On the Randomized Construction of the Delaunay Tree

The Delaunay tree is a hierarchical data structure which is defined from the Delaunay triangulation and, roughly speaking, represents a triangulation as a hierarchy of balls. It allows a smidynamic construction of the Delaunay triangulation of a finite set of n points in any dimension. In this paper, we prove that a randomized construction of the Delaunay tree (and, thus, of the Delaunay triangulation) can be done in O(n log n) expected time in the plane and in O(n⌈d2⌉) expected time in d-dimensional space. These results are optimal for fixed d. The algorithm is extremely simple and experimental results are given.

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