ON LOCALITY-SENSITIVE ORDERINGS AND THEIR APPLICATIONS

For any constant d and parameter epsilon is an element of (0, 1/2], we show the existence of (roughly) 1/epsilon(d )orderings on the unit cube [0, 1)(d )such that for any two points p, q is an element of [0, 1)(d) close together under the Euclidean metric, there is a linear ordering in which all points between p and q in the ordering are "close" to p or q. More precisely, the only points that could lie between p and q in the ordering are points with Euclidean distance at most epsilon parallel to p - q parallel to from either 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.