cse101_10_31_11

cse101_10_31_11 - DFS in directed graphs 1 From last time a...

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DFS in directed graphs 1. From last time a. A back edge is an edge that leads to an ancestor in the DFS tree. b. Fact (u,w) is a back edge post[u] < post[w] c. Fact: A directed graph is acyclic there are no back edges 2. Directed acyclic graphs (DAGs) a. Ex. Scheduling problem Wake up (1) -> dress (3) -> eat breakfast (4) -> shower (2) -> get kids ready (5) -> get in car (6) Wake up points to all nodes Shower points to dress, dress points to get kids ready In what order to perform these tasks? b. Given a DAG, we want a topological ordering: number the nodes so that every edge goes from a smaller to a larger number. c. It is always possible to topologically order a DAG; order nodes by decreasing post number (linear time) Eat breakfast (1, 4) get in car (2, 3) Shower (5, 10), dress (6, 9) get kids ready (7,8) Wake up (11, 12) Start at highest post number Claim: In any DAG, every edge goes from a bigger post # to a smaller post #. Proof: the only type of edge (u, w) which has POST[u] > POST[u] is a back edge… and a DAG has no back edges.
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