Counting Colours in Compressed Strings

Suppose we are asked to preprocess a string $s [1..n]$ 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 $n H_0 (s) + \Oh{n} + \oh{n H_0 (s)}$ 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 $\epsilon > 0$. We also show how our data structure can be made partially dynamic.