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

Prim's Algorithm
inserted edge (x,y), consisted of a set of edges that was a subtree of a minimum spanning tree , but choosing edge
(x,y) took us away from a minimum spanning tree. But how could it? There must be a path p from x to y in , using an edge
, where is in but is not. This edge must have weight at least that of (x,y), or else Prim's algorithm would
have selected it instead of (x,y) when it had the chance. Inserting (x,y) and deleting from leaves a spanning tree no
larger than before, meaning that Prim's algorithm could not have made a fatal mistake in selecting edge (x,y). Therefore, by
contradiction, Prim's algorithm has to construct a minimum spanning tree.
Prim's algorithm is correct, but how efficient is it? That depends on which data structures are used to implement it, but it
should be clear that O(nm) time suffices. In each of n iterations, we will scan through all the m edges and test whether the
current edge joins a tree with a non-tree vertex and whether this is the smallest edge seen thus far. By maintaining a Boolean
flag along with each vertex to denote whether it is in the tree or not, this test can be performed in constant time. In fact, better
data structures lead to a faster, , implementation by avoiding the need to sweep through more than n edges in any
iteration.
Next: Kruskal's Algorithm Up: Minimum Spanning Trees Previous: Minimum Spanning Trees
Algorithms
Mon Jun 2 23:33:50 EDT 1997
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