Fully Dynamic Point Location in a Monotone Subdivision

In this paper a dynamic technique for locating a point in a monotone planar subdivision, whose current number of vertices is n, is presented. The (complete set of) update operations are insertion of a point on an edge and of a chain of edges between two vertices, and their reverse operations. The data structure uses space O(n). The query time is O(log n), the time for insertion/deletion of a point is O(log n), and the time for insertion/deletion of a chain with k edges is O(log n + k), all worst-case. The technique is conceptually a special case of the chain method of Lee and Preparata and uses the same query algorithm. The emergence of full dynamic capabilities is afforded by a subtle choice of the chain set (separators), which induces a total order on the set of regions of the planar subdivision.