The.Algorithm.Design.Manual.Springer-Verlag.1998
The.Algorithm.Design.Manual.Springer-Verlag.1998 The.Algorithm.Design.Manual.Springer-Verlag.1998
Lecture 15 - DFS and BFS This tree defines a shortest path from the root to every other node in the tree. The proof is by induction on the length of the shortest path from the root: ● Length = 1 First step of BFS explores all neighbors of the root. In an unweighted graph one edge must be the shortest path to any node. ● Length = s Assume the BFS tree has the shortest paths up to length s-1. Any node at a distance of will first be discovered by expanding a distance s-1 node. Listen To Part 16-11 The key idea about DFS A depth-first search of a graph organizes the edges of the graph in a precise way. In a DFS of an undirected graph, we assign a direction to each edge, from the vertex which discover it: In a DFS of a directed graph, every edge is either a tree edge or a black edge. file:///E|/LEC/LECTUR16/NODE15.HTM (2 of 8) [19/1/2003 1:35:08]
Lecture 15 - DFS and BFS In a DFS of a directed graph, no cross edge goes to a higher numbered or rightward vertex. Thus, no edge from 4 to 5 is possible: Listen To Part 16-12 Edge Classification for DFS What about the other edges in the graph? Where can they go on a search? Every edge is either: On any particular DFS or BFS of a directed or undirected graph, each edge gets classified as one of the above. Listen To Part 17-3 DFS Trees The reason DFS is so important is that it defines a very nice ordering to the edges of the graph. file:///E|/LEC/LECTUR16/NODE15.HTM (3 of 8) [19/1/2003 1:35:08]
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Lecture 15 - DFS and BFS<br />
In a DFS of a directed graph, no cross edge goes to a higher numbered or rightward vertex. Thus, no<br />
edge from 4 to 5 is possible:<br />
Listen To Part 16-12<br />
Edge Classification for DFS<br />
What about the other edges in the graph? Where can they go on a search?<br />
Every edge is either:<br />
On any particular DFS or BFS of a directed or undirected graph, each edge gets classified as one of the<br />
above.<br />
Listen To Part 17-3<br />
DFS Trees<br />
<strong>The</strong> reason DFS is so important is that it defines a very nice ordering to the edges of the graph.<br />
file:///E|/LEC/LECTUR16/NODE15.HTM (3 of 8) [19/1/2003 1:35:08]
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