C. Lemaire et Jm. Moreau, A probabilistic result on multi-dimensional Delaunay triangulations, and its application to the 2D case, COMP GEOM, 17(1-2), 2000, pp. 69-96
This paper exploits the notion of "unfinished site", introduced by Katajain
en and Koppinen (1998) in the analysis of a two-dimensional Delaunay triang
ulation algorithm, based on a regular grid. We generalize the notion and it
s properties to any dimension k greater than or equal to 2: in the case of
uniform distributions, the expected number of unfinished sites in a k-recta
ngle is O(N1-1/k). This implies, under some specific assumptions, the linea
rity of a class of divide-and-conquer schemes based on balanced k-d trees.
This general result is then applied to the analysis of a new algorithm for
constructing Delaunay triangulations in the plane. According to Su and Drys
dale (1995, 1997), the best known algorithms for this problem run in linear
expected time, thanks in particular to the use of bucketing techniques to
partition the domain. In our algorithm, the partitioning is based on a 2-d
tree instead, the construction of which takes Theta (N log N) time, and we
show that the rest of the algorithm runs in linear expected time. This "pre
processing" allows the algorithm to adapt efficiently to irregular distribu
tions, as the domain is partitioned using point coordinates, as opposed to
a fixed, regular basis (buckets or grid). We checked that even for the larg
est data sets that could fit in internal memory lover 10 million points), c
onstructing the 2-d tree takes noticeably less CPU time than triangulating
the data. With this in mind, our algorithm is only slightly slower than the
reputedly best algorithms on uniform distributions, and is even the most e
fficient for data sets of up to several millions of points distributed in c
lusters, (C) 2000 Elsevier Science B.V. All rights reserved.