Balancing Graph Voronoi Diagrams

Abstract

Abstract-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 of a 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 establish that 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.

Parallel Network Mapping Algorithms

“…Even for sequential algorithms, this result may prove useful for applications where minimizing the number of center points is a primary optimization goal. For instance, one can apply our construction to the problems studied by Honiden, Houle, Sommer [18] for balancing graph-theoretic Voronoi diagrams to shave a (log ) factor of the number of centers. It seems likely, therefore, that this result will have other applications as well.…”

“…The key idea of our parallel network mapping algorithm is to first find a set, , of ( , )-balanced centers, using our parallel algorithm from the previous section, and then use this set of centers to compute a graph-theoretic Voronoi diagram [13,18] for , from which we may efficiently then perform a brute-force querying step for each Voronoi region. This initial center-finding step runs in (log ) rounds and builds a set, , of size ( log ).…”

Graph Reconstruction via Distance Oracles

“…In the second phase, Algorithm Local-Reconstruction(V, A) takes as input the vertex set V and the set A computed by Modified-Center(V, s), and returns the edge set of G. It partitions the graph into slightly overlapped components according to the centers in A, and proceeds by exhaustive search within each component. Inspired by the Voronoi diagram partitioning in [15], we show that these components together cover every edge of the graph. The expected query complexity in this phase is O(s log n(n + ∆ 4 (n/s) 2 )).…”

“…To reconstruct graphs of bounded degree, we apply some algorithmic ideas previously developed for compact routing [21] and ideas for Voronoi cells [15]. A closely related model in network discovery and verification provides queries which, upon receiving a node q, returns the distances from q to all other nodes in the graph [12], instead of the distance between a pair of nodes in our model.…”

“…To reconstruct graphs of bounded degree, we apply some algorithmic ideas previously developed for compact routing [21] and ideas for Voronoi cells [15].…”