A Fast Planar Partition Algorithm, I

In this paper we give a fast randomized algorithm for finding a partition of the plane induced by a given set of line segments. The planar partition problem is interesting because it has important practical implications, especially in computer graphics, which indeed served as an inspiration for our investigation. Our algorithm is ideally suited for a practical use because: (i) it is extremely simple and robust, and (ii) despite this simplicity (or rather because of it) the algorithm is optimal; its expected running time is O(m+n log n), where n is the number of input segments and m is the number of points of intersection. The storage requirement is O(m + n). Though the algorithm itself is simple, the global evolution of the underlying partition is non-trivial, which makes the analysis of the algorithm theoretically interesting in its own right.