The.Algorithm.Design.Manual.Springer-Verlag.1998

Prim's Algorithm
Next: Kruskal's Algorithm Up: Minimum Spanning Trees Previous: Minimum Spanning Trees
Prim's Algorithm
Every vertex will appear in the minimum spanning tree of any connected graph G. Prim's minimum spanning tree algorithm
starts from one vertex and grows the rest of the tree one edge at a time.
In greedy algorithms, we make the decision of what to do next by selecting the best local option from all available choices
without regard to the global structure. Since we seek the tree of minimum weight, the natural greedy algorithm for minimum
spanning tree repeatedly selects the smallest weight edge that will enlarge the tree.
Figure: Where Prim's algorithm goes bad? No, because
Prim-MST(G)
non-tree vertex
Select an arbitrary vertex to start the tree from.
While (there are still non-tree vertices)
Select the edge of minimum weight between a tree and
Add the selected edge and vertex to the tree .
Prim's algorithm clearly creates a spanning tree, because no cycle can be introduced by adding edges between tree and non-tree
vertices. However, why should it be of minimum weight over all spanning trees? We have seen ample evidence of other
natural greedy heuristics that do not yield a global optimium. Therefore, we must be particularly careful to demonstrate any
such claim.
Suppose that there existed a graph G for which Prim's algorithm did not return a minimum spanning tree. Since we are
building the tree incrementally, this means that there must have been some particular instant where we went wrong. Before we
file:///E|/BOOK/BOOK2/NODE74.HTM (1 of 2) [19/1/2003 1:29:14]