, Volume 80, Issue 11, pp 3316–3334 | Cite as

Data Structures for Halfplane Proximity Queries and Incremental Voronoi Diagrams

  • Boris Aronov
  • Prosenjit Bose
  • Erik D. DemaineEmail author
  • Joachim Gudmundsson
  • John Iacono
  • Stefan Langerman
  • Michiel Smid


We consider preprocessing a set S of n points in convex position in the plane into a data structure supporting queries of the following form: given a point q and a directed line \(\ell \) in the plane, report the point of S that is farthest from (or, alternatively, nearest to) the point q among all points to the left of line \(\ell \). We present two data structures for this problem. The first data structure uses \(O(n^{1+\varepsilon })\) space and preprocessing time, and answers queries in \(O(2^{1/\varepsilon }\log n)\) time, for any \(0< \varepsilon < 1\). The second data structure uses \(O(n \log ^3 n)\) space and polynomial preprocessing time, and answers queries in \(O(\log n)\) time. These are the first solutions to the problem with \(O(\log n)\) query time and \(o(n^2)\) space. The second data structure uses a new representation of nearest- and farthest-point Voronoi diagrams of points in convex position. This representation supports the insertion of new points in clockwise order using only \(O(\log n)\) amortized pointer changes, in addition to \(O(\log n)\)-time point-location queries, even though every such update may make \(\Theta (n)\) combinatorial changes to the Voronoi diagram. This data structure is the first demonstration that deterministically and incrementally constructed Voronoi diagrams can be maintained in o(n) amortized pointer changes per operation while keeping \(O(\log n)\)-time point-location queries.


Voronoi diagrams Data structures Trees Flarbs 



This work was initiated at the Schloss Dagstuhl Seminar 04091 on Data Structures, organized by Susanne Albers, Robert Sedgewick, and Dorothea Wagner, and held February 22–27, 2004 in Germany. This work continued at the Korean Workshop on Computational Geometry and Geometric Networks, organized by Hee-Kap Ahn, Christian Knauer, Chan-Su Shin, Alexander Wolff, and René van Oostrum, and held July 25–30, 2004 at Schloss Dagstuhl in Germany; and at the 2nd Bertinoro Workshop on Algorithms and Data Structures, organized by Andrew Goldberg and Giuseppe Italiano, and held May 29–June 4, 2005 in Italy. We thank the organizers and institutions hosting these workshops for providing a productive research atmosphere. We also thank Alexander Wolff for introducing the problem to us.


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© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Department of Computer Science and Engineering, Polytechnic School of EngineeringNew York UniversityBrooklynUSA
  2. 2.School of Computer ScienceCarleton UniversityOttawaCanada
  3. 3.Computer Science and Artificial Intelligence LaboratoryMITCambridgeUSA
  4. 4.National ICT AustraliaSydneyAustralia
  5. 5.Départment d’InformatiqueUniversité Libre de BruxellesBrusselsBelgium

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