The Inverse Voronoi Problem in Graphs II: Trees

We consider the inverse Voronoi diagram problem in trees: given a tree T with positive edge-lengths and a collection $$\mathbb {U}$$ of subsets of vertices of V(T), decide whether $${\mathbb {U}}$$ is a Voronoi diagram in T with respect to the shortest-path metric. We show that the problem can be solved in $$O(N+n \log ^2 n)$$ time, where n is the number of vertices in T and $$N=n+\sum _{U\in {\mathbb {U}}}|U|$$ is the size of the description of the input. We also provide a lower bound of $$\Omega (n \log n)$$ time for trees with n vertices.