Connecting Red Cells in a Bicolour Voronoi Diagram

Let S be a set of n + m sites, of which n are red and have weight w R , and m are blue and weigh w B . The objective of this paper is to calculate the minimum value of the red sites’ weight such that the union of the red Voronoi cells in the weighted Voronoi diagram of S is a connected region. This problem is solved for the multiplicatively-weighted Voronoi diagram in \(\mathcal{O}((n+m)^2 \log (nm))\) time and for both the additively-weighted and power Voronoi diagram in \(\mathcal{O}(nm \log (nm))\) time.