Minimum Weight Euclidean Matching and Weighted Relative Neighborhood Graphs

The Minimum Weight Euclidean Matching (MWEM) problem is: given 2n point sites in the plane with Euclidean metric for interpoint distances, match the sites into n pairs so that the sum of the n distances between matched pairs is minimized. The graph theoretic version of this problem has been extensively studied since the pioneering work of Edmonds. The best time bound known for MEWM is O(n 2:5 (log n) 4) due to Vaidya. His algorithm requires O(n log n) space. We investigate new geometric properties of the problem and propose an O(n) space, O((n 2 +F) log n) time algorithm based on the Weighted Voronoi Diagram (WVD) of the sites, where F is the number of edge-ips in the diagram as the weights change during the matching algorithm. We conjecture that F is close to O(n 2). The new geometric results established in this paper include the following: We introduce Weighted Relative Neighborhood Graphs (WRNG) and Weighted Gabriel Graphs (WGG). These are generalizations of their unweighted versions studied in the literature. We show WRNG and WGG are straight-line planar graphs; WRNG is a subgraph of WGG; and WGG is a subgraph of the dual of WVD. Furthermore, we show that the admissible edges (and hence, the matching edges) in Edmonds' primal-dual algorithm form a subgraph of WRNG.