Lecture16-sccomponents

# Sccs you need to understand the following pre and

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SCCs, you need to understand the following: § Pre and post numbering in DFS § Meta-graph created by representing each SCC with a single meta-node. § The meta-graph is a DAG and could therefore be linearized using DFS.

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Finding SCCs § Goal: Given a directed graph, find all of the SCCs § Big Idea: § A: Find a (vertex in a) sink SCC § B: Work back to find the other SCCs § How do you find a sink SCC? § Finding a sink (node) in a DAG is easy § But what about when there are cycles? § Problem A seems hard, so solve problem A’ first: § Finding a node in a source SCC of a directed graph could be easier. § A’: find a source SCC on the “reverse graph”
Reverse Graph G To find a sink, identify a source SCC of the reverse graph GR! GR Metagraph of GR

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Insights § Property : The node that receives the highest post number in a depth-first search must lie in a source strongly connected component. § Property : If C and C’ are strongly connected components, and there is an edge from a node in C to a node in C’, then the highest post number in C is bigger than the highest post number in C’.
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• Spring '08
• Jones,M
• Graph Theory, Depth-first search, Directed acyclic graph, connected component, Strongly connected component

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