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Excerpt from Partitioning Arrangements of Lines: I. An Efficient Deterministic Algorithm
Implicit point location problem: Given a collection of m points and a collection Of (possibly intersecting) n triangles in the plane, find which points lie in the union of the triangles. This turns out to be a special case of a general problem of implicit point location of planar maps formed by overlapping figures. We present a deterministic algorithm with O(m2/3n2/3 log3 n (m n) log n) time complexity.
Approximate half-plane range query: Given a set S of n points in the plane and a parameter (not necessarily constant) e 0, preprocess them so that, for any query line 2, we can approximately count the number of points lying above 8 with an error of at most i:en. We give an algorithm that preprocesses S, in time O(% log n log f), into a data structure of size O(f), so that a query can be answered in C(log n) time.
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