Elementary Graph Algorithms

Comp 122, Fall 2004

BFS(G,s)1. for each vertex u in V[G] – {s}2 do color[u] white3 d[u] 4 [u] null5 color[s] gray6 d[s] 07 [s] null8 Q 9 enqueue(Q,s)10 while Q 11 do u dequeue(Q)12 for each v in Adj[u]13 do if color[v] = white14 then color[v]

gray15 d[v] d[u] + 116 [v] u17 enqueue(Q,v)18 color[u] black

BFS(G,s)1. for each vertex u in V[G] – {s}2 do color[u] white3 d[u] 4 [u] null5 color[s] gray6 d[s] 07 [s] null8 Q 9 enqueue(Q,s)10 while Q 11 do u dequeue(Q)12 for each v in Adj[u]13 do if color[v] = white14 then color[v]

gray15 d[v] d[u] + 116 [v] u17 enqueue(Q,v)18 color[u] black

Comp 122, Fall 2004

white: undiscoveredgray: discoveredblack: finished

Q: a queue of discovered verticescolor[v]: color of vd[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 2004

1 0

1

r s t u

v w x y

Q: w r 1 1

Example (BFS)

Comp 122, Fall 2004

2

2 t u

x y

Q: r t x 1 2 2

1 0

1

r s

v w

Example (BFS)

Comp 122, Fall 2004

2

u

v y

Q: t x v 2 2 2

2

2

t

1 0

1

r s

xw

Example (BFS)

Comp 122, Fall 2004

3

u

y

Q: x v u 2 2 3

2

2

t

1 0

1

r s

xw

2

v

Example (BFS)

Comp 122, Fall 2004

3

y

Q: v u y 2 3 3

2

2

t

1 0

1

r s

2

3

u

xwv

Example (BFS)

Comp 122, Fall 2004

Q: u y 3 3

3

y

2

2

t

1 0

1

r s

2

3

u

xwv

Example (BFS)

Comp 122, Fall 2004

Q: y 3

3

y

2

2

t

1 0

1

r s

2

3

u

xwv

Example (BFS)

Comp 122, Fall 2004

Q:

3

y

2

2

t

1 0

1

r s

2

3

u

xwv

Example (BFS)

Comp 122, Fall 2004

1 0

1 2 3

2 3

2

r s t u

v w x y

BFS Tree

DFSDFS(G)1. for each vertex u V[G]2. do color[u] white3. [u] NULL4. time 05. for each vertex u V[G]6. do if color[u] = white7. then DFS-Visit(u)

DFS(G)1. for each vertex u V[G]2. do color[u] white3. [u] NULL4. time 05. for each vertex u V[G]6. do if color[u] = white7. then DFS-Visit(u)

Comp 122, Fall 2004

Uses a global timestamp time.

DFS-Visit(u)

1. color[u] GRAY

2. time time + 1

3. 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 + 1

10. f[u] time

DFS-Visit(u)

1. color[u] GRAY

2. time time + 1

3. 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 + 1

10. f[u] time

Example (DFS)

Comp 122, Fall 2004

1/

u v w

x y z

Example (DFS)

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u v w

x y z

Example (DFS)

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3/

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Example (DFS)

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4/ 3/

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u v w

x y z

Example (DFS)

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4/ 3/

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B

Example (DFS)

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4/5 3/

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u v w

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B

Example (DFS)

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4/5 3/6

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u v w

x y z

B

Example (DFS)

Comp 122, Fall 2004

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4/5 3/6

2/7

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x y z

B

Example (DFS)

Comp 122, Fall 2004

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4/5 3/6

2/7

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x y z

BF

Example (DFS)

Comp 122, Fall 2004

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2/7

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x y z

BF

Example (DFS)

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BF

Example (DFS)

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BF C

Example (DFS)

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BF C

Example (DFS)

Comp 122, Fall 2004

1/8

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2/7 9/

u v w

x y z

BF C

B

Example (DFS)

Comp 122, Fall 2004

1/8

4/5 3/6 10/11

2/7 9/

u v w

x y z

BF C

B

Example (DFS)

Comp 122, Fall 2004

1/8

4/5 3/6 10/11

2/7 9/12

u v w

x y z

BF C

B