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Lecture 8

Lecture 8. Depth First Search (DFS) Breadth First Search (BFS). Topics. Reference: Introduction to Algorithm by Cormen Chapter 23: Elementary Graph Algorithm. Graph Search (traversal). How do we search a graph? At a particular vertices, where shall we go next? Two common framework:

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Lecture 8

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  1. Lecture 8 • Depth First Search (DFS) • Breadth First Search (BFS) Topics Reference: Introduction to Algorithm by Cormen Chapter 23: Elementary Graph Algorithm Data Structure and Algorithm

  2. Graph Search (traversal) • How do we search a graph? • At a particular vertices, where shall we go next? • Two common framework: • the depth-first search (DFS) • the breadth-first search (BFS) and • In DFS, go as far as possible along a single path until reach a dead end (a vertex with no edge out or no neighbor unexplored) then backtrack • In BFS, one explore a graph level by level away (explore all neighbors first and then move on) Data Structure and Algorithm

  3. Depth-First Search (DFS) d a c b e f g h i j • The basic idea behind this algorithm is that it traverses the graph using recursion • Go as far as possible until you reach a deadend • Backtrack to the previous path and try the next branch • The graph below, started at node a, would be visited in the following order: a, b, c, g, h, i, e, d, f, j Data Structure and Algorithm

  4. DFS: Color Scheme • Vertices initially colored white • Then colored gray when discovered • Then black when finished Data Structure and Algorithm

  5. DFS: Time Stamps • Discover time d[u]: when u is first discovered • Finish time f[u]: when backtrack from u • d[u] < f[u] Data Structure and Algorithm

  6. DFS Example sourcevertex Data Structure and Algorithm

  7. DFS Example sourcevertex d f 1 | | | | | | | | Data Structure and Algorithm

  8. DFS Example sourcevertex d f 1 | | | 2 | | | | | Data Structure and Algorithm

  9. DFS Example sourcevertex d f 1 | | | 2 | | 3 | | | Data Structure and Algorithm

  10. DFS Example sourcevertex d f 1 | | | 2 | | 3 | 4 | | Data Structure and Algorithm

  11. DFS Example sourcevertex d f 1 | | | 2 | | 3 | 4 5 | | Data Structure and Algorithm

  12. DFS Example sourcevertex d f 1 | | | 2 | | 3 | 4 5 | 6 | Data Structure and Algorithm

  13. DFS Example sourcevertex d f 1 | | | 2 | 7 | 3 | 4 5 | 6 | Data Structure and Algorithm

  14. DFS Example sourcevertex d f 1 | 8 | | 2 | 7 | 3 | 4 5 | 6 | Data Structure and Algorithm

  15. DFS Example sourcevertex d f 1 | 8 | | 2 | 7 9 | 3 | 4 5 | 6 | Data Structure and Algorithm

  16. DFS Example sourcevertex d f 1 | 8 | | 2 | 7 9 |10 3 | 4 5 | 6 | Data Structure and Algorithm

  17. DFS Example sourcevertex d f 1 | 8 |11 | 2 | 7 9 |10 3 | 4 5 | 6 | Data Structure and Algorithm

  18. DFS Example sourcevertex d f 1 |12 8 |11 | 2 | 7 9 |10 3 | 4 5 | 6 | Data Structure and Algorithm

  19. DFS Example sourcevertex d f 1 |12 8 |11 13| 2 | 7 9 |10 3 | 4 5 | 6 | Data Structure and Algorithm

  20. DFS Example sourcevertex d f 1 |12 8 |11 13| 2 | 7 9 |10 3 | 4 5 | 6 14| Data Structure and Algorithm

  21. DFS Example sourcevertex d f 1 |12 8 |11 13| 2 | 7 9 |10 3 | 4 5 | 6 14|15 Data Structure and Algorithm

  22. DFS Example sourcevertex d f 1 |12 8 |11 13|16 2 | 7 9 |10 3 | 4 5 | 6 14|15 Data Structure and Algorithm

  23. DFS: Algorithm • DFS(G) • for each vertex u in V, • color[u]=white; p[u]=NIL • time=0; • for each vertex u in V • if (color[u]=white) • DFS-VISIT(u) Data Structure and Algorithm

  24. DFS: Algorithm (Cont.) sourcevertex • DFS-VISIT(u) • color[u]=gray; • time = time + 1; • d[u] = time; • for each v in Adj(u) do • if (color[v] = white) • p[v] = u; • DFS-VISIT(v); • color[u] = black; • time = time + 1; f[u]= time; Data Structure and Algorithm

  25. DFS: Complexity Analysis Initialization complexity is O(V) DFS_VISIT is called exactly once for each vertex And DFS_VISIT scans all the edges which causes cost of O(E) Overall complexity is O(V + E) Data Structure and Algorithm

  26. DFS: Application • Topological Sort • Strongly Connected Component Data Structure and Algorithm

  27. Breadth-first Search (BFS) • Search for all vertices that are directly reachable from the root (called level 1 vertices) • After mark all these vertices, visit all vertices that are directly reachable from any level 1 vertices (called level 2 vertices), and so on. • In general, level k vertices are directly reachable from a level k – 1 vertices Data Structure and Algorithm

  28. BFS: the Color Scheme • 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 Data Structure and Algorithm

  29. An Example A B C D E F G H I J K L M N O P Data Structure and Algorithm

  30. A B C D 0 E F G H I J K L M N O P Data Structure and Algorithm

  31. A B C D 0 1 E F G H 1 1 I J K L M N O P Data Structure and Algorithm

  32. A B C D 0 2 1 E F G H 1 1 I J K L 2 M N O P Data Structure and Algorithm

  33. A B C D 0 3 2 1 E F G H 1 1 3 3 I J K L 2 M N O P 3 3 Data Structure and Algorithm

  34. A B C D 0 3 2 1 E F G H 1 1 3 4 3 I J K L 2 4 4 M N O P 3 3 Data Structure and Algorithm

  35. A B C D 0 3 2 1 E F G H 1 1 3 4 3 I J K L 2 4 4 5 M N O P 3 3 5 Data Structure and Algorithm

  36. A B C D 0 3 2 1 E F G H 1 1 3 4 3 I J K L 2 4 4 5 M N O P 3 3 5 Data Structure and Algorithm

  37. A B C D 0 3 2 1 E F G H 1 1 3 4 3 I J K L 2 4 4 5 M N O P 3 3 5 Data Structure and Algorithm

  38. BFS: Algorithm BFS(G, s) • For each vertex u in V – {s}, • color[u] = white; • d[u] = infinity; • p[u] = NIL • color[s] = GRAY; d[s] = 0; p[s] = NIL; Q = empty queue • ENQUEUE(Q,s) • while (Q not empty) • u = DEQUEUE(Q) • for each v  Adj[u] • if color[v] = WHITE • then color[v] = GREY • d[v] = d[u] + 1; p[v] = u • ENQUEUE(Q, v); • color[u] = BLACK; Data Structure and Algorithm

  39. Example r s t u         v w x y Data Structure and Algorithm

  40. Example r s t u  0       v w x y Q: s Data Structure and Algorithm

  41. Example r s t u 1 0    1   v w x y Q: w r Data Structure and Algorithm

  42. Example r s t u 1 0 2   1 2  v w x y Q: r t x Data Structure and Algorithm

  43. Example r s t u 1 0 2  2 1 2  v w x y Q: t x v Data Structure and Algorithm

  44. Example r s t u 1 0 2 3 2 1 2  v w x y Q: x v u Data Structure and Algorithm

  45. Example r s t u 1 0 2 3 2 1 2 3 v w x y Q: v u y Data Structure and Algorithm

  46. Example r s t u 1 0 2 3 2 1 2 3 v w x y Q: u y Data Structure and Algorithm

  47. Example r s t u 1 0 2 3 2 1 2 3 v w x y Q: y Data Structure and Algorithm

  48. Example r s t u 1 0 2 3 2 1 2 3 v w x y Q: Ø Data Structure and Algorithm

  49. BFS: Complexity Analysis • Queuing time is O(V) and scanning all edges requires O(E) • Overhead for initialization is O (V) • So, total running time is O(V+E) Data Structure and Algorithm

  50. BFS: Application • Shortest path problem Data Structure and Algorithm

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