Adaptive Point Location in Planar Convex Subdivisions

We present a planar point location structure for a convex subdivision S. Given a query sequence of length m, the total running time is O(OPT + m log log n + n), where n is the number of vertices in S and OPT is the minimum time required by any linear decision tree for answering planar point location queries in S to process the same query sequence. The running time includes the preprocessing time. Therefore, for m ≥ n, our running time is only worse than the best possible bound by O(log log n) per query, which is much smaller than the O(log n) query time offered by a worst-case optimal planar point location structure.