Orthogonal point location and rectangle stabbing queries in 3-d

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.

• Orthogonal point location. 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. Our technique also improves upon the result of de Berg, van Kreveld, and Snoeyink [Journal of Algorithms 1995] for 3-d point location in  rectangular subdivisions: we obtain a linear-space data structure with $O(\log^2\log U)$ query time.
• Rectangle stabbing. 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.