Breadth-First Search 广度优先搜索

1. The shortest-paths problem

• Distance δ(s, v) : The length of the shortest path from s to v. For example δ(s, c)=2.
• The problem：
  • Input： A graph G = (V, E) and a source vertex s∈V
  • Question： A shortest path from s to each vertex v ∈V and the distance δ(s, v) .

2. Algorithm

• Some details：

  • White：represented “undiscovered” vertices
  • Gray：represented “discovered” but not “processed” vertices
  • Black：represented “processed” vertices

• Given a graph G = , the BFS return：

  • d[v]：proved to be the shortest distance from s to v
  • π[v]：the predecessor of v in the search, which can be used to derive a shortest path from s to vertex v
  • BFS acturally returns a shortest path tree in which the unique simple path from s to node v is a shortest path from s to v in the original graph.

3. Example of Breadth-First Search

• vertex s and a graph G
  

• step 1: initialization
  

• step 2
  

• step 3
  

• step 4
  

• step 5
  

• step 6
  

• step 7
  

• step 8
  

• step 9
  

4. What does BFS produce

5. Our goal

• δ(s,v) = d[v]