Chapter 9 - Spanning Trees and Spanners

Abstract

The most basic and well-studied network design problem is that of finding minimum spanning trees: connect the sites into a tree with minimum total weight. By constructing a complete geometric graph, a geometric minimum spanning tree in time O(n2) can be found, in any metric space for which distances can be computed quickly. Minimum spanning trees can be found by solving a collection of bichromatic closest pair problems for the pairs of a well-separated pair decomposition, then computing a minimum spanning tree on the resulting graph. However for this approach to be useful, the complexity of the well separated pair decomposition must be bounded. By considering nonplanar graphs, it is possible to find sparse graphs approximating the complete Euclidean graph arbitrarily closely. Variations of the Yao graph construction—in which one partitions the space around each point into wedges with a given fixed opening angle, and connects the point to the nearest neighbor in each wedge—produce graphs with dilation arbitrarily close to 1, with O(n) edges, and that can be constructed in time O(n log n).