Fully dynamic techniques for point location and transitive closure in planar structures

It is shown that a planar st-graph G admits two total orders on the set V∪E∪F, where V, E, and F are, respectively, the sets of vertices, edges and faces of G, with |V|=n. An O(n) space data structure for the maintenance of the two orders is exhibited that supports an update of G (insertion of an edge and expansion of a vertex, and their inverses) in time O(log n). This data structure also supports transitive-closure queries in O(log n). Moreover, planar st-graphs provide the topological underpinning of a fully dynamic planar point location technique in monotone subdivisions, which is an interesting (unique) specialization of the chain method of Lee-Preparata (1977). While maintaining storage O(n) and query time O(log² n), insertion/deletion of a chain with k edges can be done in time O(log² n+k), and insertion/deletion of a vertex on an edge can be done in time O(log n)