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Directed Acyclic Graph. DAG – directed graph with no directed cycles. Getting Dressed. Underwear. Socks. Watch. Pants. Shoes. Shirt. Belt. Tie. Jacket. 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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## Directed Acyclic Graph

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**Directed Acyclic Graph**• DAG – directed graph with no directed cycles**Getting Dressed**Underwear Socks Watch Pants Shoes Shirt Belt Tie Jacket**Topological Sort**• Linear ordering of the vertices of G, such that if (u,v)E, then u appears smewhere before v.**Getting Dressed**Underwear Socks Watch Pants Shoes Shirt Belt Tie Jacket Socks Underwear Pants Shoes Watch Shirt Belt Tie Jacket**Topological Sort**Topological-Sort (G) • call DFS(G)to compute finishing times f [v] for all v V • as each vertex is finished, insert it onto the front of a linked list • return the linked list of vertices Time:(|V|+|E|).**Getting Dressed**Underwear Socks Watch Pants Shoes Shirt 1 | Belt Undiscovered Tie Unfinished Active Finished Jacket**Getting Dressed**Underwear Socks Watch Pants Shoes Shirt 1 | Belt Undiscovered Tie Unfinished 2 | Active Finished Jacket**Getting Dressed**Underwear Socks Watch Pants Shoes Shirt 1 | Belt Undiscovered Tie Unfinished 2 | Active Finished Jacket 3 |**Getting Dressed**Underwear Socks Watch Pants Shoes Shirt 1 | Belt Undiscovered Tie Unfinished 2 | Active Finished Jacket 3 | 4 Jacket**Getting Dressed**Underwear Socks Watch Pants Shoes Shirt 1 | Belt Undiscovered Tie Unfinished 2 | 5 Active Finished Jacket 3 | 4 Tie Jacket**Getting Dressed**Underwear Socks Watch Pants Shoes Shirt 1 | Belt Undiscovered 6 | Tie Unfinished 2 | 5 Active Finished Jacket 3 | 4 Tie Jacket**Getting Dressed**Underwear Socks Watch Pants Shoes Shirt 1 | Belt Undiscovered 6 | 7 Tie Unfinished 2 | 5 Active Finished Jacket 3 | 4 Belt Tie Jacket**Getting Dressed**Underwear Socks Watch Pants Shoes Shirt 1 | 8 Belt Undiscovered 6 | 7 Tie Unfinished 2 | 5 Active Finished Jacket 3 | 4 Shirt Belt Tie Jacket**Getting Dressed**Underwear Socks Watch Pants Shoes 9 | Shirt 1 | 8 Belt Undiscovered 6 | 7 Tie Unfinished 2 | 5 Active Finished Jacket 3 | 4 Shirt Belt Tie Jacket**Getting Dressed**Underwear Socks Watch Pants Shoes 9 |10 Shirt 1 | 8 Belt Undiscovered 6 | 7 Tie Unfinished 2 | 5 Active Finished Jacket 3 | 4 Watch Shirt Belt Tie Jacket**Getting Dressed**Underwear Socks 11 | Watch Pants Shoes 9 |10 Shirt 1 | 8 Belt Undiscovered 6 | 7 Tie Unfinished 2 | 5 Active Finished Jacket 3 | 4 Watch Shirt Belt Tie Jacket**Getting Dressed**Underwear Socks 11 | Watch Pants Shoes 12 | 9 |10 Shirt 1 | 8 Belt Undiscovered 6 | 7 Tie Unfinished 2 | 5 Active Finished Jacket 3 | 4 Watch Shirt Belt Tie Jacket**Getting Dressed**Underwear Socks 11 | Watch Pants Shoes 12 | 13 | 9 |10 Shirt 1 | 8 Belt Undiscovered 6 | 7 Tie Unfinished 2 | 5 Active Finished Jacket 3 | 4 Watch Shirt Belt Tie Jacket**Getting Dressed**Underwear Socks 11 | Watch Pants Shoes 12 | 13 |14 9 |10 Shirt 1 | 8 Belt Undiscovered 6 | 7 Tie Unfinished 2 | 5 Active Finished Jacket 3 | 4 Shoes Watch Shirt Belt Tie Jacket**Getting Dressed**Underwear Socks 11 | Watch Pants Shoes 12 |15 13 |14 9 |10 Shirt 1 | 8 Belt Undiscovered 6 | 7 Tie Unfinished 2 | 5 Active Finished Jacket 3 | 4 Pants Shoes Watch Shirt Belt Tie Jacket**Getting Dressed**Underwear Socks 11 | 16 Watch Pants Shoes 12 |15 13 |14 9 |10 Shirt 1 | 8 Belt Undiscovered 6 | 7 Tie Unfinished 2 | 5 Active Finished Jacket 3 | 4 Underwear Pants Shoes Watch Shirt Belt Tie Jacket**Getting Dressed**Underwear Socks 11 | 16 17 | Watch Pants Shoes 12 |15 13 |14 9 |10 Shirt 1 | 8 Belt Undiscovered 6 | 7 Tie Unfinished 2 | 5 Active Finished Jacket 3 | 4 Underwear Pants Shoes Watch Shirt Belt Tie Jacket**Getting Dressed**Underwear Socks 11 | 16 17 | 18 Watch Pants Shoes 12 |15 13 |14 9 |10 Shirt 1 | 8 Belt Undiscovered 6 | 7 Tie Unfinished 2 | 5 Active Finished Jacket 3 | 4 Socks Underwear Pants Shoes Watch Shirt Belt Tie Jacket**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. Not Strongly Connected Strongly Connected**Strongly Connected Components**• 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.**Graph of Strongly Connected Components**• 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**• GSCC has a topological ordering**Kinds of Edges**sourcevertex d f 1 |12 8 |11 13|16 C C F 2 | 7 B 9 |10 C C 3 | 4 5 | 6 14|15 C C Tree edges Back edges Forward edges Cross edges

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