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

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

BFSBFS

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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9

AnalysisAnalysis

Time = (m + n)

Applications: Computing the distances

Graph Traversal

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