Nearly Optimal Planar $k$ Nearest Neighbors Queries under General Distance Functions

We study the $k$ nearest neighbors problem in plane for general, convex, pairwise disjoint sites of constant description complexity such as line segments, disks, and quadrilaterals under a general family distance functions including $L_p$ norms additively weighted Euclidean distances. compose static data structure this setting with nearly optimal $O(n\log\log n)$ space, $O(\log n+k)$ query time, expected $O(n\log preprocessing time. also devise dynamic (that allows insertions deletions sites) $O(\log^2n+k\log amortized $O(\log^2 insertion time $O(\log^4n)$ deletion matching best known complexities point metric improving many applications minimum spanning trees planar connectivity disk intersection graphs. Our results, to some extent, indicate that problem, may share same metric. To achieve progress, we design vertical shallow cuttings linear size functions. Vertical are key technique tackle metric, while existing generalizations either not or size. innovation is new random sampling analysis geometric structures, since provides way develop algorithms, believe it independent interest.