ON LOCALITY-SENSITIVE ORDERINGS AND THEIR APPLICATIONS.

For any constant d and parameter ε ∈(0, 1/2], we show the existence of (roughly) 1/ εd orderings on the unit cube [0, 1)d such that for any two points p, q ∈ [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 ε ||p - q|| 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 lowdimensional 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. [ABSTRACT FROM AUTHOR]