Counting Colours in Compressed Strings

TL;DR

This paper introduces a space-efficient data structure for counting distinct characters in substrings of a string, achieving near-optimal compression and fast query times, with some support for updates.

Contribution

It presents a novel compressed data structure that efficiently supports substring color counting with near-optimal space and sub-logarithmic query time, including partial dynamism.

Findings

Uses space close to the zero-order entropy of the string

Supports substring color counting in 0(\u221a{ ext{log} n}) time

Can be made partially dynamic for updates

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.