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

TL;DR

This paper introduces a new dynamic data structure for nearest neighbor queries in the plane with respect to various distance functions, improving efficiency and enabling new applications like spanner construction and connectivity queries.

Contribution

A novel dynamic data structure supporting general distance functions and convex sites with polylogarithmic update and query times, surpassing previous methods.

Findings

Supports $L_p$-norms and additively weighted Euclidean distances.

Uses $O(n ext{log}^3 n)$ storage with polylogarithmic update/query time.

Enables faster algorithms for applications like spanner construction and dynamic connectivity.

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.