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Timothy M. Chan, Sariel Har-Peled, and Mitchell Jones.
For any constant $d$ and parameter $\eps > 0$, we show the existence of (roughly) $1/\eps^d$ orderings on the unit cube $[0,1)^d$, such that any two points $\pt,\ptq\in [0,1)^d$ 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 $\pt$ and $\ptq$ in the ordering are points with Euclidean distance at most $\eps\distY{\pt}{\ptq}$ from $\pt$ or $\ptq$. 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.

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