2 research outputs found
An Optimal Algorithm for Higher-Order Voronoi Diagrams in the Plane: The Usefulness of Nondeterminism
We present the first optimal randomized algorithm for constructing the
order- Voronoi diagram of points in two dimensions. The expected running
time is , which improves the previous, two-decades-old result
of Ramos (SoCG'99) by a 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- 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 -level of
lines in two dimensions in time, and constructing
the -level of planes in three dimensions in
time. These time bounds (ignoring the term) match the current best
upper bounds on the combinatorial complexity of the -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