Prims Algorithm

Minimal spanning tree.

In the mathematical field of graph theory, a spanning tree of a connected, undirected graph is a tree which includes every vertex of that graph. More generally, a spanning forest of an arbitrary undirected graph is a forest which includes every vertex of the graph. Spanning forests always exist, and can always be constructed so as to have exactly one tree for each connected component. In certain fields of graph theory, involving weighted graphs, it is often useful to find a minimal spanning tree.

Prim's algorithm builds a tree while having the graph connected at all times.

Prim's algorithm maintains two lists, EV which is the vertices already in the tree, and E, the list of edges that makes up the spanning tree. In determining current edges for the tree, we look for a node that's in EV, and on that isn't, such that its path is minimum.
EV = { 0 }

E = { }

while( E has < n-1 edges ) {

find (u,v) with least cost, such that
u is in EV and v isn't in EV

if no such edge exists, break

add v to EV

add (u,v) to E
}
On termination, if the graph is connected, EV will contain all the nodes in the graph, and E will contain the set edges comprising the minimum spanning tree.