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'15] 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 $n-\mathcal{O}(1)$. We confirm this conjecture by proving that the expected value is, in fact, $n-\mathcal{O}({\sigma }^{-1})$, 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 $\mathrm{\Omega }(\sqrt{n})\cap \mathcal{O}(\sqrt{n{\log }_{\sigma }n})$ due to analogous bounds by Czumaj and Gąsieniec [CPM'00] for the problem of computing the shortest period of a uniformly random string.

字符串的边框是字符串的非空前缀，也是字符串的后缀，并且如果字符串除自身之外没有其他边界，则它是无边界的。Loptev，Kucherov和Starikovskaya [CPM'15]推测以下内容：如果我们从固定的非一元字母中随机地均匀地选择一个长度为n的字符串，则其无边界因子的预期最大长度为$ñ-\mathcal{Ø}（\mathrm{1个}）$。我们通过证明期望值实际上是$ñ-\mathcal{Ø}（{\sigma }^{-\mathrm{1个}}）$，其中σ是字母的大小。这立即意味着我们平均可以在线性时间中找到这样一个最大的无边界因子。但是，我们进一步证明，最佳平均情况下的运行时间为$\mathrm{\Omega }（\sqrt{ñ}）\cap \mathcal{Ø}（\sqrt{ñ{\mathrm{日志}}_{\sigma }ñ}）$ 由于Czumaj和Gąsieniec[CPM'00]的界线相似，因此无法计算均匀随机字符串的最短周期。