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CS 3343: Analysis of Algorithms

CS 3343: Analysis of Algorithms. Lecture 24: Graph searching, Topological sort. Midterm 2 overview. Midterm 2 overview: overall. Semester overview. D. C. B. A. Review of MST and shortest path problem. Run Kruskal’s algorithm Run Prim’s algorithm Run Dijkstra’s algorithm. 2. f. c. 4.

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CS 3343: Analysis of Algorithms

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  1. CS 3343: Analysis of Algorithms Lecture 24: Graph searching, Topological sort

  2. Midterm 2 overview

  3. Midterm 2 overview: overall

  4. Semester overview D C B A

  5. Review of MST and shortest path problem • Run Kruskal’s algorithm • Run Prim’s algorithm • Run Dijkstra’s algorithm 2 f c 4 7 8 8 a d g 6 7 6 1 3 5 6 b e

  6. Graph Searching • Given: a graph G = (V, E), directed or undirected • Goal: methodically explore every vertex (and every edge) • Ultimately: build a tree on the graph • Pick a vertex as the root • Find (“discover”) its children, then their children, etc. • Note: might also build a forest if graph is not connected • Here we only consider that the graph is connected

  7. Breadth-First Search • “Explore” a graph, turning it into a tree • Pick a source vertex to be the root • Expand frontier of explored vertices across the breadth of the frontier

  8. Breadth-First Search • 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

  9. Breadth-First Search BFS(G, s) { initialize vertices; // mark all vertices as white Q = {s}; // Q is a queue; initialize to s while (Q not empty) { u = Dequeue(Q); for each v  adj[u] 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?

  10. Breadth-First Search: Example r s t u         v w x y

  11. Breadth-First Search: Example r s t u  0       v w x y Q: s

  12. Breadth-First Search: Example r s t u 1 0    1   v w x y Q: w r

  13. Breadth-First Search: Example r s t u 1 0 2   1 2  v w x y Q: r t x

  14. Breadth-First Search: Example r s t u 1 0 2  2 1 2  v w x y Q: t x v

  15. Breadth-First Search: Example r s t u 1 0 2 3 2 1 2  v w x y Q: x v u

  16. Breadth-First Search: Example r s t u 1 0 2 3 2 1 2 3 v w x y Q: v u y

  17. Breadth-First Search: Example r s t u 1 0 2 3 2 1 2 3 v w x y Q: u y

  18. Breadth-First Search: Example r s t u 1 0 2 3 2 1 2 3 v w x y Q: y

  19. Breadth-First Search: Example r s t u 1 0 2 3 2 1 2 3 v w x y Q: Ø

  20. Touch every vertex: Θ(n) u = every vertex, but only once (Why?) v = every vertex that appears in some other vert’s adjacency list Total: Θ(m) BFS: The Code Again BFS(G, s) { initialize vertices; Q = {s}; while (Q not empty) { u = Dequeue(Q); for each v  adj[u] 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: Θ(n+m)

  21. Depth-First Search • Depth-first search is another strategy for exploring a graph • Explore “deeper” in the graph whenever possible • Edges are explored out of the most recently discovered vertex v that still has unexplored edges • When all of v’s edges have been explored, backtrack to the vertex from which v was discovered

  22. Depth-First Search • Vertices initially colored white • Then colored gray when discovered • Then black when finished

  23. DFS(G) { for each vertex u  G->V { u->color = WHITE; } time = 0; for each vertex u  G->V { if (u->color == WHITE) DFS_Visit(u); } } DFS_Visit(u) { u->color = GREY; time = time+1; u->d = time; for each v  u->Adj[] { if (v->color == WHITE) DFS_Visit(v); } u->color = BLACK; time = time+1; u->f = time; } Depth-First Search: The Code

  24. DFS(G) { for each vertex u  G->V { u->color = WHITE; } time = 0; for each vertex u  G->V { if (u->color == WHITE) DFS_Visit(u); } } DFS_Visit(u) { u->color = GREY; time = time+1; u->d = time; for each v  u->Adj[] { if (v->color == WHITE) DFS_Visit(v); } u->color = BLACK; time = time+1; u->f = time; } Depth-First Search: The Code What does u->d represent?

  25. DFS(G) { for each vertex u  G->V { u->color = WHITE; } time = 0; for each vertex u  G->V { if (u->color == WHITE) DFS_Visit(u); } } DFS_Visit(u) { u->color = GREY; time = time+1; u->d = time; for each v  u->Adj[] { if (v->color == WHITE) DFS_Visit(v); } u->color = BLACK; time = time+1; u->f = time; } Depth-First Search: The Code What does u->f represent?

  26. DFS(G) { for each vertex u  G->V { u->color = WHITE; } time = 0; for each vertex u  G->V { if (u->color == WHITE) DFS_Visit(u); } } DFS_Visit(u) { u->color = GREY; time = time+1; u->d = time; for each v  u->Adj[] { if (v->color == WHITE) DFS_Visit(v); } u->color = BLACK; time = time+1; u->f = time; } Depth-First Search: The Code Will all vertices eventually be colored black? (How about in BFS?)

  27. DFS(G) { for each vertex u  G->V { u->color = WHITE; } time = 0; for each vertex u  G->V { if (u->color == WHITE) DFS_Visit(u); } } DFS_Visit(u) { u->color = GREY; time = time+1; u->d = time; for each v  u->Adj[] { if (v->color == WHITE) DFS_Visit(v); } u->color = BLACK; time = time+1; u->f = time; } Depth-First Search: The Code What will be the running time?

  28. DFS(G) { for each vertex u  G->V { u->color = WHITE; } time = 0; for each vertex u  G->V { if (u->color == WHITE) DFS_Visit(u); } } DFS_Visit(u) { u->color = GREY; time = time+1; u->d = time; for each v  u->Adj[] { if (v->color == WHITE) DFS_Visit(v); } u->color = BLACK; time = time+1; u->f = time; } Depth-First Search: The Code How many times will DFS_Visit() be called?

  29. DFS(G) { for each vertex u  G->V { u->color = WHITE; } time = 0; for each vertex u  G->V { if (u->color == WHITE) DFS_Visit(u); } } DFS_Visit(u) { u->color = GREY; time = time+1; u->d = time; for each v  u->Adj[] { if (v->color == WHITE) DFS_Visit(v); } u->color = BLACK; time = time+1; u->f = time; } Depth-First Search: The Code How much time is needed within each DFS_Visit()?

  30. DFS(G) { for each vertex u  G->V { u->color = WHITE; } time = 0; for each vertex u  G->V { if (u->color == WHITE) DFS_Visit(u); } } DFS_Visit(u) { u->color = GREY; time = time+1; u->d = time; for each v  u->Adj[] { if (v->color == WHITE) DFS_Visit(v); } u->color = BLACK; time = time+1; u->f = time; } Depth-First Search: The Code So, running time of DFS = O(V+E)

  31. DFS Example sourcevertex

  32. DFS Example sourcevertex d f 1 | | | | | | | |

  33. DFS Example sourcevertex d f 1 | | | 2 | | | | |

  34. DFS Example sourcevertex d f 1 | | | 2 | | 3 | | |

  35. DFS Example sourcevertex d f 1 | | | 2 | | 3 | 4 | |

  36. DFS Example sourcevertex d f 1 | | | 2 | | 3 | 4 5 | |

  37. DFS Example sourcevertex d f 1 | | | 2 | | 3 | 4 5 | 6 |

  38. DFS Example sourcevertex d f 1 | | | 2 | 7 | 3 | 4 5 | 6 |

  39. DFS Example sourcevertex d f 1 | 8 | | 2 | 7 | 3 | 4 5 | 6 |

  40. DFS Example sourcevertex d f 1 | 8 | | 2 | 7 9 | 3 | 4 5 | 6 | What is the structure of the grey vertices? What do they represent?

  41. DFS Example sourcevertex d f 1 | 8 | | 2 | 7 9 |10 3 | 4 5 | 6 |

  42. DFS Example sourcevertex d f 1 | 8 |11 | 2 | 7 9 |10 3 | 4 5 | 6 |

  43. DFS Example sourcevertex d f 1 |12 8 |11 | 2 | 7 9 |10 3 | 4 5 | 6 |

  44. DFS Example sourcevertex d f 1 |12 8 |11 13| 2 | 7 9 |10 3 | 4 5 | 6 |

  45. DFS Example sourcevertex d f 1 |12 8 |11 13| 2 | 7 9 |10 3 | 4 5 | 6 14|

  46. DFS Example sourcevertex d f 1 |12 8 |11 13| 2 | 7 9 |10 3 | 4 5 | 6 14|15

  47. DFS Example sourcevertex d f 1 |12 8 |11 13|16 2 | 7 9 |10 3 | 4 5 | 6 14|15

  48. DFS and cycles in graph • A graph G is acyclic iff a DFS of G yields no back edges sourcevertex d f 1 | | | 2 | | 3 | | |

  49. Directed Acyclic Graphs • A directed acyclic graph or DAG is a directed graph with no directed cycles: Cyclic Acyclic

  50. Topological Sort • Topological sort of a DAG: • Linear ordering of all vertices in graph G such that vertex u comes before vertex v if edge (u, v)  G • Real-world example: getting dressed

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