Constructing levels in arrangements and higher order Voronoi diagrams

Pankaj K. Agarwal, Mark de Berg, Jiří Matoušek, Otfried Schwarzkopf
<span title="">1994</span> <i title="ACM Press"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/zqbrzpfj5rc3pcxpllbrit3n4q" style="color: black;">Proceedings of the tenth annual symposium on Computational geometry - SCG &#39;94</a> </i> &nbsp;
We give simple randomized incremental algorithms for computing the ≤k-level in an arrangement of n lines in the plane or in an arrangement of n planes in R 3 . The expected running time of our algorithms is O(nk + nα(n) log n) for the planar case and O(nk 2 + n log 3 n) for the three-dimensional case. Both bounds are optimal unless k is very small. The algorithm generalizes to computing the ≤k-level in an arrangement of discs or x-monotone Jordan curves in the plane. Our approach can also
more &raquo; ... e the k-level; this yields a randomized algorithm for computing the order-k Voronoi diagram of n points in the plane in expected time O(k(n − k) log n + n log 3 n).
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