Abstract
We revisit the range sampling problem: the input is a set of points where each point is associated with a realvalued 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 worstcase 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 nearlinear space and polylogarithmic query time whereas the lower bound shows with nearlinear space the worstcase 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 worstcase query time of a range searching problem.
BibTeX  Entry
@InProceedings{afshani_et_al:LIPIcs:2019:10408,
author = {Peyman Afshani and Jeff M. Phillips},
title = {{Independent Range Sampling, Revisited Again}},
booktitle = {35th International Symposium on Computational Geometry (SoCG 2019)},
pages = {4:14:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959771047},
ISSN = {18688969},
year = {2019},
volume = {129},
editor = {Gill Barequet and Yusu Wang},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2019/10408},
URN = {urn:nbn:de:0030drops104088},
doi = {10.4230/LIPIcs.SoCG.2019.4},
annote = {Keywords: Range Searching, Data Structures, Sampling}
}
Keywords: 

Range Searching, Data Structures, Sampling 
Collection: 

35th International Symposium on Computational Geometry (SoCG 2019) 
Issue Date: 

2019 
Date of publication: 

11.06.2019 