The normalized substring complexity $δ$ of a string is defined as $\max_k \{c[k]/k\}$, where $c[k]$ is the number of \textit{distinct} substrings of length $k$. This simply defined measure has recently attracted attention due to its established relationship to popular string compression algorithms. We consider the problem of computing $δ$ online, when the string is provided from a stream. We present two algorithms solving the problem: one working in $O(\log n)$ amortized time per character, and the other in $O(\log^3 n)$ worst-case time per character. To our knowledge, this is the first polylog-time online solution to this problem.

翻译：字符串的归一化子串复杂度 $δ$ 定义为 $\max_k \{c[k]/k\}$，其中 $c[k]$ 是长度为 $k$ 的\textit{互异}子串的数量。这一简洁定义的方法因其与主流字符串压缩算法之间已确立的关系，近来受到关注。我们研究了在字符串以流形式提供时在线计算 $δ$ 的问题。我们提出了两种解决该问题的算法：一种算法在每个字符上的均摊时间为 $O(\log n)$，另一种算法在每个字符上的最坏情况时间为 $O(\log^3 n)$。据我们所知，这是该问题的首个多对数时间在线解决方案。