CISC 621 - Homework Problem R
Model Solution
a) Kruskal’s algorithm produces a MSF for G even when G is
unconnected because it adds safe edges to the set A in such a way that
the edges in the set do not need to form a single tree.
The algorithm in the first step considers a forest of |V| trees. Then, at
each step, it finds a safe edge to add to the growing forest by finding,
of all the edges that connect any two trees in the forest, an edge (u,v)
of least weight. Thus, the algorithm ends up producing a set of
connected components, with each connected component a MST.
b) MSF-Kruskal (G,w)
1 A <- 0
2 num_vertices <- 0 \\ variable to store number of vertices in
the graph
3 num_msf_edges<- 0 \\ variable to store the number of MSF
edges for G
4 num_components <- 0 \\ variable to store the number of
components in G
5 for each vertext v ε V[G]
6 do Make-set (v)
7 num_vertices <- num_vertices + 1 \\** new addition**
\\ After the loop, the variable num_vertices contains the number of
vertices in the graph
8 sort the edges of E by nondecreasing weight w
9 for each edge (u,v) ε E , in order by nondecreasing weight
10 do if FIND-SET(u) ≠ FIND-SET (v)
11 then A<- A U {(u,v)}
12 UNION (u,v)
13 num_msf_edges = num_msf_edges + 1
\\** new addition **//
\\ After the ‘for’ loop, the variable num_msf_edges contains
the number of edges in MSF
