# Independent Range Sampling, Revisited Again

## TL;DR

This paper revisits the range sampling problem, proposing new data structures that achieve efficient expected or approximate worst-case query times, and establishes a lower bound showing the necessity of these concessions.

## Contribution

It introduces data structures with expected or approximate worst-case query times for range sampling, and provides a lower bound demonstrating the need for these approaches.

## Key findings

- Efficient data structures with expected query time for range sampling.
- Approximate proportional sampling enables worst-case query bounds.
- A lower bound shows the trade-off is necessary for 3D range sampling.

## Abstract

We revisit the range sampling problem: the input is a set of points where each point is associated with a real-valued weight. The goal is to store them in a structure such that given a query range and an integer $k$, we can extract $k$ independent random samples from the points inside the query range, where the probability of sampling a point is proportional to its weight. This line of work was initiated in 2014 by Hu, Qiao, and Tao and it was later followed up by Afshani and Wei. The first line of work mostly studied unweighted but dynamic version of the problem in one dimension whereas the second result considered the static weighted problem in one dimension as well as the unweighted problem in 3D for halfspace queries. We offer three main results and some interesting insights that were missed by the previous work: We show that it is possible to build efficient data structures for range sampling queries if we allow the query time to hold in expectation (the first result), or obtain efficient worst-case query bounds by allowing the sampling probability to be approximately proportional to the weight (the second result). The third result is a conditional lower bound that shows essentially one of the previous two concessions is needed. For instance, for the 3D range sampling queries, the first two results give efficient data structures with near-linear space and polylogarithmic query time whereas the lower bound shows with near-linear space the worst-case query time must be close to $n^{2/3}$, ignoring polylogarithmic factors. Up to our knowledge, this is the first such major gap between the expected and worst-case query time of a range searching problem.