Download - Graph Traversal
Graph TraversalGraph Traversal
Chapter 9
ProblemProblem
Given a directed or undirected graph G=(V,E), find a way to traverse all of its vertices.
SolutionSolution
Depth First Search (DFS) Breadth First Search (BFS)
DFSDFS
DFSDFS
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And so on ...
DFSDFS
We need also to record
– the birth time of any node: the time of the first visit to the node.
– the death time: the time after all of its children are visited and ..also dead
Algorithm: DFSAlgorithm: DFSInput: graph G=(V,E) directed or undirected
Output: preordering of the vertices in DFS spanning tree (or forest)
1. predfn 0; postdfn 0; % global variables
2. For each vertex vV3. mark v unvisited 4. End for5. For each vV % for connected components
6. if v is unvisited then dfs(v)7. End for
Procedure dfs(v)Procedure dfs(v)
1. Mark v visited
2. predfn predfn + 1 % birth time
3. For each edge (v,w) E
4. if w is marked unvisited then dfs(w)
5. End for
6. postdfn postdfn + 1 % death time
DefinitionsDefinitions
In a directed graph, (u,v) is– a tree edge, if it is in the DFS tree, i.e., v is
marked unvisited – back edge, if v is marked visited &
ancestor of u– Forward edge, if v marked visited &
descendant of u– Cross edge, if it is otherwise.
Tree edge
back edge
Forward edge
Cross edge
DefinitionsDefinitions
In an undirected graph, (u,v) is– a tree edge if it is in the DFS tree – back edges, otherwise.
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Analysis of DFSAnalysis of DFS
Time = (m + n)
Applications: Testing acyclicity (is G a tree?) Strongly connected components
BFSBFS
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And so on ..
Algorithm: BFSAlgorithm: BFSInput: graph G=(V,E) directed or undirected
Output: ordering of the vertices in BFS spanning tree (or forest)
1. bfn 0; % global variable
2. For each vertex vV3. mark v unvisited 4. End for5. For each vV % for connected components
6. if v is unvisited then bfs(v)7. End for
Procedure bfs(v)Procedure bfs(v)
1. Q { v }
2. Mark v visited3. While Q { }4. v pop(Q)5. bfn bfn + 1 % birth time= time to give birth to all of its children
6. For each edge (v,w) E7. if w is marked unvisited then 8. push(w, Q)9. mark w visited 10. end if11. End for12. End while
NoteNote
Order in which nodes are popped
= order in which they are pushed
= order in which are born
= order in which they give birth
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AnalysisAnalysis
Time = (m + n)
Applications: Computing the distances
