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Graphs: Adjacency Matrix

Graphs: Adjacency Matrix. Example:. 1. a. d. 2. 4. b. c. 3. 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

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Graphs: Adjacency Matrix

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  1. Graphs: Adjacency Matrix • Example: 1 a d 2 4 b c 3 zhengjin 12014/11/7

  2. 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. zhengjin 22014/11/7

  3. 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 zhengjin 32014/11/7

  4. 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->d represent? What does v->p represent? zhengjin 42014/11/7

  5. Breadth-First Search: Example r s t u         v w x y zhengjin 52014/11/7

  6. Breadth-First Search: Example r s t u  0       v w x y Q: s zhengjin 62014/11/7

  7. Breadth-First Search: Example r s t u 1 0    1   v w x y Q: w r zhengjin 72014/11/7

  8. Breadth-First Search: Example r s t u 1 0 2   1 2  v w x y Q: r t x zhengjin 82014/11/7

  9. Breadth-First Search: Example r s t u 1 0 2  2 1 2  v w x y Q: t x v zhengjin 92014/11/7

  10. Breadth-First Search: Example r s t u 1 0 2 3 2 1 2  v w x y Q: x v u zhengjin 102014/11/7

  11. Breadth-First Search: Example r s t u 1 0 2 3 2 1 2 3 v w x y Q: v u y zhengjin 112014/11/7

  12. Breadth-First Search: Example r s t u 1 0 2 3 2 1 2 3 v w x y Q: u y zhengjin 122014/11/7

  13. Breadth-First Search: Example r s t u 1 0 2 3 2 1 2 3 v w x y Q: y zhengjin 132014/11/7

  14. Breadth-First Search: Example r s t u 1 0 2 3 2 1 2 3 v w x y Q: Ø zhengjin 142014/11/7

  15. 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 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? Total running time: O(V+E) zhengjin 152014/11/7

  16. 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 zhengjin 162014/11/7

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