Dynamic planar point location with optimal query time

Abstract

We present a new dynamic technique for locating a point in a convex planar subdivision whose n vertices lie on a fixed set of N horizontal lines. The supported update operations are insertion/deletion of vertices and edges, and (horizontal) translation of vertices. Our method achieves query time O n + log IV, space O( N+ n log N). and insertion/deletion time O( log n log N). Hence, for N=O( n), the query time is O( log n), which is optimal. The proposed technique, based on the trapezoid method, provides an efficient solution to many significant applications where the most frequent operation is the point location query, while updates are more rarely executed.