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.

中文翻译：

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图中的逆 Voronoi 问题 II：树

我们考虑树中的逆 Voronoi 图问题：给定具有正边长的树 T 和 V(T) 顶点子集的集合 $$\mathbb {U}$$，决定 $${\mathbb {U }}$$ 是 T 中关于最短路径度量的 Voronoi 图。我们证明该问题可以在 $$O(N+n \log ^2 n)$$ 时间内解决，其中 n 是 T 中的顶点数，$$N=n+\sum _{U\in {\ mathbb {U}}}|U|$$ 是输入描述的大小。我们还为具有 n 个顶点的树提供了 $$\Omega (n \log n)$$ 时间的下限。