The graph Voronoi diagram with applications

Abstract

The Voronoi diagram is a famous structure of computational geometry. We show that there is a straightforward equivalent in graph theory which can be efficiently computed. In particular, we give two algorithms for the computation of graph Voronoi diagrams, prove a lower bound on the problem, and identify cases where the algorithms presented are optimal. The space requirement of a graph Voronoi diagram is modest, since it needs no more space than does the graph itself. The investigation of graph Voronoi diagrams is motivated by many applications and problems on networks that can be easily solved with their help. This includes the computation of nearest facilities, all nearest neighbors and closest pairs, some kind of collision free moving, and anticenters and closest points. © 2000 John Wiley & Sons, Inc.

Indexing Network Voronoi Diagrams

“…The network Voronoi diagrams can be constructed using parallel Dijkstra algorithm [2] with the Voronoi generators as multiple sources. Specifically, one can expand shortest path trees from each Voronoi generator simultaneously and stop the expansions when the shortest path trees meet.…”

Balancing Graph Voronoi Diagrams

“…Given a set of points (the Voronoi sites), the Voronoi decomposition leads to regions (the Voronoi regions) consisting of all points that are closest to a specific site. Mehlhorn [21] and Erwig [22] proposed an analogous decomposition, the Graph Voronoi Diagram, for undirected and directed graphs respectively. Definition 1 (Graph Voronoi Diagram [21], [22]).…”

“…Mehlhorn [21] and Erwig [22] proposed an analogous decomposition, the Graph Voronoi Diagram, for undirected and directed graphs respectively. Definition 1 (Graph Voronoi Diagram [21], [22]). In a graph G = (V, E, ω), the Voronoi diagram for a set of nodes…”

“…Erwig [22,Thm. 2] shows that the graph Voronoi diagram can be constructed via a single Dijkstra search in time O(m + n · log n).…”

“…For the case of general graphs, one may further relax the region size constraints so as to compute a partition for which the size of every region is bounded only from above, but the number of regions is not too large. The graph Voronoi diagram [21], [22] would constitute a solution to this relaxed partitioning problem if the region sizes could be made to respect the balancing condition.…”