Question

Algorithm Question：

The following questions are on minimum spanning tree.

(a) Suppose we have an undirected graph with weights that can be either positive or negative. Do Prim’s and Kruskal’s algorithim produce a MST for such a graph? Explain.

(b) Prove that for any weighted undirected graph such that the weights are distinct (no two edges have the same weight), the minimal spanning tree is unique.

Answer 1

a) Let us consider the following Graph

Edge(u,v)	Weight
b-g	3
c-f	3
a-f	4
e-g	4
a-c	5
b-e	5
d-f	5
a-d	6
b-d	6
d-g	6
f-g	6

Kruskal’s Algorithm to find the Minimum Cost Spanning Tree (MCST) of a graph G as follows:

Step 1:             

Edge(u,v	Weight
a-b	2(min)
c-f	3
b-g	3
a-f	4
e-g	4
a-c	5
b-e	5
d-f	5
a-d	6
b-d	6
d-g	6
f-g	6

                                                    

Step 2:              

Edge(u,v	Weight
b-g	3(min)
c-f	3
a-f	4
e-g	4
a-c	5
b-e	5
d-f	5
a-d	6
b-d	6
d-g	6
f-g	6

  

                               

Step 3:

Edge(u,v	Weight
c-f	3(min)
a-f	4
e-g	4
a-c	5
b-e	5
d-f	5
a-d	6
b-d	6
d-g	6
f-g	6

Step 4:         

Edge(u,v	Weight
a-f	4(min)
e-g	4
a-c	5
b-e	5
d-f	5
a-d	6
b-d	6
d-g	6
f-g	6

            

                                           

Step 5:        

Edge(u,v	Weight
e-g	4(min)
a-c	5
b-e	5
d-f	5
a-d	6
b-d	6
d-g	6
f-g	6

           

            

                

Step 6:

Edge(u,v	Weight
a-c	5(forms Loop)
b-e	5(forms Loop)
d-f	5(min)
a-d	6
b-d	6
d-g	6
f-g	6

For n- Vertices we get (n-1) Edges in MCST. Here in graph having 7 Vertices so we get 6 Edges

Which is Required MCST of a graph

b) The minimum cost spanning tree generated by the algorithm is

Prim’s Algorithm to find the minimum cost spanning tree of a graph starting at vertex (a) as follows

Step 1: All the vertices connected to (a) will be considered

Edge	Cost
a-b	2(min)
a-c	5
a-d	6
a-f	4

Step 2: All the vertices connected to (a) & (b) will be considered

Edge	Cost
a-c	5
a-d	6
a-f	4
b-d	6
b-e	5
b-g	3(min)

Step 3: All the vertices connected to (a), (b) & (g) will be considered

.Edge	Cost
a-c	5
a-d	6
a-f	4(min)
b-d	6
b-e	5
g-d	6
g-e g-f	4 6

Step 4: All the vertices connected to (a), (b), (f) & (g) will be considered

Edge	Cost
a-c	5
a-d	6
b-d	6
b-e	5
g-d	6
g-e g-f	4 6
f-c	3(min)
f-d	5

Step 5: All the vertices connected to (a), (b), (c) ,(f) & (g) will be considered

Edge	Cost
a-c	5
a-d	6
b-d	6
b-e	5
g-d	6
g-e g-f	4(min) 6
f-d	5

Step 6: All the vertices connected to (a), (b), (c), (e), (f) & (g) will be considered

Edge	Cost
a-c	5(loop)
a-d	6
b-d	6
b-e	5
g-d	6
g-f	6
f-d	5(min)

From the above weighted undirected graph Prim’s and Kruskal’s algorithim produce the Same minimal spanning tree

(b) Let us conider the Graph

Kruskal's Algorithm

Edge: Cost:
B-F 4
E-F 6
G-F 10
A-D 11
A-C 15
B-G 19
D-E 20
A-B 23
B-C 25
C-F 32
A-E 34
C-E 41

From above Prim’s and Kruskal’s algorithim weighted undirected graph such that the weights are distinct (no two edges have the same weight), the minimal spanning tree is unique.