d1,l5 kruskal's and prim's algorithms.ppt

8/9/2019 D1,L5 Kruskal's and Prim's algorithms.ppt

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Minimum spanning trees


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Minimum Connector Algorithms

Kruskal’s algorithm

1. Select the shortest edge in anetwork

2. Select the next shortest edgewhich does not create a cycle

3. Repeat step 2 until all verticeshave been connected

Prim’s algorithm

1. Select any vertex

2. Select the shortest edgeconnected to that vertex

3. Select the shortest edgeconnected to any vertexalready connected

4. Repeat step 3 until allvertices have beenconnected


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A cable company want to connect ive villages to their networkwhich currently extends to the market town o Avon ord. !hat is theminimum length o cable needed"

Avon ord #ingley

$rinleigh %ornwell

&onster

'dan

2

7

45

8 6 4

5

3

8

Example


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

A #

$ %

&

'

2

7

45

8 6 4

5

3

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AF

BC

D

E

2

7

45

8 6

4

5

3

8

(ist the edges inorder o si)e*

'& 2 A$ 3 A' 4

%& 4$% +'# +%# ,

A# -

$# %#

Kruskal’s Algorithm


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Select the shortestedge in the network

ED 2


AF

BC

D

E

2

7

45

8 6

4

5

3

8


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Select the next shortestedge which does notcreate a cycle

ED 2AB 3


AF

BC

D

E

2

7

45

8 6

4

5

3

8


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ED 2AB 3

CD 4 (or AE 4)


AF

BC

D

E

2

7

45

8 6

4

5

3

8


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ED 2AB 3

CD 4AE 4


AF

BC

D

E

2

7

45

8 6

4

5

3

8


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ED 2AB 3

CD 4AE 4BC 5 – orms a !"!leE# 5


AF

BC

D

E

2

7

4 5

8 6

4

5

3

8


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All vertices have beenconnected.

/he solution is

ED 2

AB 3CD 4AE 4E# 5

/otal weight o tree* 1


AF

BC

D

E

2

7

4 5

8 6

4

5

3

8


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AF

BC

D

E

2

7

4 5

8 6

4

5

3

8

Select any vertex

A

Select the shortestedge connected to

that vertex

A$ 3

Prim’s Algorithm


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AF

BC

D

E

2

7

4 5

8 6

4

5

3

8

Select the shortest

edge connected toany vertex alreadyconnected.

A' 4

Prim’s Algorithm


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Select the shortest


'& 2

Prim’s Algorithm

AF

BC

D

E

2

7

4 5

8 6

4

5

3

8


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Select the shortest


&% 4

Prim’s Algorithm

AF

BC

D

E

2

7

4 5

8 6

4

5

3

8


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Select the shortest


%$ + 0 orms a cycle

'# +

Prim’s Algorithm

AF

BC

D

E

2

7

4 5

8 6

4

5

3

8


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Prim’s Algorithm

AF

BC

D

E

2

7

4 5

8 6

4

5

3

8

All vertices have been

connected.

/he solution is

ED 2

AB 3CD 4AE 4E# 5

/otal weight o tree* 1


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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. n this case there ismore than one minimum connector for thenetwork.

!ome points to note

d1,l5 kruskal's and prim's algorithms.ppt

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