Random sampling, halfspace range reporting, and construction of (≤ k)-levels in three dimensions

Given n points in three dimensions, we show how to answer halfspace range reporting queries in O(log n + k) expected time for an output size k. Our data structure can be preprocessed in optimal O(n log n) expected time. We apply this result to obtain the first optimal randomized algorithm for the construction of the (≤ k)-level in an arrangement of n planes in three dimensions. The algorithm runs in O(n log n+nk²) expected time. Our techniques are based on random sampling. Applications in two dimensions include an improved data structure for "k nearest neighbors" queries and an algorithm that constructs the order-k Voronoi diagram in O(n log n + nk log k) expected time.