Maintenance of Configurations in the Plane

For a number of common configurations of points (lines) in the plane, we develop data structures in which insertions and deletions of points (or lines, respectively) can be processed rapidly, without sacrificing much of the efficiency of query answering which known static structures for these configurations attain. As a main result we establish a fully dynamic maintenance algorithm for convex hulls that can process insertions and deletions of single points in only O(log* n) steps per transaction, where n is the number of points currently in the set. The algorithm has several intriguing applications, including the fact that the “trimmed” mean of a set of n points in the plane can be determined in only O(n log2 n) steps. Likewise, efficient algorithms are obtained for dynamically maintaining the common intersection of a set of halfspaces and for dynamically maintaining the maximal elements of a set of points. The results are all derived by means of one master technique, which is applied repeatedly and which captures an appropriate notion of “decomposability” for configurations closely related to the existence of divide-and-conquer solutions.