Colored range searching in linear space

In colored range searching, we are given a set of n colored points in d ≥ 2 dimensions to store, and want to support orthogonal range queries taking colors into account. In the colored range counting problem, a query must report the number of distinct colors found in the query range, while an answer to the colored range reporting problem must report the distinct colors in the query range. We give the first linear space data structure for both problems in two dimensions (d = 2) with o(n) worst case query time. We also give the first data structure obtaining almost-linear space usage and o(n) worst case query time for points in d > 2 dimensions. Finally, we present the first dynamic solution to both counting and reporting with o(n) query time for d ≥ 2 and d ≥ 3 dimensions, respectively