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Unformatted text preview: CS 473: Algorithms Chandra Chekuri chekuri@cs.illinois.edu 3228 Siebel Center University of Illinois, UrbanaChampaign Fall 2009 Chekuri CS473ug Strong Connected Components (SCCs) A B C D E F G H Algorithmic Problem Find all SCCs of a given directed graph. Previous lecture: saw an O ( n ( n + m )) time algorithm. This lecture: O ( n + m ) time algorithm. Chekuri CS473ug Graph of SCCs A B C D E F G H Figure: Graph G B , E , F G H A , C , D Figure: Graph of SCCs G SCC Metagraph of SCCs Let S 1 , S 2 , . . . S k be the SCCs of G . The graph of SCCs is G SCC Vertices are S 1 , S 2 , . . . S k There is an edge ( S i , S j ) if there is some u S i and v S j such that ( u , v ) is an edge in G . Chekuri CS473ug Reversal and SCCs Proposition For any graph G, the graph of SCCs of G rev is the same as the reversal of G SCC . Proof. Exercise. Chekuri CS473ug SCCs and DAGs Proposition For any graph G, the graph G SCC has no directed cycle. Chekuri CS473ug SCCs and DAGs Proposition For any graph G, the graph G SCC has no directed cycle. Proof. If G SCC has a cycle S 1 , S 2 , . . . , S k then S 1 S 2 S k is an SCC in G . Formal details: exercise. Chekuri CS473ug Part I Directed Acyclic Graphs Chekuri CS473ug Directed Acyclic Graphs Definition A directed graph G is a directed acyclic graph (DAG) if there is no directed cycle in G . 1 2 3 4 Chekuri CS473ug Sources and Sinks source sink 1 2 3 4 Definition A vertex u is a source if it has no incoming edges. A vertex u is a sink if it has no outgoing edges. Chekuri CS473ug Simple DAG Properties Every DAG G has at least one source and at least one sink. Chekuri CS473ug Simple DAG Properties Every DAG G has at least one source and at least one sink. If G is a DAG if and only if G rev is a DAG. Chekuri CS473ug Simple DAG Properties Every DAG G has at least one source and at least one sink. If G is a DAG if and only if G rev is a DAG. G is a DAG if and only each node is in its own strong component. Chekuri CS473ug Simple DAG Properties Every DAG G has at least one source and at least one sink. If G is a DAG if and only if G rev is a DAG. G is a DAG if and only each node is in its own strong component. Chekuri CS473ug Simple DAG Properties Every DAG G has at least one source and at least one sink. If G is a DAG if and only if G rev is a DAG. G is a DAG if and only each node is in its own strong component. Formal proofs: exercise. Chekuri CS473ug Topological Ordering/Sorting 1 2 3 4 Figure: Graph G 1 2 3 4 Figure: Topological Ordering of G Definition A topological ordering/sorting of G = ( V , E ) is an ordering < on V such that if ( u , v ) E then u < v . Chekuri CS473ug DAGs and Topological Sort Lemma A directed graph G can be topologically ordered iff it is a DAG....
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 Fall '08
 Chekuri,C
 Algorithms

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