构造具有渐近最优长度的$k$元可定向序列
Constructing $k$-ary Orientable Sequences with Asymptotically Optimal Length
AI总结：
针对大字母表下可定向序列构造缺乏通用高效方法的问题，提出适用于$k$元不对称手镯的父规则构建环连接树，进而得到以$O(n)$时间每符号、$O(n)$空间构造渐近最优$k$元可定向序列的后继规则，并给出$n=2$时最大长度序列的简单构造。
摘要：
字母表$\u003c0,1,\ldots, k{-}1\u003e$上的$n$阶可定向序列是一种循环序列，满足每个长度为$n$的子串在**任一方向**上至多出现一次。当$k=2$时，已有高效算法可通过应用经典的环连接技术，构造出具有渐近最优长度的二元可定向序列。该构造的关键在于定义一种父规则，以构建不对称手镯的环连接树。遗憾的是，该父规则无法推广到更大的字母表。此外，与二元情形不同，当$k \geq 3$时，环连接树无法直接导出简单的后继规则，除非该树具备特定性质。本文中，我们推导得到一种父规则，用于构建$k$元不对称手镯的环连接树。由此可得到一种后继规则，能够以每个符号$O(n)$的时间、$O(n)$的空间构造出渐近最优的$k$元可定向序列。在$n=2$的特殊情形下，我们给出了一种构造最大长度$k$元可定向序列的简单方法。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
