Approximate Range Counting Revisited

TL;DR

This paper revisits approximate range counting for colored objects, providing optimal solutions for orthogonal range-searching and rectangle stabbing, mainly through reductions and introducing nested shallow cuttings.

Contribution

It introduces optimal solutions for approximate colored range counting problems and the novel concept of nested shallow cuttings.

Findings

Achieved optimal and near-optimal data structures for the problems.

Reduced colored range counting to uncolored versions for efficiency.

Introduced nested shallow cuttings as a new technique.

Abstract

We study range-searching for colored objects, where one has to count (approximately) the number of colors present in a query range. The problems studied mostly involve orthogonal range-searching in two and three dimensions, and the dual setting of rectangle stabbing by points. We present optimal and near-optimal solutions for these problems. Most of the results are obtained via reductions to the approximate uncolored version, and improved data-structures for them. An additional contribution of this work is the introduction of nested shallow cuttings.