CHAPTER 25 Weighted Graphs and Applications Objectives • To ...

Figure 25.5
The tree in (c) and (d) are minimum spanning trees of the graph in (a).
The problem of finding a minimum spanning tree has many applications. Consider a company with
branches in many cities. The company wants to lease telephone lines to connect all branches together. The
phone company charges different amounts of money to connect different pairs of cities. There are many
ways to connect all branches together. The cheapest way is to find a spanning tree with the minimum total
rates.
25.4.1 Minimum Spanning Tree Algorithms
How do you find a minimum spanning tree? There are several well-known algorithms for doing so. This
section introduces Prim’s algorithm. Prim’s algorithm starts with a spanning tree T that contains an
arbitrary vertex. The algorithm expands the tree by adding a vertex with the smallest edge incident to a
vertex already in the tree. The algorithm is described in Listing 25.4.
Listing 25.4 Prim’s Minimum Spanning Tree Algorithm
Input: A connected undirected weighted G = (V, E)
Output: MST (a minimum spanning tree)
1 MST minimumSpanningTree()
2 {
3 Let T be a set for the vertices in the spanning tree;
4 Set cost[v] = infinity for each vertex v;
5 Pick an arbitrary vertex, say s and set cost[s] = 0 and parent[s]
= -1;
6
7 while (size of T < n)
8 {
9 Find u not in T with the smallest cost[u];
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