随机字符串的最大无界因子
Maximal Unbordered Factors of Random Strings
摘要：
一个字符串的边界是字符串的非空前缀，也是字符串的后缀，如果一个字符串除了它自己之外没有边界，那么它就是无边界的。Loptev，Kucherov和Starikovskaya [CPM 2015]证明了以下内容：如果我们从固定的非一元字母表中随机均匀地选择一个长度为$n$的字符串，那么其无边界因子的预期最大长度是$n - O(1)$。我们证实了这一猜想，证明预期值是，事实上，${n - Θ(σ^{-1})}$，其中$σ$是字母表的大小。这直接意味着我们可以在线性时间平均中找到这样一个最大的无边界因子。然而，我们进一步表明，最佳的平均情况下的运行时间是在$Ω(\sqrt{n}) \cap O (\sqrt{n \log_σn})$由于类似的界限Czumaj和Grachsieniec [CPM 2000]的问题，计算一个均匀随机字符串的最短周期。
英文摘要：
A border of a string is a non-empty prefix of the string that is also a suffix of the string, and a string is unbordered if it has no border other than itself. Loptev, Kucherov, and Starikovskaya [CPM 2015] conjectured the following: If we pick a string of length $n$ from a fixed non-unary alphabet uniformly at random, then the expected maximum length of its unbordered factors is $n - O(1)$. We confirm this conjecture by proving that the expected value is, in fact, ${n - Θ(σ^{-1})}$, where $σ$ is the size of the alphabet. This immediately implies that we can find such a maximal unbordered factor in linear time on average. However, we go further and show that the optimum average-case running time is in $Ω(\sqrt{n}) \cap O (\sqrt{n \log_σn})$ due to analogous bounds by Czumaj and Gąsieniec [CPM 2000] for the problem of computing the shortest period of a uniformly random string.
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