History of Graphs

ClassS04CS141 | recent changes | Preferences

Difference (from prior major revision) (no other diffs)

Changed: 4c4

Added: 26a27

Changed: 60,124c61,62

Depth-first search (DFS)

Depth-first search is a "backtracking" method of exploring a graph.
It searches from vertex to vertex, backtracking when it encounters
vertices that have already been discovered. Here is pseudo-code:


 DFS(graph G, vertex v)
    Array<int> visited(0);
    DFS1(G, v, visited);
 end
 
 DFS1(graph G, vertex v, Array<int>& visited)
    if (visited[v] == 1)  return
    visited[v] = 1
    for each neighbor w of v do
       DFS1(G, w, visited)
 end

We demonstrated DFS on an example.
DFS on classifies the edge into two kinds:
* DFS tree edges -- these are the edges (v,w) such that DFS1(G, w, visited) is \
called directly from within DFS1(G, v, visited) when w is not yet visited.
* back edges -- these are the other edges.

The tree edges form a tree (essentially a recursion tree for the DFS).
For any back edge (u,v), either u is a descendant of v in the tree or
vice versa.

In a graph with N vertices and M edges, the running time for DFS
is O(N+M), because the work done for each vertex is proportional
to the degree of the vertex, and the sum of the vertex degrees
is 2M.

Breadth-first search

DFS is called "depth first" because it descends into the graph as quickly as possible.

Breadth-search first, in contrast, explores nodes by following a "wave front". It explores
in order of their distance from the start vertex.


 BFS(graph G, vertex v)
 
    FIFO_Queue Q;
    Array<int> distance(-1);
    
    distance[v] = 0;
    insert vertex v into the queue Q.
 
    while (Q is not empty)
       take the next vertex W off the front of the queue Q
       for each neighbor X of W do
          if (distance[X] == -1) then
             distance[X] = distance[W] + 1
             add vertex X to the tail of the queue Q
 end

We demonstrated BFS on an example,
and noted that it ran in time O(N+M)
on any graph with N vertices and M edges.

References

* Chapter 6 of GoodrichAndTomassia .

	Revision 2 . . May 4, 2004 7:54 pm by Neal
	Revision 1 . . May 4, 2004 7:52 pm by Neal