Next: Problem 4: Union of
Up: The Open Problems Project
Previous: Problem 2: Voronoi Diagram
Problem 3: Voronoi Diagram of Lines in 3D
- What is the combinatorial complexity of the Voronoi diagram
of a set of lines (or line segments) in three dimensions?
- Uncertain, pending investigation.
Conjectured to be nearly quadradic.
- Partial and Related Results
- There is a gap between
a lower bound of
Ω(n2) and an upper bound that is
essentially cubic [Sha94] for the Euclidean case
(and yet is
quadratic for polyhedral metrics [BSTY98]).
A recent advance shows that
the ``level sets'' of the Voronoi diagram of lines, given by the union of
a set of cylinders,
indeed has near-quadratic complexity [AS00b].
- Related Open Problems
- This problem is closely related to Problem 2,
because points moving in the plane with constant velocity yield
straight-line trajectories in space-time.
- Voronoi diagrams
- Entry Revision History
- J. O'Rourke, 2 Aug. 2001; 13 Dec. 2001.
Pankaj K. Agarwal and Micha Sharir.
Pipes, cigars, and kreplach: The union of Minkowski sums in three
Discrete Comput. Geom., 24(4):645-685, 2000.
Jean-Daniel Boissonnat, Micha Sharir, Boaz Tagansky, and Mariette Yvinec.
Voronoi diagrams in higher dimensions under certain polyhedral
Discrete Comput. Geom., 19(4):473-484, 1998.
J. S. B. Mitchell and Joseph O'Rourke.
Computational geometry column 42.
Internat. J. Comput. Geom. Appl., 11(5):573-582, 2001.
Also in SIGACT News 32(3):63-72 (2001), Issue 120.
Almost tight upper bounds for lower envelopes in higher dimensions.
Discrete Comput. Geom., 12:327-345, 1994.
The Open Problems Project - September 19, 2017