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.

翻译：在字母表{0,1,…,k-1}上，一个n阶可定向序列是指满足以下条件的循环序列：每个长度为n的子串至多出现一次（在任意方向上）。当k=2时，通过应用经典的环连接技术，已知存在高效算法来构造具有渐近最优长度的二进制可定向序列。该构造的关键在于定义父规则以构建非对称手镯的环连接树。遗憾的是，该父规则无法直接推广到更大的字母表。此外，与二进制情况不同，当k≥3时，环连接树并不能直接导出简单的后继规则，除非该树具备特定性质。本文推导出一种父规则，用于构建k元非对称手镯的环连接树。这导出了一个后继规则，能以每个符号O(n)时间、O(n)空间构造渐近最优的k元可定向序列。在n=2的特殊情况下，我们提供了一种构造最大长度k元可定向序列的简单方法。