Counting Colours in Compressed Strings

Abstract: 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.