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arXiv · 1604.03654

Dynamic Planar Voronoi Diagrams for General Distance Functions and their Algorithmic Applications

Haim Kaplan·Wolfgang Mulzer·Liam Roditty·Paul Seiferth·Micha Sharir
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Abstract

We describe a new data structure for dynamic nearest neighbor queries in the plane with respect to a general family of distance functions. These include $L_p$-norms and additively weighted Euclidean distances. Our data structure supports general (convex, pairwise disjoint) sites that have constant description complexity (e.g., points, line segments, disks, etc.). Our structure uses $O(n \log^3 n)$ storage, and requires polylogarithmic update and query time, improving an earlier data structure of Agarwal, Efrat and Sharir that required $O(n^\varepsilon)$ time for an update and $O(\log n)$ time for a query [SICOMP, 1999]. Our data structure has numerous applications. In all of them, it gives faster algorithms, typically reducing an $O(n^\varepsilon)$ factor in the previous bounds to polylogarithmic. In addition, we give here two new applications: an efficient construction of a spanner in a disk intersection graph, and a data structure for efficient connectivity queries in a dynamic disk graph.

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Haim Kaplan, Wolfgang Mulzer, Liam Roditty, Paul Seiferth, Micha Sharir. 2020-10-01. Dynamic Planar Voronoi Diagrams for General Distance Functions and their Algorithmic Applications. https://doi.org/10.1007/s00454-020-00243-7

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