**Next:** Problem 3: Voronoi Diagram of Lines in 3D

**Previous:** Problem 1: Minimum Weight Triangulation

- Statement
What is the maximum number of combinatorial changes possible in a Euclidean Voronoi diagram of a set of n points each moving along a line at unit speed in two dimensions?

- Origin
Unknown (to JOR). Perhaps Michael Atallah?

- Status/Conjectures
Long conjectured to be nearly quadratic. Solved now: [Rub15]. Natan Rubin proved an upper bound of O(n^{2+\epsilon}), and a quadratic lower bound is known.

- Partial and Related Results
See [Rub15] for a review of earlier work, now superceded.

- Appearances
- Categories
Voronoi diagrams; Delaunay triangulations

- Entry Revision History
J. O’Rourke, 1 Aug. 2001; 19Sep2017.

- [MO01]
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.- [Rub15]
Natan Rubin. On kinetic Delaunay triangulations: A near-quadratic bound for unit speed motions.

*Journal of the ACM (JACM)*, 62(3):25, 2015.