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Directed Acyclic Graph

• DAG – directed graph with no directed cycles

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Getting Dressed

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Topological Sort

• Linear ordering of the vertices of G, such that if (u,v)E, then u appears smewhere before v.

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Getting Dressed

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Topological Sort

Topological-Sort (G)1. call DFS(G) to compute finishing times f [v] for all v V2. as each vertex is finished, insert it onto the front of a linked list

3. return the linked list of vertices

Topological-Sort (G)1. call DFS(G) to compute finishing times f [v] for all v V2. as each vertex is finished, insert it onto the front of a linked list

3. return the linked list of vertices

Time: (|V|+|E|).

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1 |

Undiscovered

Active

Finished

Unfinished

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1 |

2 |

Undiscovered

Active

Finished

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Getting Dressed

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ShoesPants

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1 |

2 |

3 |

Undiscovered

Active

Finished

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Getting Dressed

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Jacket

1 |

2 |

3 | 4

Undiscovered

Active

Finished

Unfinished

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Getting Dressed

Underwear Socks

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Belt

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Tie Jacket

1 |

2 | 5

3 | 4

Undiscovered

Active

Finished

Unfinished

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Getting Dressed

Underwear Socks

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Belt

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Tie

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Tie Jacket

6 |

1 |

2 | 5

3 | 4

Undiscovered

Active

Finished

Unfinished

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Getting Dressed

Underwear Socks

ShoesPants

Belt

Shirt

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Tie

Jacket

Belt Tie Jacket

6 | 7

1 |

2 | 5

3 | 4

Undiscovered

Active

Finished

Unfinished

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Getting Dressed

Underwear Socks

ShoesPants

Belt

Shirt

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Tie

Jacket

Shirt Belt Tie Jacket

6 | 7

1 | 8

2 | 5

3 | 4

Undiscovered

Active

Finished

Unfinished

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Getting Dressed

Underwear Socks

ShoesPants

Belt

Shirt

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Tie

Jacket

Shirt Belt Tie Jacket

6 | 7

1 | 8

2 | 5

3 | 4

9 |

Undiscovered

Active

Finished

Unfinished

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Getting Dressed

Underwear Socks

ShoesPants

Belt

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Watch Shirt Belt Tie Jacket

6 | 7

1 | 8

2 | 5

3 | 4

9 |10

Undiscovered

Active

Finished

Unfinished

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Getting Dressed

Underwear Socks

ShoesPants

Belt

Shirt

Watch

Tie

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Watch Shirt Belt Tie Jacket

11 |

6 | 7

1 | 8

2 | 5

3 | 4

9 |10

Undiscovered

Active

Finished

Unfinished

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Getting Dressed

Underwear Socks

ShoesPants

Belt

Shirt

Watch

Tie

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Watch Shirt Belt Tie Jacket

11 |

12 |

6 | 7

1 | 8

2 | 5

3 | 4

9 |10

Undiscovered

Active

Finished

Unfinished

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Getting Dressed

Underwear Socks

ShoesPants

Belt

Shirt

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Tie

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Watch Shirt Belt Tie Jacket

11 |

12 |

6 | 7

13 |

1 | 8

2 | 5

3 | 4

9 |10

Undiscovered

Active

Finished

Unfinished

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Getting Dressed

Underwear Socks

ShoesPants

Belt

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Tie

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Shoes Watch Shirt Belt Tie Jacket

11 |

12 |

6 | 7

13 |14

1 | 8

2 | 5

3 | 4

9 |10

Undiscovered

Active

Finished

Unfinished

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Getting Dressed

Underwear Socks

ShoesPants

Belt

Shirt

Watch

Tie

Jacket

Pants Shoes Watch Shirt Belt Tie Jacket

11 |

12 |15

6 | 7

13 |14

1 | 8

2 | 5

3 | 4

9 |10

Undiscovered

Active

Finished

Unfinished

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Getting Dressed

Underwear Socks

ShoesPants

Belt

Shirt

Watch

Tie

Jacket

Underwear Pants Shoes Watch Shirt Belt Tie Jacket

11 | 16

12 |15

6 | 7

13 |14

1 | 8

2 | 5

3 | 4

9 |10

Undiscovered

Active

Finished

Unfinished

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Getting Dressed

Underwear Socks

ShoesPants

Belt

Shirt

Watch

Tie

Jacket

Underwear Pants Shoes Watch Shirt Belt Tie Jacket

11 | 16

12 |15

6 | 7

13 |14

17 |

1 | 8

2 | 5

3 | 4

9 |10

Undiscovered

Active

Finished

Unfinished

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Getting Dressed

Underwear Socks

ShoesPants

Belt

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Tie

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Socks Underwear Pants Shoes Watch Shirt Belt Tie Jacket

11 | 16

12 |15

6 | 7

13 |14

17 | 18

1 | 8

2 | 5

3 | 4

9 |10

Undiscovered

Active

Finished

Unfinished

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Strongly-Connected

• Graph G is strongly connected if, for every u and v in V, there is some path from u to v and some path from v to u.

Strongly Connected

Not Strongly Connected

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• A strongly connected component (SCC) of G is a maximal set of vertices C V such that for all u, v C, both u v and v u exist.

Strongly Connected Components

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• GSCC=(VSCC, ESCC): one vertex for each component– (u, v) ESCC if there exists at least one directed

edge from the corresponding components

Graph of Strongly Connected Components

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• GSCC has a topological ordering

Graph of Strongly Connected Components

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1 |12 8 |11 13|16

14|155 | 63 | 4

2 | 7 9 |10

sourcevertex

d f

Tree edges Back edges Forward edges Cross edges

Kinds of Edges

B F

C

C

C

C

C

C