d1,l5 1 kruskal's and prim's algorithms

7/29/2019 D1,L5 1 Kruskal's and Prim's Algorithms

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Minimum spanning trees


2/18

Minimum Connector Algorithms

Kruskals algorithm

1. Select the shortest edge in a

network

2. Select the next shortest edge

which does not create a cycle

3. Repeat step 2 until all vertices

have been connected

Prims algorithm

1. Select any vertex

2. Select the shortest edgeconnected to that vertex

3. Select the shortest edge

connected to any vertex

already connected

4. Repeat step 3 until all

vertices have been

connected


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A cable company want to connect five villages to their network

which currently extends to the market town of Avonford. What is

the minimum length of cable needed?

Avonford Fingley

Brinleigh Cornwell

Donster

Edan

2

7

4

5

8 64

5

3

8

Example


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We model the situation as a network, then theproblem is to find the minimum connector for the

network

A F

B C

D

E

2

7

4

5

8 64

5

3

8


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AF

BC

D

E

2

7

45

86

4

5

3

8

List the edges in

order of size:

ED 2

AB 3

AE 4

CD 4BC 5

EF 5

CF 6

AF 7

BF 8DF 8

Kruskals Algorithm


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Select the shortest

edge in the network

ED 2

Kruskals Algorithm

AF

BC

D

E

2

7

45

86

4

5

3

8


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Select the next shortest

edge which does notcreate a cycle

ED 2

AB 3

Kruskals Algorithm

AF

BC

D

E

2

7

45

86

4

5

3

8


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ED 2

AB 3

CD 4 (or AE 4)

Kruskals Algorithm

AF

BC

D

E

2

7

45

86

4

5

3

8


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ED 2

AB 3

CD 4AE 4

Kruskals Algorithm

AF

BC

D

E

2

7

45

86

4

5

3

8


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ED 2

AB 3

CD 4AE 4

BC 5 forms a cycle

EF 5

Kruskals Algorithm

AF

BC

D

E

2

7

45

86

4

5

3

8


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All vertices have been

connected.

The solution is

ED 2

AB 3CD 4

AE 4

EF 5

Total weight of tree: 18

Kruskals Algorithm

AF

BC

D

E

2

7

45

86

4

5

3

8


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AF

BC

D

E

2

7

45

86

4

5

3

8

Select any vertex

A

Select the shortest

edge connected to

that vertex

AB 3

Prims Algorithm


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AF

BC

D

E

2

7

45

86

4

5

3

8

Select the shortest

edge connected toany vertex already

connected.

AE 4

Prims Algorithm


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Select the shortest


connected.

ED 2

Prims Algorithm

AF

BC

D

E

2

7

45

86

4

5

3

8


15/18

Select the shortest


connected.

DC 4

Prims Algorithm

AF

BC

D

E

2

7

45

86

4

5

3

8


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Select the shortest


connected.

CB 5 forms a cycle

EF 5

Prims Algorithm

AF

BC

D

E

2

7

45

86

4

5

3

8


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Prims Algorithm

AF

BC

D

E

2

7

45

86

4

5

3

8

All vertices have been

connected.

The solution is

ED 2

AB 3CD 4

AE 4

EF 5

Total weight of tree: 18


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Both algorithms will always give solutions with thesame length.

They will usually select edges in a different order

you must show this in your workings.

Occasionally they will use different edges thismay happen when you have to choose betweenedges with the same length. In this case there is

more than one minimum connector for thenetwork.

Some points to note

d1,l5 1 kruskal's and prim's algorithms

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