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Graph Searching Algorithms

Graph Searching Algorithms. Tree. Breadth-First Search (BFS). Breadth-First Search (BFS). Not discovered. white. u. ∞. 0. x. Discovered, adjacent white nodes. gray. v. y. ∞. ∞. Discovered, no adjacent white nodes. black. w. ∞. ∞. z. Breadth-First Search (BFS). u. ∞. 0.

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Graph Searching Algorithms

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  1. Graph Searching Algorithms

  2. Tree

  3. Breadth-First Search (BFS)

  4. Breadth-First Search (BFS) Not discovered white u ∞ 0 x Discovered, adjacent white nodes gray v y ∞ ∞ Discovered, no adjacent white nodes black w ∞ ∞ z

  5. Breadth-First Search (BFS) u ∞ 0 x BFS(G, u): 1. Initialize the graph color[u]  gray π[u]  Nil d[u]  0 for each other vertex color[u]  white v y ∞ ∞ w ∞ ∞ z

  6. Breadth-First Search (BFS) Q u ∞ 0 x u BFS(G, u): 2. Initialize the queue Q  Ø Enqueue(Q, u) v y ∞ ∞ w ∞ ∞ z

  7. Breadth-First Search (BFS) t = u Q u ∞ 0 x BFS(G, u): 3. While Q ≠ Ø 1) tDequeue(Q) v y ∞ ∞ w ∞ ∞ z

  8. Breadth-First Search (BFS) t = u r = x, v Q u 1 0 x v x BFS(G, u): 3. While Q ≠ Ø 2) for each radj to t if color[r] = white color[r]  gray π[r] t d[r] d[t] + 1 Enqueue(Q, r) v y ∞ 1 w ∞ ∞ z

  9. Breadth-First Search (BFS) t = u r = x, v Q u 1 0 x v x BFS(G, u): 3. While Q ≠ Ø 3) color[t]  black v y ∞ 1 w ∞ ∞ z

  10. Breadth-First Search (BFS) t = v Q u 1 0 x x BFS(G, u): 3. While Q ≠ Ø 1) tDequeue(Q) 2) for each radj to t … 3) color[t]  black v y ∞ 1 w ∞ ∞ z

  11. Breadth-First Search (BFS) t = v r = y Q u 1 0 x x y BFS(G, u): 3. While Q ≠ Ø 1) tDequeue(Q) 2) for each radj to t … 3) color[t]  black v y 2 1 w ∞ ∞ z

  12. Breadth-First Search (BFS) t = v r = y Q u 1 0 x x y BFS(G, u): 3. While Q ≠ Ø 1) tDequeue(Q) 2) for each radj to t … 3) color[t]  black v y 2 1 w ∞ ∞ z

  13. Breadth-First Search (BFS) t = x r = Q u 1 0 x y BFS(G, u): 3. While Q ≠ Ø 1) tDequeue(Q) 2) for each radj to t … 3) color[t]  black v y 2 1 w ∞ ∞ z

  14. Breadth-First Search (BFS) t = y r = w Q u 1 0 x w BFS(G, u): 3. While Q ≠ Ø 1) tDequeue(Q) 2) for each radj to t … 3) color[t]  black v y 2 1 w 3 ∞ z

  15. Breadth-First Search (BFS) t = w r = z Q u 1 0 x z BFS(G, u): 3. While Q ≠ Ø 1) tDequeue(Q) 2) for each radj to t … 3) color[t]  black v y 2 1 w 3 4 z

  16. Breadth-First Search (BFS) t = z r = Q u 1 0 x BFS(G, u): 3. While Q ≠ Ø 1) tDequeue(Q) 2) for each radj to t … 3) color[t]  black v y 2 1 w 3 4 z

  17. Breadth-First Search (BFS) u 1 0 x BFS(G, u): - the shortest-path distance from u v y 2 1 w 3 4 z

  18. Breadth-First Search (BFS) u 1 0 x BFS(G, u): - the shortest-path distance from u - construct a tree v y 2 1 w 3 4 z

  19. Breadth-First Search (BFS) u 1 0 x BFS(G, u): - Initialization: |V| - Enqueuing/dequeuing: |V| - Scanning adj vertices: |E| v y 2 1 w 3 4 z

  20. Breadth-First Search (BFS) u 1 0 x BFS(G, u): - Initialization: O(|V|) - Enqueuing/dequeuing: O(|V|) - Scanning adjacent vertices: O(|E|) => total running time: O(|V| + |E|) v y 2 1 w 3 4 z

  21. Depth-First Search (DFS)

  22. Depth-First Search (DFS) d[u]: when u is discovered f[u]: when searching adj of u is finished u v w

  23. Depth-First Search (DFS) timestamp: t d[u]: when u is discovered f[u]: when searching adj of u is finished d[u] = t u v w

  24. Depth-First Search (DFS) timestamp: t+1 d[u]: when u is discovered f[u]: when searching adj of u is finished d[u] = t u v w d[v] = t+1

  25. Depth-First Search (DFS) timestamp: t+2 d[u]: when u is discovered f[u]: when searching adj of u is finished d[u] = t u v w d[v] = t+1 f[v] = t+2

  26. Depth-First Search (DFS) timestamp: t+3 d[u]: when u is discovered f[u]: when searching adj of u is finished d[u] = t u v w d[v] = t+1 f[v] = t+2 d[w] = t+3

  27. Depth-First Search (DFS) timestamp: t+4 d[u]: when u is discovered f[u]: when searching adj of u is finished d[u] = t u v w d[v] = t+1 f[v] = t+2 d[w] = t+3 f[v] = t+4

  28. Depth-First Search (DFS) timestamp: t+5 d[u]: when u is discovered f[u]: when searching adj of u is finished d[u] = t f[u] = t+5 u v w d[v] = t+1 f[v] = t+2 d[w] = t+3 f[w] = t+4

  29. Depth-First Search (DFS) d[u]: when u is discovered f[u]: when searching adj of u is finished d[u] = t f[u] = t+5 u d[u] < f[u] [ d[u], f[u] ] entirely contains [ d[v], f[v] ] [ d[v], f[v] ] and [ d[w], f[w] ] are entirely disjoint v w d[v] = t+1 f[v] = t+2 d[w] = t+3 f[w] = t+4

  30. Depth-First Search (DFS) Not discovered white u x Discovered, adjacent white nodes gray v y Discovered, no adjacent white nodes black w z

  31. Depth-First Search (DFS) Not discovered u x Discovered, adjacent white nodes d / v y Discovered, no adjacent white nodes d / f w z

  32. Depth-First Search (DFS) u x DFS(G): 1. Initialization for each u V[G], color[u]  white π[u]  Nil time  0 v y w z

  33. Depth-First Search (DFS) u 1/ x DFS(G): 1. Initialization 2. For each u V[G] if color[u] = white DFS-Visit(u) v y DFS-Visit(u): 1. Initial Setting color[u]  gray d[u]  time  time + 1 w z

  34. Depth-First Search (DFS) u 1/ x DFS(G): 1. Initialization 2. For each u V[G] if color[u] = white DFS-Visit(u) v y 2/ DFS-Visit(u): 1. Initial Setting 2. for each adjv of white π[v] u DFS-Visit[v] w z

  35. Depth-First Search (DFS) u 1/ x DFS(G): 1. Initialization 2. For each u V[G] if color[u] = white DFS-Visit(u) v y 3/ 2/ DFS-Visit(u): 1. Initial Setting 2. for each adjv of white π[v] u DFS-Visit[v] w z

  36. Depth-First Search (DFS) u 4/ 1/ x DFS(G): 1. Initialization 2. For each u V[G] if color[u] = white DFS-Visit(u) v y 3/ 2/ DFS-Visit(u): 1. Initial Setting 2. for each adjv of white π[v] u DFS-Visit[v] w z

  37. Depth-First Search (DFS) u 4/5 1/ x DFS(G): 1. Initialization 2. For each u V[G] if color[u] = white DFS-Visit(u) v y 3/ 2/ DFS-Visit(u): 1. Initial Setting 2. Handling adj vertices 3. color[u]  black f[u]  time  time + 1 w z

  38. Depth-First Search (DFS) u 4/5 1/ x DFS(G): 1. Initialization 2. For each u V[G] if color[u] = white DFS-Visit(u) v y 3/6 2/ DFS-Visit(u): 1. Initial Setting 2. Handling adj vertices 3. color[u]  black f[u]  time  time + 1 w z

  39. Depth-First Search (DFS) u 4/5 1/ x DFS(G): 1. Initialization 2. For each u V[G] if color[u] = white DFS-Visit(u) v y 3/6 2/7 DFS-Visit(u): 1. Initial Setting 2. Handling adj vertices 3. color[u]  black f[u]  time  time + 1 w z

  40. Depth-First Search (DFS) u 4/5 1/8 x DFS(G): 1. Initialization 2. For each u V[G] if color[u] = white DFS-Visit(u) v y 3/6 2/7 DFS-Visit(u): 1. Initial Setting 2. Handling adj vertices 3. color[u]  black f[u]  time  time + 1 w z

  41. Depth-First Search (DFS) u 4/5 1/8 x DFS(G): 1. Initialization 2. For each u V[G] if color[u] = white DFS-Visit(u) v y 3/6 2/7 DFS-Visit(u): 1. Initial Setting 2. Handling adj vertices 3. color[u]  black f[u]  time  time + 1 w 9/ z

  42. Depth-First Search (DFS) u 4/5 1/8 x DFS(G): 1. Initialization 2. For each u V[G] if color[u] = white DFS-Visit(u) v y 3/6 2/7 DFS-Visit(u): 1. Initial Setting 2. Handling adj vertices 3. color[u]  black f[u]  time  time + 1 w 9/ 10/ z

  43. Depth-First Search (DFS) u 4/5 1/8 x DFS(G): 1. Initialization 2. For each u V[G] if color[u] = white DFS-Visit(u) v y 3/6 2/7 DFS-Visit(u): 1. Initial Setting 2. Handling adj vertices 3. color[u]  black f[u]  time  time + 1 w 9/ 10/11 z

  44. Depth-First Search (DFS) u 4/5 1/8 x DFS(G): 1. Initialization 2. For each u V[G] if color[u] = white DFS-Visit(u) v y 3/6 2/7 DFS-Visit(u): 1. Initial Setting 2. Handling adj vertices 3. color[u]  black f[u]  time  time + 1 w 9/12 10/11 z

  45. Depth-First Search (DFS) u 4/5 1/8 x DFS(G): - construct a forest v y 3/6 2/7 w 9/12 10/11 z

  46. Depth-First Search (DFS) u 4/5 1/8 x DFS(G): - Initialization: O(|V|) - Traversing vertices: O(|V|) - Scanning adjacent vertices: O(|E|) => total running time: O(|V| + |E|) v y 3/6 2/7 w 9/12 10/11 z

  47. Topological Sorting m n o s q r t u m n q o s r u t

  48. Topological Sorting Brute-Force way 1. Find a vertex without edges. 2. Put it onto the front of the list, and remove it from G. 3. Remove all edges to the removed edge. 4. Repeat 1~3. m n o s q r t u O(|V|2 + |V||E|) Or O(|V|2)

  49. Topological Sorting Using DFS 1. Call DFS(G) 2. When a vertex is finished, put it onto the front of the list. m n o s q r t u O(|V| + |E|)

  50. Topological Sorting v enters the list before u? u Using DFS 1. Call DFS(G) 2. When a vertex is finished, put it onto the front of the list. v 1) v is white: d[u] < d[v] < f[u] 2) v is black: f[v] < d[u] 3) v is gray: d[v] < d[u] < f[v] At d[u]:

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