Adaptive Planar Point Location

Abstract: We present self-adjusting data structures for answering point location queries in convex and connected subdivisions. Let $n$ be the number of vertices in a convex or connected subdivision. Our structures use $O(n)$ space. For any convex subdivision $S$, our method processes any online query sequence $\sigma$ in $O(\mathrm{OPT} + n)$ time, where $\mathrm{OPT}$ is the minimum time required by any linear decision tree for answering point location queries in $S$ to process $\sigma$. For connected subdivisions, the processing time is $$O(\mathrm{OPT} + n + |\sigma|\log(\log^* n))$$. In both cases, the time bound includes the $O(n)$ preprocessing time.