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Orthogonal Point Location and Rectangle Stabbing Queries in 3-d

Authors Timothy M. Chan, Yakov Nekrich, Saladi Rahul, Konstantinos Tsakalidis

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Author Details

Timothy M. Chan
  • Dept. of Computer Science, University of Illinois at Urbana-Champaign, USA
Yakov Nekrich
  • Cheriton School of Computer Science, University of Waterloo, Canada
Saladi Rahul
  • Dept. of Computer Science, University of Illinois at Urbana-Champaign, USA
Konstantinos Tsakalidis
  • Dept. of Computer and Information Science, Tandon School of Engineering, New York University, USA

Funding

  • Tsakalidis, Konstantinos: Partially supported by NSF grants CCF-1319648 and CCF-1533564

Cite As Get BibTex

Timothy M. Chan, Yakov Nekrich, Saladi Rahul, and Konstantinos Tsakalidis. Orthogonal Point Location and Rectangle Stabbing Queries in 3-d. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 31:1-31:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018) https://doi.org/10.4230/LIPIcs.ICALP.2018.31

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(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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • geometric data structures
  • orthogonal point location
  • rectangle stabbing
  • pointer machines
  • I/O model
  • word RAM model

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