Constructing $k$-ary Orientable Sequences with Asymptotically Optimal Length

TL;DR

This work advances the construction of long orientable sequences over a $k$-ary alphabet by extending the cycle-joining framework to $k\ge 3$. It introduces a new parent rule yielding a cycle-joining tree on asymmetric bracelets with the Chain Property, enabling an $O(n)$-time, $O(n)$-space successor rule that produces an $OS_k(n)$ of asymptotically maximal length, for all $n\ge 3$ and $k\ge 3$. For the special case $n=2$, it provides a simple maximal-length construction, achieving $M_k(2)=k\lfloor (k-1)/2\rfloor$. The paper also derives a Möbius-inversion based formula for the exact lower bound $L_k(n)$ on the maximal orientable sequence length and shows $L_k(n)$ is asymptotically tight with $M_k(n)$, underscoring the practical efficiency of the proposed generator. These results enable scalable generation of long orientable sequences with applications in sensing, DNA-inspired computation, and related combinatorial design problems.

Abstract

An orientable sequence of order $n$ over an alphabet $\{0,1,\ldots, k{-}1\}$ is a cyclic sequence such that each length-$n$ substring appears at most once \emph{in either direction}. When $k= 2$, efficient algorithms are known to construct binary orientable sequences, with asymptotically optimal length, by applying the classic cycle-joining technique. The key to the construction is the definition of a parent rule to construct a cycle-joining tree of asymmetric bracelets. Unfortunately, the parent rule does not generalize to larger alphabets. Furthermore, unlike the binary case, a cycle-joining tree does not immediately lead to a simple successor-rule when $k \geq 3$ unless the tree has certain properties. In this paper, we derive a parent rule to derive a cycle-joining tree of $k$-ary asymmetric bracelets. This leads to a successor rule that constructs asymptotically optimal $k$-ary orientable sequences in $O(n)$ time per symbol using $O(n)$ space. In the special case when $n=2$, we provide a simple construction of $k$-ary orientable sequences of maximal length.

Figures (5)

• Figure 1: Cycle-joining trees for $\mathbf{N}_3(3)$ induced by four different parent rules.
• Figure 2: Illustrating the construction of $U_7$ and $U_8$.
• Figure 3: The cycle-joining tree $\mathbb{T}_{3}(6)$. Each node differs from its parent (cyclically) at the highlighted symbol. The symbols highlighted in blue indicate that $\mathop{\mathrm{par}}\nolimits(\alpha) = \mathop{\mathrm{lastNonMax}}\nolimits(\alpha)$; the symbols highlighted in red indicate that $\mathop{\mathrm{par}}\nolimits(\alpha) = \mathop{\mathrm{lastSymbol}}\nolimits(\alpha)$; the symbols highlighted in bold black indicate that $\mathop{\mathrm{par}}\nolimits(\alpha) = \mathop{\mathrm{firstNonMin}}\nolimits(\alpha)$. There are no nodes in this tree such that $\mathop{\mathrm{par}}\nolimits(\alpha) = \mathop{\mathrm{secondLastNonMax}}\nolimits(\alpha)$; the first instance of such a case arises when $n=12$.
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Theorems & Definitions (15)

• Theorem 1
• Theorem 2
• Lemma 3: GS-Orientable:2024
• Lemma 4
• Lemma 5
• Lemma 6
• Lemma 7
• Lemma 8
• Lemma 9
• Theorem 10