Optimal Color Range Reporting in One Dimension

TL;DR

This paper introduces an optimal data structure for one-dimensional color range reporting that efficiently reports distinct colors within a query range using linear space and optimal query time, with extensions to dynamic and external memory models.

Contribution

It presents a linear-space data structure achieving optimal query time for one-dimensional color range reporting, improving previous solutions and supporting dynamic updates and external memory extensions.

Findings

Achieves O(N) space and O(k+1) query time for color reporting.

Supports dynamic updates and external memory extensions.

Provides a theoretically optimal solution for the problem.

Abstract

Color (or categorical) range reporting is a variant of the orthogonal range reporting problem in which every point in the input is assigned a \emph{color}. While the answer to an orthogonal point reporting query contains all points in the query range $Q$, the answer to a color reporting query contains only distinct colors of points in $Q$. In this paper we describe an O(N)-space data structure that answers one-dimensional color reporting queries in optimal $O(k+1)$ time, where $k$ is the number of colors in the answer and $N$ is the number of points in the data structure. Our result can be also dynamized and extended to the external memory model.