Kinetic and dynamic data structures for convex hulls and upper envelopes

Giora Alexandron, Haim Kaplan*, Micha Sharir

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


Let S be a set of n moving points in the plane. We present a kinetic and dynamic (randomized) data structure for maintaining the convex hull of S. The structure uses O(n) space, and processes an expected number of O( n2βs+ 2(n)logn) critical events, each in O( log2n) expected time, including O(n) insertions, deletions, and changes in the flight plans of the points. Here s is the maximum number of times where any specific triple of points can become collinear, βs(q) = λs(q)/q, and λs(q) is the maximum length of Davenport-Schinzel sequences of order s on n symbols. Compared with the previous solution of Basch, Guibas and Hershberger [J. Basch, L.J. Guibas, J. Hershberger, Data structures for mobile data, J. Algorithms 31 (1999) 1-28], our structure uses simpler certificates, uses roughly the same resources, and is also dynamic.

Original languageEnglish
Pages (from-to)144-158
Number of pages15
JournalComputational Geometry: Theory and Applications
Issue number2
StatePublished - Feb 2007


  • Convex hull
  • Dynamic data structures
  • Kinetic data structures
  • Lower envelope
  • Treaps


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