Minimum Spanning Trees: Chapter 6s The Modern Corp. Problem

• Modern Corporation has decided to have a state-of-the-art fiber-optic network installed to provide high-speed communication (data, voice, and video) between its major centers.

• Any pair of centers do not need to have a cable directly connecting them in order to take advantage of the technology. All that is necessary is to have a series of cables that connect the centers.

Question: Which cables should be installed to provide high-speed communications between every pair of centers.

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Modern Corporation’s Major Centers

B

A

D

C

F

E

G2 2

1 3 1

5

7

5

4

4

7

4

6s-2

The Optimal Solution

B

A

D

C

F

E

G2 2

1 3 1

5

6s-3

Assumptions of a Minimum-Spanning Tree Problem

1. You are given the nodes of a network but not the links. Instead, you are given the potential links and the positive cost (or a similar measure) for each if it is inserted into the network.

2. You wish to design the network by inserting enough links to satisfy the requirement that there be a path between every pair of nodes.

3. The objective is to satisfy this requirement in a way that minimizes the total cost of doing so.

6s-4

Algorithm for a Minimum-Spanning-Tree Problem

1. Choice of the first link: Select the cheapest potential link.

2. Choice of the next link: Select the cheapest potential link between a node that already is touched by a link and a node that does not yet have such a link.

3. Repeat step 2 over and over until every node is touched by a link (perhaps more than one). At that point, an optimal solution (a minimum spanning tree) has been obtained.

(Ties for the cheapest potential link at each step may be broken arbitrarily.)

6s-5

Application of Algorithm to Modern Corp.: First Link

B

A

D

C

F

E

G2 2

1 3 1

5

7

45

7

4

4

6s-6

Application of Algorithm to Modern Corp.: Second Link

B

A

D

C

F

E

G2 2

1 3 1

5

7

45

7

4

4

6s-7

Application of Algorithm to Modern Corp.: Third Link

B

A

D

C

F

E

G2 2

1 3 1

5

7

45

7

4

4

6s-8

Application of Algorithm to Modern Corp.: Fourth Link

B

A

D

C

F

E

G2 2

1 3 1

5

7

45

7

4

4

6s-9

Application of Algorithm to Modern Corp.: Fifth Link

B

A

D

C

F

E

G2 2

1 3 1

5

7

45

7

4

4

6s-10

Application of Algorithm to Modern Corp.: Final Link

B

A

D

C

F

E

G2 2

1 3 1

5

7

45

7

4

4

6s-11

Applications of Minimum-Spanning-Tree Problems

1. Design of telecommunication networks (computer networks, lease-line telephone networks, cable television networks, etc.)

2. Design of a lightly-used transportation network to minimize the total cost of providing the links (rail lines, roads, etc.)

3. Design of a network of high-voltage electrical power transmission lines.

4. Design of a network of wiring on electrical equipment (e.g., a digital computer system) to minimize the total length of the wire.

5. Design of a network of pipelines to connect a number of locations.

6s-12