Adaptive Planar Point Location

We present a self-adjusting point location structure for convex subdivisions. Let n be the number of vertices in a convex subdivision S. Our structure for S uses O(n) space and processes any online query sequence σ in O(n+ OPT) time, where OPT is the minimum time required by any linear decision tree for answering point location queries in S to process σ. The O(n + OPT) time bound includes the preprocessing time. Our result is a two-dimensional analog of the static optimality property of splay trees. For connected subdivisions, we achieve a processing time of O(|σ| log logn+ n+ OPT). 1998 ACM Subject Classification F.2.2 [Analysis of Algorithms and Problem Complexity] Nonnumerical Algorithms and Problems – Geometrical Problems and Computations