Elementary Graph Algorithms Comp 122, Fall 2004

BFS(G,s) 1.for each vertex u in V[G] – {s} 2do color[u]  white 3 d[u]   4  [u]  null 5color[s]  gray 6d[s]  0 7  [s]  null 8Q   9enqueue(Q,s) 10while Q   11do u  dequeue(Q) 12for each v in Adj[u] 13do if color[v] = white 14then color[v]  gray 15 d[v]  d[u]  [v]  u 17 enqueue(Q,v) 18color[u]  black BFS(G,s) 1.for each vertex u in V[G] – {s} 2do color[u]  white 3 d[u]   4  [u]  null 5color[s]  gray 6d[s]  0 7  [s]  null 8Q   9enqueue(Q,s) 10while Q   11do u  dequeue(Q) 12for each v in Adj[u] 13do if color[v] = white 14then color[v]  gray 15 d[v]  d[u]  [v]  u 17 enqueue(Q,v) 18color[u]  black Comp 122, Fall 2004 white: undiscovered gray: discovered black: finished Q: a queue of discovered vertices color[v]: color of v d[v]: distance from s to v  [u]: predecessor of v Example: animation.

Example (BFS) Comp 122, Fall 2004  0       r s t u v w x y Q: s 0

Example (BFS) Comp 122, Fall      r s t u v w x y Q: w r 1 1

Example (BFS) Comp 122, Fall  2  t u x y Q: r t x  r s v w

Example (BFS) Comp 122, Fall 2004   2 u v y Q: t x v t r s x w

Example (BFS) Comp 122, Fall 2004  3 u y Q: x v u t r s x w 2 v

Example (BFS) Comp 122, Fall y Q: v u y t r s 2 3 u x w v

Example (BFS) Comp 122, Fall 2004 Q: u y y 2 2 t r s 2 3 u x w v

Example (BFS) Comp 122, Fall 2004 Q: y 3 3 y 2 2 t r s 2 3 u x w v

Example (BFS) Comp 122, Fall 2004 Q:  3 y 2 2 t r s 2 3 u x w v

Example (BFS) Comp 122, Fall r s t u v w x y BFS Tree

DFS DFS(G) 1. for each vertex u  V[G] 2. do color[u]  white 3.  [u]  NULL 4. time  0 5. for each vertex u  V[G] 6. do if color[u] = white 7. then DFS-Visit(u) DFS(G) 1. for each vertex u  V[G] 2. do color[u]  white 3.  [u]  NULL 4. time  0 5. for each vertex u  V[G] 6. do if color[u] = white 7. then DFS-Visit(u) Comp 122, Fall 2004 Uses a global timestamp time. DFS-Visit(u) 1.color[u]  GRAY 2.time  time d[u]  time 4. for each v  Adj[u] 5. do if color[v] = WHITE 6. then  [v]  u 7. DFS-Visit(v) 8. color[u]  BLACK 9.time  time f[u]  time DFS-Visit(u) 1.color[u]  GRAY 2.time  time d[u]  time 4. for each v  Adj[u] 5. do if color[v] = WHITE 6. then  [v]  u 7. DFS-Visit(v) 8. color[u]  BLACK 9.time  time f[u]  time

Example (DFS) Comp 122, Fall / u v w x y z

Example (DFS) Comp 122, Fall / 2/ u v w x y z

Example (DFS) Comp 122, Fall / 3/ 2/ u v w x y z

Example (DFS) Comp 122, Fall / 4/ 3/ 2/ u v w x y z

Example (DFS) Comp 122, Fall / 4/ 3/ 2/ u v w x y z B

Example (DFS) Comp 122, Fall / 4/5 3/ 2/ u v w x y z B

Example (DFS) Comp 122, Fall / 4/5 3/6 2/ u v w x y z B

Example (DFS) Comp 122, Fall / 4/5 3/6 2/7 u v w x y z B

Example (DFS) Comp 122, Fall / 4/5 3/6 2/7 u v w x y z B F

Example (DFS) Comp 122, Fall /8 4/5 3/6 2/7 u v w x y z B F

Example (DFS) Comp 122, Fall /8 4/5 3/6 2/7 9/ u v w x y z B F

Example (DFS) Comp 122, Fall /8 4/5 3/6 2/7 9/ u v w x y z B F C

Example (DFS) Comp 122, Fall /8 4/5 3/6 10/ 2/7 9/ u v w x y z B F C

Example (DFS) Comp 122, Fall /8 4/5 3/6 10/ 2/7 9/ u v w x y z B F C B

Example (DFS) Comp 122, Fall /8 4/5 3/6 10/11 2/7 9/ u v w x y z B F C B

Example (DFS) Comp 122, Fall /8 4/5 3/6 10/11 2/7 9/12 u v w x y z B F C B