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Directed Graphs

Directed Graphs. Types of Edges. Forward, back, cross and tree edges. DAGs. Odd to mention them now, but they will come up later and now’s as good a place as any. They have unique search characteristics. acyclic == linearizability == no back edges. Strongly Connected Components.

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Directed Graphs

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  1. Directed Graphs

  2. Types of Edges Forward, back, cross and tree edges.

  3. DAGs • Odd to mention them now, but they will come up later and now’s as good a place as any. • They have unique search characteristics. acyclic == linearizability == no back edges.

  4. Strongly Connected Components Set of vertices such that each vertex is reachable from every other vertex—including itself. Directed analogue of a biconnected component.

  5. Finding SCCs • Book assumes you know G^R • We’ll assume you don’t.

  6. Tarjan’s Algorithm Input: Graph G = (V, E) index = 0 // DFS node number counter S = empty // An empty stack of nodes forall v in V do if (v.index is undefined) // Start a DFS at each node tarjan(v) // we haven't visited yet procedure tarjan(v) v.index = index // Set the depth index for v v.lowlink = index index = index + 1 S.push(v) // Push v on the stack forall (v, v') in E do // Consider successors of v if (v'.index is undefined) // Was successor v' visited? tarjan(v') // Recurse v.lowlink = min(v.lowlink, v'.lowlink) else if (v' is in S) // Was successor v' in stack S? v.lowlink = min(v.lowlink, v'.index) if (v.lowlink == v.index) // Is v the root of an SCC? print "SCC:" repeat v' = S.pop print v' until (v' == v)

  7. Tarjan’s Algorithm • Does it work? • How fast is it? • Can we do better? • Is it parallelizable?

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