Hoc And Sensor Networks Sample Solution to Exercise 3

From the four considered algorithms the Minimum Spanning Tree (MST) is the only one resulting in a bounded-degree topology. Figure 1 depicts worst-case instances for the different graphs. Figure 1(a) gives thereby an idea why the degree of the MST is bounded to six. If the angle between two adjacent edges (u, v) and (u,w) is less than 60 degrees it would be cheaper to abandon one of these edges and add (v, w) to the final topology. By the definition of the Relative Neighborhood Graph (RNG), an edge is discarded if a node is situated inside the lune of this edge (see Figure 1(b)). However, the boundary is not included. If we therefore arrange n nodes on a circle around another node, we end up with a maximum degree of n. The same worst-case example also exists for the Gabriel Graph (GG), and the Delaunay Triangulation (DT) (Figure 1(c) and (d)). With these topologies the critical areas depicted in Figure 1 do not contain any other nodes on the circle around the center node. Thus, we also obtain a graph with unbounded degree in the worst case.