Higher-order Voronoi diagrams on triangulated surfaces

We study the complexity of higher-order Voronoi diagrams on triangulated surfaces under the geodesic distance, when the sites may be polygonal domains of constant complexity. More precisely, we show that on a surface defined by n triangles the sum of the combinatorial complexities of the order-j Voronoi diagrams of m sites, for j=1,…,k, is O(k²n²+k²m+knm), which is asymptotically tight in the worst case.