INCREMENTAL ALGORITHMS FOR FINDING THE CONVEX HULLS OF CIRCLES AND THE LOWER ENVELOPES OF PARABOLAS

The existing O(n log n) algorithms for finding the convex hulls of circles and the lower envelope of parabolas follow the divide-and-conquer paradigm. The difficulty with developing incremental algorithms for these problems is that the introduction of a new circle or parabola can cause Theta(n) structural changes, leading to Theta(n(2)) total structural changes during the running of the algorithm. In this note we examine the geometry of these problems and show that, if the circles or parabolas are first sorted by appropriate parameters before constructing the convex hull or lower envelope incrementally, then each new addition may cause at most 3 changes in an amortized sense. These observations are then used to develop O(n log n) incremental algorithms for these problems.