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 sigma 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 sigma. 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(|sigma| log log n + n + OPT)