Document Type

Article

Publication Date

9-27-2022

Department

Department of Computer Science

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 sup- ports 3-d point location queries on n disjoint axis-aligned boxes with optimal O (log") query time in the (arithmetic) pointer machine model. This improves the previous 0 (\ogi/2 n^ bound of Rahul \SODA 201o|. 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 HAM model with sub-logarithmic query time. Our tech- nique also improves upon the result of de Berg, van Kreveld, and Snoeyink \ Journal of Algorithms I995| for 3-d point location in (space filling) box subdivisions: we obtain a linear-space data structure with G{\og2\ogU) 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(\ogwn 4- k) time in the word RAM model, where k is the number of rectangles reported and w is the number of bits in a word. We similarly obtain the first optimal data structure for the closely related problem of 2-d top-/: 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.

Publisher's Statement

Copyright (c) 2022 Timothy M. Chan, Yakov Nekrich, Saladi Rahul, Konstantinos Tsakalidis. Publisher’s version of record: https://doi.org/10.20382/jocg.v13i1a15

Publication Title

Journal of Computational Geometry

Creative Commons License

Creative Commons Attribution 4.0 International License
This work is licensed under a Creative Commons Attribution 4.0 International License.

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