TODAY:  Depth First Traversal of a graph
 	Breadth First Traversal of a graph


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Depth First Traversal of a graph:


  
                    5              ADJACENCY LIST:
	            |				  0: 1
	            |				  1: 0,2,3,7
                    4---2---3			  2: 1,3,4,7
                   / \  |\ /|			  3: 1,2,7
		  /   \ |/ \|			  4: 2,5,6,7
		 6      7---1			  5: 4
			    |			  6: 4
			    |			  7: 1,2,3,4
                            0

Above graph:  start vertex v=6 or DFS(6)
nodes visited in order:  6,4,2,1,0,3,7,5



Depth First Search(DFS):
	-decides connectivity
	-provides DFS numbering
	-computes a spanning tree

  Begin DFS by first visiting the start vertex v.  Next, an unvisited 
  vertex w adjacent to v is selected and a DFS from w is initiated.  
  When a vertex u is reached such that all its adjacent verticies have
  been visited, back up to the last vertex visited that has an 
  unvisited vertex w adjacent to it and initiate a DFS from w.  The
  search terminates when no unvisited vertex can be reached from any
  of the visited verticies.




IMPLEMENTATION:

for each w in V:  VISITED[w] = 0; //initializes the search
DFS(v)                            // v is the start vertex
	VISIT(v);  VISITED[v] = 1;  
	for every w in Adj[v]
		if(! VISITED[w])
		    DFS(w);

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Breadth First Search Traversal of a graph



        	5
		|
		|
		4---2---3
	       / \  |\ /|
	      /   \ |/ \|
	     6      7   1
			|
			|
			0

  order of visitation, BFS(6):  6,4,2,5,7,1,3,0

  an array is used to keep track of which nodes have been visited
  queue representation:  enqueue a node's adjacent nodes
			     (that aren't already in the queue)
			 dequeue a node that has been visited

  example for BFS(6):

  front of queue->	6  (enqueue 6)
		->	4  (dequeue 6, enqueue 4)
		->	2,5,7 (dequeue 4, enqueue 2,5,7)
		->	5,7,1,3 (dequeue 2, enqueue 1,3)
		->	7,1,3 (dequeue 5)
		->	1,3 (dequeue 7)
		->	3,0 (dequeue 1, enqueue 0)
		->	0 (dequeue 3)
		->	  (dequeue 0)--DONE


Breadth First Search(BFS):
	Begin by visiting the start vertex v.  Next, all unvisited verticies
	adjacent to v are visited.  Unvisited verticies adjacent to these
	newly visited verticies are then visited, and so on.




IMPLEMENTATION:

BFS(v)
  for each w in V, visited[w]=0;
  Queue q;
	q.insert(v);  visited[v]=1;
	while(! q.empty())
	  v<-q.dequeue();
	  VISIT(v);
	  for each w in Adj[v]
	    if(! visited[w])
	       {
		q.insert(w);
		visited[w]=1;
	       }

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Time Bounds

DFS & BFS:  O(n+m) time   (n=verticies, m=edges)
	    same as O(m)--graph connected

	    O(m) space