Orthogonal Point Location and Rectangle Stabbing Queries in 3-d.
ICALP(2018)
摘要
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 $Oleft( log nright)$ query time in the (arithmetic) pointer machine model. This improves the previous $Oleft( log^{3/2} n right)$ 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. 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 Rauheu0027s grid-based recursive technique (FOCS 2000), combined with a number of new ideas.
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