Independent Range Sampling, Revisited Again

by   Peyman Afshani, et al.

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.


page 1

page 2

page 3

page 4


A New Lower Bound for Semigroup Orthogonal Range Searching

We report the first improvement in the space-time trade-off of lower bou...

A limit field for orthogonal range searches in two-dimensional random point search trees

We consider the cost of general orthogonal range queries in random quadt...

An Optimal Lower Bound for Simplex Range Reporting

We give a simplified and improved lower bound for the simplex range repo...

For Kernel Range Spaces a Constant Number of Queries Are Sufficient

We introduce the notion of an ε-cover for a kernel range space. A kernel...

Optimal Dynamic Subset Sampling: Theory and Applications

We study the fundamental problem of sampling independent events, called ...

New bounds for range closest-pair problems

Given a dataset S of points in R^2, the range closest-pair (RCP) problem...

Distribution Privacy Under Function Recoverability

A user generates n independent and identically distributed data random v...

Please sign up or login with your details

Forgot password? Click here to reset