions.",{"name":82,"@type":73,"acceptedAnswer":83},"How does the proposed algorithm achieve O(1)-amortized time per bit?",{"text":84,"@type":76},"It first builds the desired sequences with a cycle-joining tree using O(n) time per bit and O(n) space, then applies a concatenation-tree framework to convert the construction to O(1)-amortized time per bit using O(n^2) 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