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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
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Some details：
- White：represented “undiscovered” vertices
- Gray：represented “discovered” but not “processed” vertices
- Black：represented “processed” vertices
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Given a graph G = <V,E>, 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
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vertex s and a graph G
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step 1: initialization
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step 2
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step 3
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step 4
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step 5
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step 6
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step 7
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step 8
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step 9
4. What does BFS produce
5. Our goal
- δ(s,v) = d[v]