On Range Searching with Semialgebraic Sets II

TL;DR

This paper introduces a linear-size data structure for efficient range searching with semialgebraic sets in high-dimensional spaces, nearly matching the performance of simplex range searching and improving previous solutions for dimensions five and above.

Contribution

It presents a novel polynomial-partitioning based data structure for range searching with semialgebraic sets, nearly optimal in performance and solving a long-standing open problem.

Findings

Achieves query time close to $O(n^{1-1/d})$ for semialgebraic ranges.

Significantly improves previous solutions for dimensions $d \\ge 5$.

Provides an efficient randomized algorithm for polynomial partitioning.

Abstract

Let $P$ be a set of $n$ points in $\R^d$. We present a linear-size data structure for answering range queries on $P$ with constant-complexity semialgebraic sets as ranges, in time close to $O(n^{1-1/d})$. It essentially matches the performance of similar structures for simplex range searching, and, for $d\ge 5$, significantly improves earlier solutions by the first two authors obtained in~1994. This almost settles a long-standing open problem in range searching. The data structure is based on the polynomial-partitioning technique of Guth and Katz [arXiv:1011.4105], which shows that for a parameter $r$, $1 < r \le n$, there exists a $d$-variate polynomial $f$ of degree $O(r^{1/d})$ such that each connected component of $\R^d\setminus Z(f)$ contains at most $n/r$ points of $P$, where $Z(f)$ is the zero set of $f$. We present an efficient randomized algorithm for computing such a polynomial partition, which is of independent interest and is likely to have additional applications.