🤖 AI Summary
This work addresses the inefficiency and suboptimal size of locality-sensitive orderings (LSOs) in Euclidean space, where existing constructions fall significantly short of the theoretical lower bound. We propose a novel construction based on geometric partitioning, nested grids, and hierarchical randomized ordering. In ℝᵈ, our method achieves the first ε-LSO of size O(ε¹⁻ᵈ log(1/ε)), matching the Ω(ε¹⁻ᵈ) lower bound up to a logarithmic factor—thus attaining near-optimal size. Crucially, the ordering guarantees that for any two points, all intermediate points lie within ε-distance of at least one endpoint. Theoretically, this reveals an intrinsic compactness of LSOs in low-dimensional Euclidean spaces. Practically, it enables several breakthroughs in dynamic geometric algorithms: (i) dynamic spanner maintenance with insertion/deletion cost matching the known lower bound, and (ii) improved data structures for the bichromatic closest-pair problem.
📝 Abstract
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ewcommand{pts}{s}$ For a parameter $eps in (0,1)$, we present a new construction of $eps$-locality-sensitive orderings (<LSOs) in $Re^d$ of size $M = O(Eps^{d-1} log Eps)$, where $Eps = 1/eps$. This improves over previous work by a factor of $Eps$, and is optimal up to a factor of $log Eps$. Such a set of LSOs has the property that for any two points, $p, q in [0,1]^d$, there exist an order in the set such that all the points between $p$ and $q$ in the order are $eps$-close to either $p$ or $q$. The existence of such LSOs is a fundamental property of low dimensional Euclidean space, conceptually similar to the existence of well-separated pairs decomposition, so the question of how to compute (near) optimal construction of LSOs is quite natural. 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 $log Eps$ factor) the lower bound, Thus offering a near-optimal simple dynamic data-structure for maintaining spanners under insertions and deletions.
