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zhengjin 1 2015-5-9 Graphs: Adjacency Matrix ● Example: 1 24 3 a d bc A1234 1 2 3 ?? 4

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zhengjin 2 2015-5-9 Breadth-First Search ● “Explore” a graph, turning it into a tree ■ One vertex at a time ■ Expand frontier of explored vertices across the breadth of the frontier ● Builds a tree over the graph ■ Pick a source vertex to be the root ■ Find (“discover”) its children, then their children, etc.

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zhengjin 3 2015-5-9 Breadth-First Search ● Again will associate vertex “colors” to guide the algorithm ■ White vertices have not been discovered ○ All vertices start out white ■ Grey vertices are discovered but not fully explored ○ They may be adjacent to white vertices ■ Black vertices are discovered and fully explored ○ They are adjacent only to black and gray vertices ● Explore vertices by scanning adjacency list of grey vertices

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zhengjin 4 2015-5-9 Breadth-First Search BFS(G, s) { initialize vertices; Q = {s};// Q is a queue (duh); initialize to s while (Q not empty) { u = RemoveTop(Q); for each v u->adj { if (v->color == WHITE) v->color = GREY; v->d = u->d + 1; v->p = u; Enqueue(Q, v); } u->color = BLACK; } What does v->p represent? What does v->d represent?

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zhengjin 5 2015-5-9 Breadth-First Search: Example rstu vwxy

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zhengjin 6 2015-5-9 Breadth-First Search: Example 0 rstu vwxy s Q:

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zhengjin 7 2015-5-9 Breadth-First Search: Example 1 0 1 rstu vwxy w Q: r

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zhengjin 8 2015-5-9 Breadth-First Search: Example 1 0 1 2 2 rstu vwxy r Q: tx

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zhengjin 9 2015-5-9 Breadth-First Search: Example 1 2 0 1 2 2 rstu vwxy Q: txv

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zhengjin 10 2015-5-9 Breadth-First Search: Example 1 2 0 1 2 2 3 rstu vwxy Q: xvu

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zhengjin 11 2015-5-9 Breadth-First Search: Example 1 2 0 1 2 2 3 3 rstu vwxy Q: vuy

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zhengjin 12 2015-5-9 Breadth-First Search: Example 1 2 0 1 2 2 3 3 rstu vwxy Q: uy

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zhengjin 13 2015-5-9 Breadth-First Search: Example 1 2 0 1 2 2 3 3 rstu vwxy Q: y

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zhengjin 14 2015-5-9 Breadth-First Search: Example 1 2 0 1 2 2 3 3 rstu vwxy Q: Ø

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zhengjin 15 2015-5-9 BFS: The Code Again BFS(G, s) { initialize vertices; Q = {s}; while (Q not empty) { u = RemoveTop(Q); for each v u->adj { if (v->color == WHITE) v->color = GREY; v->d = u->d + 1; v->p = u; Enqueue(Q, v); } u->color = BLACK; } What will be the running time? Touch every vertex: O(V) u = every vertex, but only once (Why?) So v = every vertex that appears in some other vert’s adjacency list Total running time: O(V+E)

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zhengjin 16 2015-5-9 Breadth-First Search: Properties ● BFS calculates the shortest-path distance to the source node ■ Shortest-path distance (s,v) = minimum number of edges from s to v, or if v not reachable from s ● BFS builds breadth-first tree, in which paths to root represent shortest paths in G ■ Thus can use BFS to calculate shortest path from one vertex to another in O(V+E) time

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Graph Traversals Visit vertices of a graph G to determine some property: Is G connected? Is there a path from vertex a to vertex b? Does G have a cycle?

Graph Traversals Visit vertices of a graph G to determine some property: Is G connected? Is there a path from vertex a to vertex b? Does G have a cycle?

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