On locality-sensitive orderings and their applications

For any constant d and parameter ε > 0, we show the existence of (roughly) 1/ε^d orderings on the unit cube [0, 1)^d, such that any two points p, q ∈ [0, 1)^d that are close together under the Euclidean metric are “close together” in one of these linear orderings in the following sense: the only points that could lie between p and q in the ordering are points with Euclidean distance at most ε‖p − q‖ from p or q. These orderings are extensions of the Z-order, and they can be efficiently computed. Functionally, the orderings can be thought of as a replacement to quadtrees and related structures (like well-separated pair decompositions). We use such orderings to obtain surprisingly simple algorithms for a number of basic problems in low-dimensional computational geometry, including (i) dynamic approximate bichromatic closest pair, (ii) dynamic spanners, (iii) dynamic approximate minimum spanning trees, (iv) static and dynamic fault-tolerant spanners, and (v) approximate nearest neighbor search.