Algorithm Theory – SWAT 2012

Volume 7357 of the series Lecture Notes in Computer Science pp 48-58

Kinetic Pie Delaunay Graph and Its Applications

  • Mohammad Ali AbamAffiliated withDept. of Computer Engineering, Sharif University of TechnologyInstitute for Research in Fundamental Sciences (IPM)
  • , Zahed RahmatiAffiliated withDept. of Computer Science, University of Victoria
  • , Alireza ZareiAffiliated withDept. of Mathematical Science, Sharif University of Technology

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We construct a new proximity graph, called the Pie Delaunay graph, on a set of n points which is a super graph of Yao graph and Euclidean minimum spanning tree (EMST). We efficiently maintain the Pie Delaunay graph where the points are moving in the plane. We use the kinetic Pie Delaunay graph to create a kinetic data structure (KDS) for maintenance of the Yao graph and the EMST on a set of n moving points in 2-dimensional space. Assuming x and y coordinates of the points are defined by algebraic functions of at most degree s, the structure uses O(n) space, O(nlogn) preprocessing time, and processes O(n 2 λ 2s + 2(n)β s + 2(n)) events for the Yao graph and O(n 2 λ 2s + 2(n)) events for the EMST, each in O(log2 n) time. Here, λ s (n) =  s (n) is the maximum length of Davenport-Schinzel sequences of order s on n symbols. Our KDS processes nearly cubic events for the EMST which improves the previous bound O(n 4) by Rahmati et al. [1].


Euclidean minimum spanning tree Yao graph Pie Delaunay triangulation kinetic data structures