关于搜索Zimin模式
On Searching Zimin Patterns
摘要：
在模式可避免性方面，中心作用是由称为“自敏模式”的特殊词汇发挥的。这些模式的符号被视为变量，模式的秩是其变量的数量。本文引入了一个词的Zimin类型$x$作为Zimin模式匹配$x$的最大秩。我们展示了如何在线性时间内在线计算一个词的Zimin类型。从而得到一个二次时间线性空间搜索词中Zimin模式的算法。然后我们讨论了无穷Fibonacci字的长度前缀$n$的Zimin型与$n$在Fibonacci数制中的表示之间的关系。利用这个关系，我们证明了这样的前缀和Zimin模式里面的Zimin类型可以在对数时间内找到。最后，我们给出了函数$f(n,k)$的一些界，使得每个长度至少为$f(n,k)$的$k$元字都有一个因子与秩$n$ Zimin模式相匹配.
英文摘要：
In the area of pattern avoidability the central role is played by special words called Zimin patterns. The symbols of these patterns are treated as variables and the rank of the pattern is its number of variables. Zimin type of a word $x$ is introduced here as the maximum rank of a Zimin pattern matching $x$. We show how to compute Zimin type of a word on-line in linear time. Consequently we get a quadratic time, linear-space algorithm for searching Zimin patterns in words. Then we how the Zimin type of the length $n$ prefix of the infinite Fibonacci word is related to the representation of $n$ in the Fibonacci numeration system. Using this relation, we prove that Zimin types of such prefixes and Zimin patterns inside them can be found in logarithmic time. Finally, we give some bounds on the function $f(n,k)$ such that every $k$-ary word of length at least $f(n,k)$ has a factor that matches the rank $n$ Zimin pattern.
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。
