spanning trees. prims mst algorithm algorithm ( this is also greedy) select an arbitrary vertex to...

Spanning Trees

Prim’s MST Algorithm

Algorithm ( this is also greedy)

Select an arbitrary vertex to start the tree,

while there are fringe vertices:

1)select an edge of minimum weight

between a tree vertex and a fringe

vertex.

2)add the selected edge and the

fringe vertex to the tree.

end.

Prim’s AlgorithmMinimal Spanning Tree

1 6

2 4

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52

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Prim’s AlgorithmMinimal Spanning Tree

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Example: start with 7Example: start with 7

Prim’s AlgorithmMinimal Spanning Tree

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3 5

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Prim’s AlgorithmMinimal Spanning Tree

1 6

2 4

3 5

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Prim’s AlgorithmMinimal Spanning Tree

1 6

2 4

3 5

3

2

1

4

4

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5

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7 8

1

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52

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Prim’s AlgorithmMinimal Spanning Tree

1 6

2 4

3 5

3

2

1

4

4

2

5

3

4

7 8

1

2

52

6

Prim’s AlgorithmMinimal Spanning Tree

1 6

2 4

3 5

3

2

1

4

4

2

5

3

4

7 8

1

2

52

6

Prim’s AlgorithmMinimal Spanning Tree

1 6

2 4

3 5

3

2

1

4

4

2

5

3

4

7 8

1

2

56

Prim’s AlgorithmMinimal Spanning Tree

1 6

2 4

3 5

3

2

1

4

4

2

5

3

4

7 8

1

2

56

Prim’s AlgorithmMinimal Spanning Tree

1 6

2 4

3 5

2

1

4

4

2

5

3

4

7 8

1

2

6

Prim’s AlgorithmMinimal Spanning Tree

1 6

2 4

3 5

2

1

4

4

2

5

3

4

7 8

1

2

6

Prim’s AlgorithmMinimal Spanning Tree

1 6

2 4

3 5

2

1 4

2

5

3

4

7 8

1

2

6

Prim’s AlgorithmMinimal Spanning Tree

1 6

2 4

3 5

2

1 4

2

5

3

4

7 8

1

2

6

Prim’s AlgorithmMinimal Spanning Tree

1 6

2 4

3 5

2

1

2

5

3

4

7 8

1

2

Prim’s AlgorithmMinimal Spanning Tree

1 6

2 4

3 5

2

1

2

5

3

4

7 8

1

2

Prim’s AlgorithmMinimal Spanning Tree

1 6

2 4

3 5

2

1

2

3

7 8

1

2

MST weight = 15MST weight = 15

4

Topological Sorting Algorithm

while (the graph has a node with no successor) do

remove one of those nodes from the graph

and add it to the end of a list

if (the graph is empty) then

the list contains the reverse of some topological order

else

the graph contains a cycle

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