Online computation of normalized substring complexity

The normalized substring complexity $\delta$ of a string is defined as ${\max }_k\{c[k]\mathrm{/}k\}$, where $c[k]$ is the number of $\mathit{\text{distinct}}$ 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 $\delta$ 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.