Sublinear Explicit Incremental Planar Voronoi Diagrams

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.

14 pages, 10 figures. Presented ant JCDCGGG 2019