Prim's algorithm finds a minimum spanning tree of a connected weighted graph. It starts with a minimum weight edge and adds the minimum weight edge incident to the growing tree at each step, as long as it does not form a cycle. The algorithm is demonstrated on a sample weighted graph, finding a minimum spanning tree of weight 10 using Prim's algorithm and alphabetical order to break ties.

1. Prim’s Algorithm
We consider a weighted connected graph G with n vertices. Prim’s algorithm finds a
minimum spanning tree of G.
procedure Prim(G: weighted connected graph with n vertices)
T := a minimum-weight edge
for i = 1 to n − 2
begin
e := an edge of minimum weight incident to a vertex in T and not forming a circuit
in T if added to T
T := T with e added
end
return(T)
Example: Use Prim’s algorithm to find a minimum spanning tree in the following
weighted graph. Use alphabetical order to break ties.
r.
....................................................................................................................................................................................................
r
.
....................................................................................................................................................................................................
r. ............................................................................................................................................................................................................................................................ r
. ............................................................................................................................................................................................................................................................ r.
....................................................................................................................................................................................................
r
.
.....................................................................................................................................................................................................
........................................................................................................................................................................
.
...................................................................................................................................................................................................................................................................................................................
.
........................................................................................................................................................................
a z
c e
b d
3 4
3
2
2 2
5
16
Solution: Prim’s algorithm will proceed as follows. First we add edge {d, e} of weight
1. Next, we add edge {c, e} of weight 2. Next, we add edge {d, z} of weight 2. Next, we
add edge {b, e} of weight 3. And finally, we add edge {a, b} of weight 2. This produces
a minimum spanning tree of weight 10. A minimum spanning tree is the following.
r.
....................................................................................................................................................................................................
r
r. ............................................................................................................................................................................................................................................................ r
r.
....................................................................................................................................................................................................
r
.
........................................................................................................................................................................
.
...................................................................................................................................................................................................................................................................................................................
a z
c e
b d
3
2
2 2
1
Gilles Cazelais. Typeset with LATEX on December 6, 2006.