A dynamic data structure for 3-D convex hulls and 2-D nearest neighbor queries

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We present a fully dynamic randomized data structure that can answer queries about the convex hull of a set of n points in three dimensions, where insertions take O(log3n) expected amortized time, deletions take O(log6n) expected amortized time, and extreme-point queries take O(log2n) worst-case time. This is the first method that guarantees polylogarithmic update and query cost for arbitrary sequences of insertions and deletions, and improves the previous O(nε)-time method by Agarwal and Matouek a decade ago. As a consequence, we obtain similar results for nearest neighbor queries in two dimensions and improved results for numerous fundamental geometric problems (such as levels in three dimensions and dynamic Euclidean minimum spanning trees in the plane).

Original languageEnglish (US)
Article number16
JournalJournal of the ACM
Issue number3
StatePublished - Mar 1 2010
Externally publishedYes


  • Computational geometry
  • Convex hulls
  • Dynamic data structures
  • Nearest neighbor search

ASJC Scopus subject areas

  • Software
  • Control and Systems Engineering
  • Information Systems
  • Hardware and Architecture
  • Artificial Intelligence


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