eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2018-07-04
31:1
31:14
10.4230/LIPIcs.ICALP.2018.31
article
Orthogonal Point Location and Rectangle Stabbing Queries in 3-d
Chan, Timothy M.
1
Nekrich, Yakov
2
Rahul, Saladi
1
Tsakalidis, Konstantinos
3
Dept. of Computer Science, University of Illinois at Urbana-Champaign, USA
Cheriton School of Computer Science, University of Waterloo, Canada
Dept. of Computer and Information Science, Tandon School of Engineering, New York University, USA
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(log n) query time in the (arithmetic) pointer machine model. This improves the previous O(log^{3/2} n) 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.
- 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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol107-icalp2018/LIPIcs.ICALP.2018.31/LIPIcs.ICALP.2018.31.pdf
geometric data structures
orthogonal point location
rectangle stabbing
pointer machines
I/O model
word RAM model