Graph Traversals
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Graph Traversals • Depth-First Traversals. – Algorithms. – Example. – Implementation. • Breadth-First Traversal. – The Algorithm. – Example. – Implementation. • Review Questions.
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Graph Traversals. Depth-First Traversals. Algorithms. Example. Implementation. Breadth-First Traversal. The Algorithm. Example. Implementation. Review Questions. Depth-First Traversal Algorithm. - PowerPoint PPT Presentation
Transcript of Graph Traversals
Graph Traversals
• Depth-First Traversals.– Algorithms.– Example.– Implementation.
• Breadth-First Traversal.– The Algorithm.– Example.– Implementation.
• Review Questions.
Depth-First Traversal Algorithm• In this method, After visiting a vertex v, which is adjacent to w1, w2, w3, ...;
Next we visit one of v's adjacent vertices, w1 say. Next, we visit all vertices adjacent to w1 before coming back to w2, etc.
• Must keep track of vertices already visited to avoid cycles.
Algorithm:-#define TRUE 1
#define FALSE 0 short int visited[20];
void dfs(int v,GNODEPTR *graph){GNODEPTR w; visited[v]=TRUE; printf(“%d\t”,v);for(w=graph[v];w;w=w->next)if(!visited[w->vertex]) dfs(w->vertex,graph);}
• Note: Adjacent vertices can be pushed in any order; but to obtain a unique traversal, we will push them in reverse alphabetical order.
Example
• Demonstrate depth-first traversal using an explicit stack.
StackA B C F E G D H I
Breadth-First Traversal Algorithm• In this method, After visiting a vertex v, we must visit all its adjacent
vertices w1, w2, w3, ..., before going down next level to visit vertices adjacent to w1 etc.
• The method can be implemented using a queue.
• A boolean array is used to ensure that a vertex is enqueued only once.
1 enqueue the starting vertex 2 while(queue is not empty){3 dequeue a vertex v from the queue;4 visit v. 5 enqueue vertices adjacent to v that were never enqueued;6 }
• Note: Adjacent vertices can be enqueued in any order; but to obtain a unique
traversal, we will enqueue them in alphabetical order.
Bfs algorithm• void bfs(int v,GNODEPTR *graph){• GNODEPTR q[20],w; int front=0,rear=-1;• printf(“%d\t”,v); visited[v]=TRUE;• addq(&front,&rear,q,graph[v]);• while(front<=rear) {• w=deleteq(&front,&rear,q);• for(; w; w=w->next)• If(!visited[w->vertex]){printf(“%d\t”,w->vertex);• addq(&front,&rear,q,graph[w->vertex]); • visited[w->vertex]=TRUE;}}}
Example
• Demonstrating breadth-first traversal using a queue.
Order ofTraversal Queue rearA B D E C G F H I
Queue front
Example
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Adjacency List
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Visited Table (T/F)
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Q = { }
Initialize visitedtable (all False)
Initialize Q to be empty
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Visited Table (T/F)
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Q = { 2 }
Flag that 2 has been visited.
Place source 2 on the queue.
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Visited Table (T/F)
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Q = {2} → { 8, 1, 4 }
Mark neighborsas visited.
Dequeue 2. Place all unvisited neighbors of 2 on the queue
Neighbors
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Visited Table (T/F)
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Q = { 8, 1, 4 } → { 1, 4, 0, 9 }
Mark new visitedNeighbors.
Dequeue 8. -- Place all unvisited neighbors of 8 on the queue. -- Notice that 2 is not placed on the queue again, it has been visited!
Neighbors
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Visited Table (T/F)
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Q = { 1, 4, 0, 9 } → { 4, 0, 9, 3, 7 }
Mark new visitedNeighbors.
Dequeue 1. -- Place all unvisited neighbors of 1 on the queue. -- Only nodes 3 and 7 haven’t been visited yet.
Neighbors
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Visited Table (T/F)
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Q = { 4, 0, 9, 3, 7 } → { 0, 9, 3, 7 }
Dequeue 4. -- 4 has no unvisited neighbors!
Neighbors
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Visited Table (T/F)
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Q = { 0, 9, 3, 7 } → { 9, 3, 7 } Dequeue 0. -- 0 has no unvisited neighbors!
Neighbors
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Visited Table (T/F)
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Q = { 9, 3, 7 } → { 3, 7 } Dequeue 9. -- 9 has no unvisited neighbors!
Neighbors
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Visited Table (T/F)
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Q = { 3, 7 } → { 7, 5 } Dequeue 3. -- place neighbor 5 on the queue.
Neighbors
Mark new visitedVertex 5.
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Q = { 7, 5 } → { 5, 6 } Dequeue 7. -- place neighbor 6 on the queue.
Neighbors
Mark new visitedVertex 6.
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Visited Table (T/F)
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Q = { 5, 6} → { 6 } Dequeue 5. -- no unvisited neighbors of 5.
Neighbors
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Visited Table (T/F)
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Q = { 6 } → { } Dequeue 6. -- no unvisited neighbors of 6.
Neighbors
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Visited Table (T/F)
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Q = { }
STOP!!! Q is empty!!!
What did we discover?
Look at “visited” tables.
There exists a path from sourcevertex 2 to all vertices in the graph.
