Counting Colours in Compressed Strings

Suppose we are asked to preprocess a string $s[\mathrm{1..}n]$ such that later, given a substring’s endpoints, we can quickly count how many distinct characters it contains. In this paper we give a data structure for this problem that takes $n{H}_0(s)+\text{Oh}n+\text{oh}n{H}_0(s)$ bits, where $H_0(s)$ is the 0th-order empirical entropy of $s$, and answers queries in $\Oh{\log^{1 + \epsilon} n}$ time for any constant $ϵ>0$. We also show how our data structure can be made partially dynamic.