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CS 332: Algorithms

CS 332: Algorithms. Graph Algorithms . Review: 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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CS 332: Algorithms

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  1. CS 332: Algorithms Graph Algorithms David Luebke 16/6/2014

  2. Review: 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 David Luebke 26/6/2014

  3. 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 = YELLOW; 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; } Review: DFS Code David Luebke 36/6/2014

  4. DFS Example sourcevertex David Luebke 46/6/2014

  5. DFS Example sourcevertex d f 1 | | | | | | | | David Luebke 56/6/2014

  6. DFS Example sourcevertex d f 1 | | | 2 | | | | | David Luebke 66/6/2014

  7. DFS Example sourcevertex d f 1 | | | 2 | | 3 | | | David Luebke 76/6/2014

  8. DFS Example sourcevertex d f 1 | | | 2 | | 3 | 4 | | David Luebke 86/6/2014

  9. DFS Example sourcevertex d f 1 | | | 2 | | 3 | 4 5 | | David Luebke 96/6/2014

  10. DFS Example sourcevertex d f 1 | | | 2 | | 3 | 4 5 | 6 | David Luebke 106/6/2014

  11. DFS Example sourcevertex d f 1 | 8 | | 2 | 7 | 3 | 4 5 | 6 | David Luebke 116/6/2014

  12. DFS Example sourcevertex d f 1 | 8 | | 2 | 7 | 3 | 4 5 | 6 | David Luebke 126/6/2014

  13. DFS Example sourcevertex d f 1 | 8 | | 2 | 7 9 | 3 | 4 5 | 6 | What is the structure of the yellow vertices? What do they represent? David Luebke 136/6/2014

  14. DFS Example sourcevertex d f 1 | 8 | | 2 | 7 9 |10 3 | 4 5 | 6 | David Luebke 146/6/2014

  15. DFS Example sourcevertex d f 1 | 8 |11 | 2 | 7 9 |10 3 | 4 5 | 6 | David Luebke 156/6/2014

  16. DFS Example sourcevertex d f 1 |12 8 |11 | 2 | 7 9 |10 3 | 4 5 | 6 | David Luebke 166/6/2014

  17. DFS Example sourcevertex d f 1 |12 8 |11 13| 2 | 7 9 |10 3 | 4 5 | 6 | David Luebke 176/6/2014

  18. DFS Example sourcevertex d f 1 |12 8 |11 13| 2 | 7 9 |10 3 | 4 5 | 6 14| David Luebke 186/6/2014

  19. DFS Example sourcevertex d f 1 |12 8 |11 13| 2 | 7 9 |10 3 | 4 5 | 6 14|15 David Luebke 196/6/2014

  20. DFS Example sourcevertex d f 1 |12 8 |11 13|16 2 | 7 9 |10 3 | 4 5 | 6 14|15 David Luebke 206/6/2014

  21. 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? David Luebke 216/6/2014

  22. DFS Example sourcevertex d f 1 |12 8 |11 13|16 2 | 7 9 |10 3 | 4 5 | 6 14|15 Tree edges David Luebke 226/6/2014

  23. 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 yellow vertex (yellow to yellow) David Luebke 236/6/2014

  24. 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 David Luebke 246/6/2014

  25. 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 • From yellow node to black node David Luebke 256/6/2014

  26. 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 David Luebke 266/6/2014

  27. 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 • From a yellow node to a black node David Luebke 276/6/2014

  28. 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 David Luebke 286/6/2014

  29. 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 David Luebke 296/6/2014

  30. DFS: Kinds Of Edges • Thm 23.9: 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? David Luebke 306/6/2014

  31. DFS: Kinds Of Edges • Thm 23.9: 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? David Luebke 316/6/2014

  32. DFS And Graph Cycles • Thm: An undirected graph is acyclic iff a DFS yields no back edges • If acyclic, no back edges (because a back edge implies a cycle • If no back edges, 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 David Luebke 326/6/2014

  33. DFS And Cycles • How would you modify the code to detect cycles? 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; } David Luebke 336/6/2014

  34. 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; } David Luebke 346/6/2014

  35. 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 count the edges: if ever see |V| distinct edges, must have seen a back edge along the way David Luebke 356/6/2014

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