On lazy randomized incremental construction

Abstract

We introduce a new type of randomized incremental algorithms. Contrary to standard randomized incremental algorithms, these lazy randomized incremental algorithms are suited for computing structures that have a 'non-local' definition. In order to analyze these algorithms we generalize some results on random sampling to such situations. We apply our techniques to obtain efficient algorithms for the computation of single cells in arrangements of segments in the plane, single cells in arrangements of triangles in space, and zones in arrangements of hyperplanes. We also prove combinatorial bounds on the complexity of what we call the k)-cell in arrangements of segments in the plane or triangles in space; this is the set of all points on the segments (triangles) that can reach the origin with a path that crosses at most k - 1 segments (triangles).