Graphs 1.7 - Properties of DFS Property 1 DFS-Visit(u)...

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Unformatted text preview: Properties of DFS Property 1 DFS-Visit(u) visits all the vertices and edges in the connected component of u A Property 2 The discovery edges labeled by DFS-Visit(u) form a spanning tree of the connected component of u CSE 2011 Prof. J. Elder B D E C - 31 - Last Updated: 4/1/10 10:16 AM DFS Algorithm Pattern DFS(G) Precondition: G is a graph Postcondition: all vertices in G have been visited for each vertex u V [G] color[u] = BLACK //initialize vertex for each vertex u V [G] if color[u] = BLACK //as yet unexplored total work = (V ) DFS-Visit(u) CSE 2011 Prof. J. Elder - 32 - Last Updated: 4/1/10 10:16 AM DFS Algorithm Pattern DFS-Visit (u) Precondition: vertex u is undiscovered Postcondition: all vertices reachable from u have been processed colour[u ] RED for each v Adj[u ] //explore edge (u,v ) if color[v ] = BLACK colour [u ] DFS-Visit(v ) GRAY total work = | Adj[v ] | = (E ) vV Thus running time = (V + E ) (assuming adjacency list structure) CSE 2011 Prof. J. Elder - 33 - Last Updated: 4/1/10 10:16 AM Variants of Depth-First Search In addition to, or instead labeling vertices with colours, they can be labeled with discovery and finishing times. ‘Time’ is an integer that is incremented whenever a vertex changes state from unexplored to discovered from discovered to finished These discovery and finishing times can then be used to solve other graph problems (e.g., computing strongly-connected components) CSE 2011 Prof. J. Elder - 34 - Last Updated: 4/1/10 10:16 AM DFS Algorithm with Discovery and Finish Times DFS(G) Precondition: G is a graph Postcondition: all vertices in G have been visited for each vertex u V [G] color[u] = BLACK //initialize vertex time 0 for each vertex u V [G] if color[u] = BLACK //as yet unexplored DFS-Visit(u) CSE 2011 Prof. J. Elder - 35 - Last Updated: 4/1/10 10:16 AM ...
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