Sublinear Explicit Incremental Planar Voronoi Diagrams

TL;DR

This paper introduces a novel data structure for maintaining Voronoi diagrams and convex hulls with sublinear expected amortized insertion time, advancing dynamic geometric data structure efficiency.

Contribution

It presents the first sublinear expected amortized time algorithm for explicit incremental updates of planar Voronoi diagrams and 3D convex hulls.

Findings

Supports insertions in  O(N^{3/4}) expected amortized time

First to achieve sublinear time insertions for these structures

Improves upon previous    bounds for dynamic updates

Abstract

A data structure is presented that explicitly maintains the graph of a Voronoi diagram of $N$ point sites in the plane or the dual graph of a convex hull of points in three dimensions while allowing insertions of new sites/points. Our structure supports insertions in $\tilde O (N^{3/4})$ expected amortized time, where $\tilde O$ suppresses polylogarithmic terms. This is the first result to achieve sublinear time insertions; previously it was shown by Allen et al. that $\Theta(\sqrt{N})$ amortized combinatorial changes per insertion could occur in the Voronoi diagram but a sublinear-time algorithm was only presented for the special case of points in convex position.