Dynamic Distribution-Sensitive Point Location

We propose a dynamic data structure for the distribution-sensitive point location problem in the plane. Suppose that there is a fixed query distribution within a convex subdivision S , and we are given an oracle that can return in O (1) time the probability of a query point falling into a polygonal region of constant complexity. We can maintain S such that each query is answered in O opt (S) ) expected time, where opt ( S ) is the expected time of the best linear decision tree for answering point location queries in S . The space and construction time are O(n log 2 n ), where n is the number of vertices of S . An update of S as a mixed sequence of k edge insertions and deletions takes O(k log 4 n) amortized time. As a corollary, the randomized incremental construction of the Voronoi diagram of n sites can be performed in O(n log 4 n ) expected time so that, during the incremental construction, a nearest neighbor query at any time can be answered optimally with respect to the intermediate Voronoi diagram at that time.

Adaptive Point Location in Planar Convex Subdivisions

We present a planar point location structure for a convex subdivision [Formula: see text]. Given a query sequence of length [Formula: see text], the total running time is [Formula: see text], where [Formula: see text] is the number of vertices in [Formula: see text] and [Formula: see text] is the minimum time required by any linear decision tree for answering planar point location queries in [Formula: see text] to process the same query sequence. The running time includes the preprocessing time. Therefore, for [Formula: see text], our running time is only worse than the best possible bound by [Formula: see text] per query, which is much smaller than the [Formula: see text] query time offered by a worst-case optimal planar point location structure.