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We give a simple lazy randomized incremental algorithm to compute ≤<italic>k</italic>-levels in arrangements of <italic>x</italic>-monotone Jordan curves in the plane, and in arrangements of planes in three-dimensional space. If each pair of curves intersects in at most <italic>s</italic> points, the expected running time of the algorithm is <italic>O</italic>(<italic>k</italic><supscrpt>2</supscrpt>λ<subscrpt><italic>s</italic></subscrpt>(<italic>n/k</italic>)+min(λ<subscrpt><italic>s</italic></subscrpt>(<italic>n</italic>)log<supscrpt>2</supscrpt><italic>n</italic>,<italic>k</italic><supscrpt>2</supscrpt>λ<italic><subscrpt>s</subscrpt></italic>(<italic>n/k</italic>)log<italic>n</italic>)). For the three-dimensional case the expected running time is <italic>O</italic>(<italic>nk</italic><supscrpt>2</supscrpt>+min(<italic>n</italic>log<supscrpt>3</supscrpt><italic>n</italic>,<italic>nk</italic><supscrpt>2</supscrpt>log<italic>n</italic>)). The algorithm also works for computing the ≤<italic>k</italic>-level in a set of discs, with an expected running time of <italic>O</italic>(<italic>nk</italic>+min(<italic>n</italic>log<supscrpt>2</supscrpt><italic>n</italic>,<italic>nk</italic>log<italic>n</italic>)). Furthermore, we give a simple algorithm for computing the order-<italic>k</italic> Voronoi diagram of a set of <italic>n</italic> points in the plane that runs in expected time <italic>O</italic>(<italic>k</italic>(<italic>n</italic>−<italic>k</italic>)log<italic>n</italic>+<italic>n</italic>log<supscrpt>3</supscrpt><italic>n</italic>).

Constructing levels in arrangements and higher order Voronoi diagrams