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1 Directed Acyclic Graph DAG – directed graph with no directed cycles

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2 Getting Dressed UnderwearSocks ShoesPants Belt Shirt Watch Tie Jacket

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3 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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4 Getting Dressed UnderwearSocks ShoesPants Belt Shirt Watch Tie Jacket SocksUnderwearPantsShoesWatchShirtBeltTieJacket

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5 Topological Sort Topological-Sort ( G ) 1.call DFS ( G ) to compute finishing times f [ v ] for all v V 2.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 V 2.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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6 Getting Dressed UnderwearSocks ShoesPants Belt Shirt Watch Tie Jacket 1 | Undiscovered Active Finished Unfinished

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7 Getting Dressed UnderwearSocks ShoesPants Belt Shirt Watch Tie Jacket 1 | 2 | Undiscovered Active Finished Unfinished

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8 Getting Dressed UnderwearSocks ShoesPants Belt Shirt Watch Tie Jacket 1 | 2 | 3 | Undiscovered Active Finished Unfinished

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9 Getting Dressed UnderwearSocks ShoesPants Belt Shirt Watch Tie Jacket 1 | 2 | 3 | 4 Undiscovered Active Finished Unfinished

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10 Getting Dressed UnderwearSocks ShoesPants Belt Shirt Watch Tie Jacket TieJacket 1 | 2 | 5 3 | 4 Undiscovered Active Finished Unfinished

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11 Getting Dressed UnderwearSocks ShoesPants Belt Shirt Watch Tie Jacket TieJacket 6 | 1 | 2 | 5 3 | 4 Undiscovered Active Finished Unfinished

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12 Getting Dressed UnderwearSocks ShoesPants Belt Shirt Watch Tie Jacket BeltTieJacket 6 | 7 1 | 2 | 5 3 | 4 Undiscovered Active Finished Unfinished

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13 Getting Dressed UnderwearSocks ShoesPants Belt Shirt Watch Tie Jacket ShirtBeltTieJacket 6 | 7 1 | 8 2 | 5 3 | 4 Undiscovered Active Finished Unfinished

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14 Getting Dressed UnderwearSocks ShoesPants Belt Shirt Watch Tie Jacket ShirtBeltTieJacket 6 | 7 1 | 8 2 | 5 3 | 4 9 | Undiscovered Active Finished Unfinished

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15 Getting Dressed UnderwearSocks ShoesPants Belt Shirt Watch Tie Jacket WatchShirtBeltTieJacket 6 | 7 1 | 8 2 | 5 3 | 4 9 |10 Undiscovered Active Finished Unfinished

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16 Getting Dressed UnderwearSocks ShoesPants Belt Shirt Watch Tie Jacket WatchShirtBeltTieJacket 11 | 6 | 7 1 | 8 2 | 5 3 | 4 9 |10 Undiscovered Active Finished Unfinished

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17 Getting Dressed UnderwearSocks ShoesPants Belt Shirt Watch Tie Jacket WatchShirtBeltTieJacket 11 | 12 | 6 | 7 1 | 8 2 | 5 3 | 4 9 |10 Undiscovered Active Finished Unfinished

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18 Getting Dressed UnderwearSocks ShoesPants Belt Shirt Watch Tie Jacket WatchShirtBeltTieJacket 11 | 12 | 6 | 7 13 | 1 | 8 2 | 5 3 | 4 9 |10 Undiscovered Active Finished Unfinished

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19 Getting Dressed UnderwearSocks ShoesPants Belt Shirt Watch Tie Jacket ShoesWatchShirtBeltTieJacket 11 | 12 | 6 | 7 13 |14 1 | 8 2 | 5 3 | 4 9 |10 Undiscovered Active Finished Unfinished

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20 Getting Dressed UnderwearSocks ShoesPants Belt Shirt Watch Tie Jacket PantsShoesWatchShirtBeltTieJacket 11 | 12 |15 6 | 7 13 |14 1 | 8 2 | 5 3 | 4 9 |10 Undiscovered Active Finished Unfinished

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21 Getting Dressed UnderwearSocks ShoesPants Belt Shirt Watch Tie Jacket UnderwearPantsShoesWatchShirtBeltTieJacket 11 | 16 12 |15 6 | 7 13 |14 1 | 8 2 | 5 3 | 4 9 |10 Undiscovered Active Finished Unfinished

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22 Getting Dressed UnderwearSocks ShoesPants Belt Shirt Watch Tie Jacket UnderwearPantsShoesWatchShirtBeltTieJacket 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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23 Getting Dressed UnderwearSocks ShoesPants Belt Shirt Watch Tie Jacket SocksUnderwearPantsShoesWatchShirtBeltTieJacket 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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24 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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25 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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26 G SCC =(V SCC, E SCC ): one vertex for each component –(u, v) E SCC if there exists at least one directed edge from the corresponding components Graph of Strongly Connected Components

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27 G SCC has a topological ordering Graph of Strongly Connected Components

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28 1 |128 |1113|16 14|155 | 63 | 4 2 | 79 |10 source vertex d f Tree edgesBack edgesForward edgesCross edges Kinds of Edges B F C C C C C C

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