MINIMAL CONNECTOR PROBLEMSProblem:

A cable TV company is installing a system

of cables to connect all the towns in a region.

The numbers in the network show

distancesin miles.

F E

D

C

B

A

10

8 14

1215

13

1920

12

KRUSKAL’S ALGORITHM

Step 1 Rank the arcs in ascending order of weight

Step 2 Select the arc of least weight and use this tostart the tree

Step 3 Choose the next smallest arc and addthis to the tree UNLESS IT

COMPLETES A CYCLE in which casereject it and proceed to the next smallest arc

Step 4 Repeat Step 3 until all vertices areincluded in the tree

F E

D

C

B

A

10

8 14

1215

13

1920

12

Smallest arc:

FE (8)

F E8

List the arcs as you choose them

FE

Next smallest arc:

FD (10)

D

10

FD

DB (12)

B12 DB

Not DE because it produces a cycle

BC (13)

C

13

BC

AB (19)

A

19

AB

Not EC or DC as they produce cycles

You now have a minimal spanning tree

Total Length = 62

PRIM’S ALGORITHM

Step 1 Choose a starting vertex

Step 2 Connect it to the nearest vertex using the

least arc

Step 3 Connect the nearest vertex not in the tree,to the tree, using the least arc

Step 4 Repeat step 3 until all of the vertices areconnected

F E

D

C

B

A

10

8 14

1215

13

1920

12

Start with vertex A

Nearest is B so add AB

A

B

19

Arcs from A and B have lengths

20, 12 and 13

So choose BD (12)

List the arcs as you choose themAB

D12 BD

Arcs from A, B & D have lengths

13, 15, 12 and 10

So choose DF (10)

F

10 DF

Arcs from A, B, D, E & F have lengths

13, 15 and 14

So choose FE (8)

E8

FE

So choose BC (13)

Arcs from A, B, D & F have lengths

13, 15 and 8

C

13BC

You now have a minimal spanning tree

Total Length = 62

Both algorithms have led to the same minimal spanning treeBut the order of choosing the arcs is different

Kruskal gives: FE, FD, DB, BC, AB

Prim gives:

AB, BD, DF, FE, BC

That is why it is important to write down the arcs as you select them so that the examiner can tell whether or not you have used the correct algorithm