**Papers Published**

- Agarwal, PK; Berg, MD; Matoušek, J; Schwarzkopf, O,
*Constructing Levels in Arrangements and Higher Order Voronoi Diagrams*, SIAM Journal on Computing, vol. 27 no. 3 (1998), pp. 654-667

(last updated on 2018/08/17)**Abstract:**

We give simple randomized incremental algorithms for computing the ≤κ-level in an arrangement of n lines in the plane or in an arrangement of n planes in ℝ3. The expected running time of our algorithms is O(nκ + nα(n) logn) for the planar case and O(nκ2 + nlog3n) for the three-dimensional case. Both bounds are optimal unless κ is very small. The algorithm generalizes to computing the ≤κ-level in an arrangement of discs or x-monotone Jordan curves in the plane. Our approach can also compute the κ-level; this yields a randomized algorithm for computing the order-κ Voronoi diagram of n points in the plane in expected time O(κ(n - κ) log n + n log3 n).