Construction of orientable sequences in $O(1)$-amortized time per bit
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866929356306120704 |
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| author | Gabric, Daniel Sawada, Joe |
| author_facet | Gabric, Daniel Sawada, Joe |
| contents | An orientable sequence of order $n$ is a cyclic binary sequence such that each length-$n$ substring appears at most once \emph{in either direction}. Maximal length orientable sequences are known only for $n\leq 7$, and a trivial upper bound on their length is $2^{n-1} - 2^{\lfloor(n-1)/2\rfloor}$. 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)]. Applying a recent concatenation-tree framework, the same sequences can be generated in $O(1)$-amortized time per bit using $O(n^2)$ space. Our sequences are applied to find new longest-known (aperiodic) orientable sequences for $n\leq 20$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_14341 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Construction of orientable sequences in $O(1)$-amortized time per bit Gabric, Daniel Sawada, Joe Data Structures and Algorithms Discrete Mathematics Information Theory Combinatorics An orientable sequence of order $n$ is a cyclic binary sequence such that each length-$n$ substring appears at most once \emph{in either direction}. Maximal length orientable sequences are known only for $n\leq 7$, and a trivial upper bound on their length is $2^{n-1} - 2^{\lfloor(n-1)/2\rfloor}$. 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)]. Applying a recent concatenation-tree framework, the same sequences can be generated in $O(1)$-amortized time per bit using $O(n^2)$ space. Our sequences are applied to find new longest-known (aperiodic) orientable sequences for $n\leq 20$. |
| title | Construction of orientable sequences in $O(1)$-amortized time per bit |
| topic | Data Structures and Algorithms Discrete Mathematics Information Theory Combinatorics |
| url | https://arxiv.org/abs/2401.14341 |