Advanced Wireless Networks - 4G Technologies

188 TELETRAFFIC MODELING AND ANALYSIS

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u1

V1

u2

V2

u3

V3

u4

Figure 7.7 Bipartite graph.

7.1.8 Trees

Let G = (V, E) be an undirected graph. The following statements are equivalent:

(1)G is a tree;

(2)any two vertices in G are connected by unique simple path;

(3)G is connected, but if any edge is removed from E, the resulting graph is disconnected;

(4)G is connected, and |E| = |V | − 1;

(5)G is acyclic, and |E| = |V | − 1;

(6)G is acyclic, but if any edge is added to E, the resulting graph contains a cycle.

For an illustration see Figure 7.8.

7.1.9 Spanning tree

A tree (T ) is said to span G = (V, E) if T = (V, E ) and E E. For the graph shown Figure 7.4, two possible spanning trees are shown in Figure 7.9. Given connected graph G with real-valued edge weights ce, a minimum spanning tree (MST) is a spanning tree of G whose sum of edge weights is minimized.

196 ADAPTIVE NETWORK LAYER

Figure 7.8 Tree.

Figure 7.9 Spanning trees.

7.1.10 MST computation

7.1.10.1 Prim’s algorithm

Select an arbitrary node as the initial tree (T ). Augment T in an iterative fashion by adding the outgoing edge (u, v), (i.e. u T and v G − T ) with minimum cost (i.e. weight).

Figure 7.10 Minimum spanning tree.

The algorithm stops after |V | − 1 iterations. Computational complexity = O(|V |2). An illustration of the algorithm is given in Figure 7.11.

7.1.10.2 Kruskal’s algorithm

Select the edge e E of minimum weight → E = {e}. Continue to add the edge e E − E of minimum weight that, when added to E , does not form a cycle. Computational complexity = O(|E|xlog|E|). An illustration of the algorithm is given in Figure 7.12.

7.1.10.3 Distributed algorithms

For these algorithms each node does not need complete knowledge of the topology. The MST is created in a distributed manner. The algorithm starts with one or more fragments consisting of single nodes. Each fragment selects its minimum weight outgoing edge and, using control messaging fragments, coordinates to merge with a neighboring fragment over its minimum weight outgoing edge.The algorithm can produce an MST in O(|V |x|V |) time provided that the edge weights are unique. If these weights are not unique, the algorithm still works by using the nodes IDs to break ties between edges with equal weight.The algorithm requires O(|V |xlog|V |) + |E|) message overhead. An illustration of the distributed algorithm is given in Figure 7.13.