Orthogonal Point Location and Rectangle Stabbing Queries in 3-d

TL;DR

This paper introduces the first linear-space data structures with optimal or sub-logarithmic query times for 3D orthogonal point location and rectangle stabbing problems, improving upon previous bounds and simplifying solutions.

Contribution

It presents novel linear-space data structures for 3D orthogonal point location and rectangle stabbing with optimal or improved query times, using new recursive and bit-packing techniques.

Findings

First linear-space 3D point location data structure with optimal $O(\log n)$ query time.

First linear-space data structures for 3D rectangle stabbing with optimal or sub-logarithmic query times.

Simplified solutions based on recursive and grid-based techniques with new ideas.

Abstract

In this work, we present a collection of new results on two fundamental problems in geometric data structures: orthogonal point location and rectangle stabbing. -We give the first linear-space data structure that supports 3-d point location queries on $n$ disjoint axis-aligned boxes with optimal $O\left( \log n\right)$ query time in the (arithmetic) pointer machine model. This improves the previous $O\left( \log^{3/2} n \right)$ bound of Rahul [SODA 2015]. We similarly obtain the first linear-space data structure in the I/O model with optimal query cost, and also the first linear-space data structure in the word RAM model with sub-logarithmic query time. -We give the first linear-space data structure that supports 3-d $4$-sided and $5$-sided rectangle stabbing queries in optimal $O(\log_wn+k)$ time in the word RAM model. We similarly obtain the first optimal data structure for the closely related problem of 2-d top-$k$ rectangle stabbing in the word RAM model, and also improved results for 3-d 6-sided rectangle stabbing. For point location, our solution is simpler than previous methods, and is based on an interesting variant of the van Emde Boas recursion, applied in a round-robin fashion over the dimensions, combined with bit-packing techniques. For rectangle stabbing, our solution is a variant of Alstrup, Brodal, and Rauhe's grid-based recursive technique (FOCS 2000), combined with a number of new ideas.