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

Input: a weighted graph G = (V, E) with non-negative weights
Output: A shortest path tree from a source vertex s
1 ShortestPathTree getShortestPath(s) {
2 Let T be a set that contains the vertices whose
3 paths to s are known;
4 Initially T contains source vertex s with cost[s] = 0;
5 for (each u in V – T)
6 cost[u] = infinity;
7
8 while (size of T < n)
{
9 Find v in V – T with the smallest cost[u] + w(u, v) value
10 among all u in T;
11 Add v to T and set cost[v] = cost[u] + w(u, v);
12 parent[v] = u;
13 }
14 }
The algorithm uses cost[v] to store the cost of the shortest path from vertex v to the source vertex s.
cost[s] is 0. Initially assign infinity to cost[v] to indicate that no path is found from v
to s. Let V denote all vertices in the graph and T denote the set of the vertices whose costs
are known. Initially, the source vertex s is in T. The algorithm repeatedly finds a vertex u in
T and a vertex v in V – T such that cost[u] + w(u, v) is the smallest, and moves v
to T. The shortest path algorithm given in the text coninously update the cost and parent for a
vertex in V – T. This algorithm initializes the cost to infinity for each vertex and then
changes the cost for a vertex only once when the vertex is added into T. This algorithm can
be implemented using priority queues to achieve the O(|E|log|V|) time complexity. Use
Listing 25.9 TestShortestPath.cpp to test your new algorithm.
49