We present the first optimal randomized algorithm for constructing the
order-k Voronoi diagram of n points in two dimensions. The expected running
time is O(nlogn+nk), which improves the previous, two-decades-old result
of Ramos (SoCG'99) by a 2O(logβk) factor. To obtain our result, we (i)
use a recent decision-tree technique of Chan and Zheng (SODA'22) in combination
with Ramos's cutting construction, to reduce the problem to verifying an
order-k Voronoi diagram, and (ii) solve the verification problem by a new
divide-and-conquer algorithm using planar-graph separators.
We also describe a deterministic algorithm for constructing the k-level of
n lines in two dimensions in O(nlogn+nk1/3) time, and constructing
the k-level of n planes in three dimensions in O(nlogn+nk3/2)
time. These time bounds (ignoring the nlogn term) match the current best
upper bounds on the combinatorial complexity of the k-level. Previously, the
same time bound in two dimensions was obtained by Chan (1999) but with
randomization.Comment: To appear in SODA 2024. 16 pages, 1 figur