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CSC 413/513: Intro to Algorithms

CSC 413/513: Intro to Algorithms. Graph Algorithms DFS. 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

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CSC 413/513: Intro to Algorithms

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  1. CSC 413/513: Intro to Algorithms Graph Algorithms DFS

  2. 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

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

  4. 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

  5. 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?

  6. 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?

  7. 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?

  8. DFS Example sourcevertex

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

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

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

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

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

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

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

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

  17. 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?

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

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

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

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

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

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

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

  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 will be the running time?

  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 Running time: O(n2) because call DFS_Visit on each vertex, and the loop over Adj[] can run as many as |V| times

  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 BUT, there is actually a tighter bound. How many times will DFS_Visit() actually be called?

  28. Depth-First Search Analysis • For a node u • DFS_Visit(u) cannot be called more than once • DFS_Visit(u) cannot be called less than once • Therefore, exactly once • DFS_Visit(u) requires c + ϴ(|Adj(u)|) time • The fact that recursive calls are made to v’s is accounted for by DFS_Visit(v) • Total= ∑u ( c + ϴ(|Adj(u)|) ) = ϴ(V+E) • Considered linear in input size, since adjacency list is a ϴ(V+E) representation

  29. DFS: Kinds of edges • DFS introduces an important distinction among edges in the original graph: • Tree edge: encounter new (white) vertex • The tree edges form a spanning forest • Can tree edges form cycles? Why or why not?

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

  31. DFS: Kinds of edges • DFS introduces an important distinction among edges in the original graph: • Tree edge: encounter new (white) vertex • Back edge: from descendent to ancestor • Encounter a grey vertex (grey to grey)

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

  33. DFS: Kinds of edges • DFS introduces an important distinction among edges in the original graph: • Tree edge: encounter new (white) vertex • Back edge: from descendent to ancestor • Forward edge: from ancestor to descendent • Not a tree edge, though • If forward edge, then grey node to black node • Converse not true; could be a cross edge (see next)

  34. DFS Example sourcevertex d f 1 |12 8 |11 13|16 2 | 7 9 |10 3 | 4 5 | 6 14|15 Tree edges Back edges Forward edges

  35. DFS: Kinds of edges • DFS introduces an important distinction among edges in the original graph: • Tree edge: encounter new (white) vertex • Back edge: from descendent to ancestor • Forward edge: from ancestor to descendent • Cross edge: between or within trees • (Also) from a grey node to a black node • No ancestor-descendent relation, if in the same DFS tree

  36. DFS Example sourcevertex d f 1 |12 8 |11 13|16 2 | 7 9 |10 3 | 4 5 | 6 14|15 Tree edges Back edges Forward edges Cross edges

  37. DFS: Kinds of edges • DFS introduces an important distinction among edges in the original graph: • Tree edge: encounter new (white) vertex • Back edge: from descendent to ancestor • Forward edge: from ancestor to descendent • Cross edge: between a tree or subtrees • Note: tree & back edges are important; most algorithms don’t distinguish forward & cross

  38. DFS: Kinds Of Edges • Thm 22.10: If G is undirected, a DFS produces only tree and back edges • Proof by contradiction: • Assume there’s a forward edge • But F? edge must actually be a back edge (why?) source F?

  39. DFS: Kinds Of Edges • Thm 22.10: If G is undirected, a DFS produces only tree and back edges • Proof by contradiction: • Assume there’s a cross edge • But C? edge cannot be cross: • must be explored from one of the vertices it connects, becoming a treevertex, before other vertex is explored • So in fact the picture is wrong…bothlower tree edges cannot in fact betree edges source C?

  40. DFS And Graph Cycles • Thm: An undirected graph is acyclic iff a DFS yields no back edges • If acyclic, then no back edges (because a back edge implies a cycle) • If no back edges, then acyclic • No back edges implies only tree edges (Why?) • Only tree edges implies we have a tree or a forest • Which by definition is acyclic • Thus, can run DFS to find whether a graph has a cycle

  41. DFS And Cycles • How would you modify the code to detect cycles? 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; } 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); } } If v->color == GREY and Pred(u)≠v, …

  42. DFS And Cycles • What will be the running time? 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; }

  43. DFS And Cycles • What will be the running time? • A: O(V+E) • We can actually determine if cycles exist in O(V) time: • In an undirected acyclic forest, |E|  |V| - 1. So if no cycle, then O(V+E) is O(V) • If cycle, count the edges: if ever see |V| distinct edges, must have seen a back edge along the way • Can halt and output “cycle exists” • O(V) time

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