Adaptive Point Location with almost No Preprocessing in Delaunay Triangulations

This paper studies adaptive point location in Delaunaytriangulations with $o(n^{1/3})$ (and practically $O(1)$) preprocessing and storage. Given $n$ pseudo-random points in a compact convex set $C$ with unit area in two dimensions (2D) and the corresponding Delaunay triangulation, assume that we know the query points are clustered into $k$ compact convex sets $C_i\subset C$, each with diameter$D(C_i)$, then we show that an adaptive version of the Jump\& Walk method(which requires $o(n^{1/3})$ preprocessing) achieves average query bound$O(n^{\frac{1-4\delta}{3}})$ when in the preprocessing$\Theta(n^{\frac{1-4\delta}{3}})$ sample points are chosen within each $C_i$, where $D(C_i)=\Theta(\frac{1}{n^\delta})$ and $0\leq\delta\leq 1/4$.Similar result holds in three dimensions (3D). Empirical results in 2Dshow that this procedure is 23\%-350\% more efficient than its predecessors under various clustered cases.