Adaptive Point Location in Planar Convex Subdivisions

Abstract

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.