Simplex Range Searching Revisited: How to Shave Logs in Multi-Level Data Structures

TL;DR

This paper improves data structures for simplex range searching and related problems in computational geometry, reducing query times and space complexity by eliminating multiple logarithmic factors through novel multi-level data structures.

Contribution

It introduces new data structures with optimal or near-optimal query times and reduced space complexity for various geometric range searching problems, surpassing decades-old bounds.

Findings

Optimal $O( ext{log} n)$ query time for simplex range counting and reporting.

Reduced space complexity to $O(n^d)$ for high-dimensional range queries.

Improved ray shooting query time to $O( oot n)$ in 2D.

Abstract

We revisit the classic problem of simplex range searching and related problems in computational geometry. We present a collection of new results which improve previous bounds by multiple logarithmic factors that were caused by the use of multi-level data structures. Highlights include the following: $\bullet$ For a set of $n$ points in a constant dimension $d$, we give data structures with $O(n^d)$ (or slightly better) space that can answer simplex range counting queries in optimal $O(\log n)$ time and simplex range reporting queries in optimal $O(\log n + k)$ time, where $k$ denotes the output size. For semigroup range searching, we obtain $O(\log n)$ query time with $O(n^d\mathop{\rm polylog}n)$ space. Previous data structures with similar space bounds by Matou\v{s}ek from nearly three decades ago had $O(\log^{d+1}n)$ or $O(\log^{d+1}n + k)$ query time. $\bullet$ For a set of $n$ simplices in a constant dimension $d$, we give data structures with $O(n)$ space that can answer stabbing counting queries (counting the number of simplices containing a query point) in $O(n^{1-1/d})$ time, and stabbing reporting queries in $O(n^{1-1/d}+k)$ time. Previous data structures had extra $\log^d n$ factors in space and query time. $\bullet$ For a set of $n$ (possibly intersecting) line segments in 2D, we give a data structure with $O(n)$ space that can answer ray shooting queries in $O(\sqrt{n})$ time. This improves Wang's recent data structure [SoCG'20] with $O(n\log n)$ space and $O(\sqrt{n}\log n)$ query time.