Counting Colours in Compressed Strings

Abstract

Motivated by the problem of counting unique visitors to a website, we consider how to preprocess a string s [1..n] such that later, given a substring’s endpoints, we can quickly count how many distinct characters that substring contains. The smallest reasonably fast previous data structure for this problem takes $n \log \sigma + \ensuremath{\mathcal{O}\!\left( {n \log \log n} \right)}$ bits and answers queries in $\ensuremath{\mathcal{O}\!\left( {\log n} \right)}$ time. We give a data structure for this problem that takes $n H_0 (s) + \ensuremath{\mathcal{O}\!\left( {n} \right)} + o (n H_0 (s))$ bits, where H ₀ (s) is the 0th-order empirical entropy of s, and answers queries in $\ensuremath{\mathcal{O}\!\left( {\log \ell} \right)}$ time, where ℓ is the length of the query substring. As far as we know, this is the first data structure, where the query time depends only on ℓ and not on n. We also show how our data structure can be made partially dynamic.