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

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... 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).