8.5 Minimum Spanning Trees
A common problem is communications networks and circuit design is that of connecting together a set of
nodes by a network of total minimum length. The length is the sum of lengths of connecting wires.
Consider, for example, laying cable in a city for cable t.v.
The computational problem is called the minimum spanning tree (MST) problem. Formally, we are given a
connected, undirected graph G=(V, E) Each edge (u,v) has numeric weight of cost. We define the cost
of a spanning tree T to be the sum of the costs of edges in the spanning tree
w(T) =
(u,v) 2T
w(u, v)
∈
∑
A minimum spanning tree is a tree of minimum weight.
Figures 8.42, 8.43 and 8.44 show three spanning trees for the same graph. The first is a spanning tree but is
not a MST; the other two are.
Figure 8.42: A spanning tree Figure 8.43: A minimum Figure 8.44: Another mini-
that is not MST spanning tree mum spanning tree
We will present two greedy algorithms (Kruskal’s and Prim’s) for computing MST. Recal l that a greedy
algorithm is one that builds a solution by repeatedly selecting the cheapest among all options at each
stage. Once the choice is made, it is never undone.
Before presenting the two algorithms, let us review facts about free trees. A free tree is a tree with no
vertex designated as the root vertex. A free tree with n vertices has exactly n- 1 edges. There exists a
unique path between any two vertices of a free tree. Adding any edge to a free tree creates a unique cycle.
Breaking any edge on this cycle restores the free tree. This is illustrated in Figure 8.45. When the edges
(b, e) or (b, d) are added to the free tree, the result is a cycle.
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