Balancing Graph Voronoi Diagrams

Many facility location problems are concerned with minimizing operation and transportation costs by partitioning territory into regions of similar size, each of which is served by a facility. For many optimization problems, the overall cost can be reduced by means ofa partitioning into balanced subsets, especially in those cases where the cost associated with a subset is superlinear in its size.In this paper, we consider the problem of generating a Voronoi partition of a discrete graph so as to achieve balance conditions on the region sizes.Through experimentation, we first establishthat the region sizes of randomly-generated graph Voronoi diagrams vary greatly in practice. We then show how to achieve a balanced partition of a graph via Voronoi site resampling. For bounded-degree graphs, where each of the $n$ nodes has degree at most $d$, and for an initial randomly-chosen set of $s$ Voronoi nodes,we prove that, by extending the set of Voronoi nodes using an algorithm by Thorup and Zwick, each Voronoi region has size at most $4dn/s+1$ nodes, and that the expected size of the extended set of Voronoi nodes is at most $2s\log n$.

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