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

Orthogonal Point Location and Rectangle Stabbing Queries in 3-d