Near Optimal Locality Sensitive Orderings in Euclidean Space

For a parameter ε ∈ (0, 1), a set of ε-locality-sensitive orderings (LSOs) has the property that for any two points, p, q ∈ [0, 1)^d, there exist an order in the set such that all the points between p and q (in the order) are ε-close to either p or q. Since the original construction of LSOs can not be (significantly) improved, we present a construction of modified LSOs, that yields a smaller set, while preserving their usefulness. Specifically, the resulting set of LSOs has size M = O(E^d−1 log E), where E = 1/ε. This improves over previous work by a factor of E, and is optimal up to a factor of log E. As a consequence we get a flotilla of improved dynamic geometric algorithms, such as maintaining bichromatic closest pair, and spanners, among others. In particular, for geometric dynamic spanners the new result matches (up to the aforementioned log E factor) the lower bound, Specifically, this is a near-optimal simple dynamic data-structure for maintaining spanners under insertions and deletions.