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Traversal A systematic way to visit vertices in a
graph Two general approaches:
breath first searching (BFS) start from one vertex, first visit all the adjacency
vertices depth first searching (DFS)
eager method
These slides assume the adjacency list representation
“graph” ADT in C: Interface// in file “graph.h”#ifndef GRAPH_H#define GRAPH_H#define T Graph_ttypedef struct T *T;typedef void (*tyVisit)(poly);
T Graph_new ();void Graph_insertVertex (T g, poly data);void Graph_insertEdge (T g, poly from, poly to);void Graph_dfs (T g, poly start, tyVisit visit);void Graph_bfs (T g, poly start, tyVisit visit);// we’d see more later…#endif
Sample Graph
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4
2
65
3
Sample Graph BFS
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3bfs (g, 1, Int_print);
Sample Graph BFS
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3bfs (g, 1, Int_print);
print 1;
Sample Graph BFS
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3bfs (g, 1, Int_print);
print 1;
// a choice
print 2;
Sample Graph BFS
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3bfs (g, 1, Int_print);
print 1;
// a choice
print 2;
print 4;
Sample Graph BFS
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3bfs (g, 1, Int_print);
print 1;
// a choice
print 2;
print 4;
print 5;
Sample Graph BFS
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4
2
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3bfs (g, 1, Int_print);
print 1;
// a choice
print 2;
print 4;
print 5;
// a choice
print 3;
Sample Graph BFS
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4
2
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3bfs (g, 1, Int_print);
print 1;
// a choice
print 2;
print 4;
print 5;
// a choice
print 3;
print 6;
BFS Algorithmvoid bfsOneVertex (Tv start, tyVisit visit){ Queue_t q = Queue_new ();
Queue_en (q, start); while (q not empty) { Tv current = Queue_de (q); visit (current); for (each adjacent vertex u of “current”) if (not visited u) Queue_en (q, u); }}
BFS Algorithmvoid bfs (T g, poly start, tyVisit visit){ Tv startV = searchVertex (g, start); bfsOneVertex (startV, visit);
for (each vertex u in graph g) if (not visited u) bfsOneVertex (q, u);}
Sample Graph BFS
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3bfs (g, 1, Int_output);
Queue: 1
Sample Graph BFS
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3bfs (g, 1, Int_output);
print 1;
Queue: 2, 4
Queue: 1
Sample Graph BFS
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3bfs (g, 1, Int_print);
print 1;
// a choice
print 2;
Queue: 2, 4
Queue: 1
Queue: 4, 5
Sample Graph BFS
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2
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3bfs (g, 1, Int_print);
print 1;
// a choice
print 2;
print 4;
Queue: 5
Queue: 2, 4
Queue: 1
Queue: 4, 5
Sample Graph BFS
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3bfs (g, 1, Int_print);
print 1;
// a choice
print 2;
print 4;
print 5;
Queue:
Queue: 5
Queue: 2, 4
Queue: 1
Queue: 4, 5
Queue: 3
Sample Graph BFS
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3bfs (g, 1, Int_print);
print 1;
// a choice
print 2;
print 4;
print 5;
// a choice
print 3;
Queue:
Queue: 5
Queue: 2, 4
Queue: 1
Queue: 4, 5
Queue: 3
Queue: 6
Sample Graph BFS
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3bfs (g, 1, Int_print);
print 1;
// a choice
print 2;
print 4;
print 5;
// a choice
print 3;
print 6;Queue:
Queue: 5
Queue: 2, 4
Queue: 1
Queue: 4, 5
Queue: 3
Queue: 6
Queue:
Moral
BFS is a generalization like the level-order traversal on trees Also maintain internally a queue to
control the visit order Obtain a BFS forest when finished
Sample Graph DFS
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3dfs (g, 1, Int_print);
Sample Graph DFS
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3dfs (g, 1, Int_print);
print 1;
Sample Graph DFS
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2
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3dfs (g, 1, Int_print);
print 1;
// a choice
print 2;
Sample Graph DFS
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4
2
65
3dfs (g, 1, Int_print);
print 1;
// a choice
print 2;
print 5;
Sample Graph DFS
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4
2
65
3dfs (g, 1, Int_print);
print 1;
// a choice
print 2;
print 5;
print 4;
Sample Graph DFS
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4
2
65
3dfs (g, 1, Int_print);
print 1;
// a choice
print 2;
print 5;
print 4;
// a choice
print 3;
Sample Graph DFS
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4
2
65
3dfs (g, 1, Int_print);
print 1;
// a choice
print 2;
print 5;
print 4;
// a choice
print 3;
print 6;
DFS Algorithmvoid dfsOneVertex (Tv start, tyVisit visit){ visit (start);
for (each adjacent vertex u of “start”) if (not visited u) dfsOneVertex (u, visit);}
DFS Algorithmvoid dfs (T g, poly start, tyVisit visit){ Tv startV = searchVertex (g, start); dfsOneVertex (startV, visit);
for (each vertex u in graph g) if (not visited u) dfsOneVertex (u, visit);}
Sample Graph DFS
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2
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3dfs (g, 1, Int_print);
print 1;
dfs(1)
Sample Graph DFS
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3dfs (g, 1, Int_print);
print 1;
// a choice
print 2;
dfs(1) => dfs(2)
Sample Graph DFS
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2
65
3dfs (g, 1, Int_print);
print 1;
// a choice
print 2;
print 5;
dfs(1) => dfs(2) => dfs(5)
Sample Graph DFS
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4
2
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3dfs (g, 1, Int_print);
print 1;
// a choice
print 2;
print 5;
print 4;dfs(1) => dfs(2) => dfs(5) => dfs(4)
Sample Graph DFS
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4
2
65
3dfs (g, 1, Int_print);
print 1;
// a choice
print 2;
print 5;
print 4;dfs(1) => dfs(2) => dfs(5) => dfs(4) => dfs(2)???
Sample Graph DFS
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4
2
65
3dfs (g, 1, Int_print);
print 1;
// a choice
print 2;
print 5;
print 4;dfs(1) => dfs(2) => dfs(5)
Sample Graph DFS
1
4
2
65
3dfs (g, 1, Int_print);
print 1;
// a choice
print 2;
print 5;
print 4;dfs(1) => dfs(2)
Sample Graph DFS
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2
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3dfs (g, 1, Int_print);
print 1;
// a choice
print 2;
print 5;
print 4;dfs(1)
Sample Graph DFS
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4
2
65
3dfs (g, 1, Int_print);
print 1;
// a choice
print 2;
print 5;
print 4;dfs(1) =>dfs(4)???
Sample Graph DFS
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4
2
65
3dfs (g, 1, Int_print);
print 1;
// a choice
print 2;
print 5;
print 4;dfs(1)
Sample Graph DFS
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2
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3dfs (g, 1, Int_print);
print 1;
// a choice
print 2;
print 5;
print 4;empty!
Sample Graph DFS
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2
65
3dfs (g, 1, Int_print);
print 1;
// a choice
print 2;
print 5;
print 4;
// a choice
print 3;
dfs(3)
Sample Graph DFS
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2
65
3dfs (g, 1, Int_print);
print 1;
// a choice
print 2;
print 5;
print 4;
// a choice
print 3;
print 6;
dfs(3) =>dfs(6)
Sample Graph DFS
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4
2
65
3dfs (g, 1, Int_print);
print 1;
// a choice
print 2;
print 5;
print 4;
// a choice
print 3;
print 6;
dfs(3) =>dfs(6) =>dfs(6)???
Sample Graph DFS
1
4
2
65
3dfs (g, 1, Int_print);
print 1;
// a choice
print 2;
print 5;
print 4;
// a choice
print 3;
print 6;
dfs(3) =>dfs(6)
Sample Graph DFS
1
4
2
65
3dfs (g, 1, Int_print);
print 1;
// a choice
print 2;
print 5;
print 4;
// a choice
print 3;
print 6;
dfs(3)
Sample Graph DFS
1
4
2
65
3dfs (g, 1, Int_print);
print 1;
// a choice
print 2;
print 5;
print 4;
// a choice
print 3;
print 6;
dfs(3) =>dfs(5)???
Sample Graph DFS
1
4
2
65
3dfs (g, 1, Int_print);
print 1;
// a choice
print 2;
print 5;
print 4;
// a choice
print 3;
print 6;
dfs(3)
Sample Graph DFS
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4
2
65
3dfs (g, 1, Int_print);
print 1;
// a choice
print 2;
print 5;
print 4;
// a choice
print 3;
print 6;
empty!
Moral
DFS is a generalization of the pre-order traversal on trees
Maintain internally a stack to control the visit order for recursion function, machine maintain
an implicit stack Obtain a DFS forest when finished
Edge Classification
Once we obtain the DFS (or BFS) spanning trees (forests), the graph edges could be classified according to the trees: tree edges: edges in the trees forward edges: ancestors to descants back edges: descants to ancestors cross edges: others
Edge Classification Example
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3• tree edges:
1->2, 2->5, 5->4, 3->6
• forward edges:
1->4
• back edges:
4->2, 6->6
• cross edges:
3->5
