bfsanddfs (1) bfs dfs

7/27/2019 BFSandDFS (1) bfs dfs

1/20

BFS and DFS 1

BFS and DFS

For many applications, one must systematically

search through a graph to determine the structure

of the graph.

Two common elementary algorithms for

tree-searching are

Breadth-first search (BFS), and

Depth-first search (DFS).

Both of these algorithms work on directed or

undirected graphs.

Many advanced graph algorithms are based on theideas of BFS or DFS.

Each of these algorithms traverses edges in the

graph, discovering new vertices as it proceeds.

The difference is in the order in which each

algorithm discovers the edges.

Instead of talking about the differences, we will

just look at each algorithm in turn.


2/20

BFS and DFS 2

Breadth-First Search

Let

be a graph.

Let

be a special vertex called the source.

A breadth-first search of

will:

Determine all vertices reachable from .

Determine the distance (that is, the fewest

number of edges) from to all vertices

reachable.

Determine a shortest path from to all vertices

reachable. Construct a breadth-first tree rooted at .

That is, a tree whose edges form the shortest

paths from to all vertices reachable.

Breadth-first search is so called because it locates

all vertices of distance

before it locates any of

distance .


3/20

BFS and DFS 3

How BFS Works

For each node in the graph we will need to store

the list of adjacent vertices,

the distance from to (

),

predecessor of (

), and

the color of (

).

The color of a vertex indicates the status of a

vertex:

A white vertex has not been discovered.

A gray vertex has been discovered, but it may

have undiscovered neighbors.

A black vertex has been discovered, and it has

no undiscovered neighbors.

The algorithm starts with

"

,

! # $ %

, and

! ' (

) 1

for 3

, and

4

,

# $ %

, and

5 6 8 @

.


4/20

BFS and DFS 4

How BFS Works Continued

Gray vertices are stored in a queue, B .

When the algorithm begins, is on the queue.

The algorithm proceeds as follows:

Remove the first element, , from the queue

( C

1 D F 1 F 1

B

).

For each neighbor@

of I

If@

is white.P Color

@

gray ( @ 5 6 8 @

).P Let the distance of

@

be one more than the

distance of

(

@

).P Let

be the predecessor of

@

(

@

).

P Place@

at the end of the queue

( V

D F 1 F 1

B

@

).I

If@

is gray or black, do nothing.

Color black.

Repeat until the queue is empty.

Now lets look at the whole algorithm.


5/20

BFS and DFS 5

BFS Algorithm

Let

be a graph, and

be some

vertex.

The breadth-first search algorithm is as follows:

BFS(G,s)

ForAll u in V-{s}

c(u)=white

d(u)=Max_Int

p(u)=NIL

c(s)=gray

d(s)=0

p(s)=NIL

Enqueue(Q,s)while(NotEmpty(Q))

u=Dequeue(Q)

ForAll v adjacent to u

if c(v) = white then

c(v)=gray

d(v)=d(u)+1

p(v)=uEnqueue(Q,v)

color(u) = black


6/20

BFS and DFS 6

BFS Example

e

s

a

d

cb

0

Q=[s] b s,a,c,e inf nil white

a s,b,c,e inf nil white

d e inf nil white

e a,b,c,d inf nil white

c b,a,e inf nil white

adjacent dp

colorv

e

s

a

d

cb

0adjacent d p colorv

1

1

2

2

a s,b,c,e 1 s black

c b,a,e 2 a gray

e a,b,c,d 2 a gray

e

s

a

d

cb


1

1

2

2

a s,b,c,e 1 s black

Q=[b,c,e]

c b,a,e 2 a gray

e a,b,c,d 2 a gray

e

s

a

d

cb

0

d e inf nil white

e a,b,c,d inf nil white

c b,a,e inf nil white

adjacent d p colorv

1

1Q=[a,b]

a s,b,c,e 1 s gray

Q=[c,e]

d e inf nil white

d e inf nil white

b s,a,c,e 1 s gray

b s,a,c,e 1 s gray

b s,a,c,e 1 s black

s a,b 0 nil gray

s a,b 0 nil black

s a,b 0 nil black

s a,b 0 nil black


7/20

BFS and DFS 7

BFS Example Continued

e

s

a

d

cb

0 adjacent dp

colorv

1

1

2

2

a s,b,c,e 1 s black

c b,a,e 2 a gray

e a,b,c,d 2 a gray

Q=[c,e] b s,a,c,e 1 s black

e

s

a

d

cb

0

adjacent d p colorv

1

1

2

2

a s,b,c,e 1 s black

e a,b,c,d 2 a gray

c b,a,e 2 a black

b s,a,c,e 1 s black

e

s

a

d

cb


1

1

2

2

a s,b,c,e 1 s black

b s,a,c,e 1 s black

3

e a,b,c,d 2 a black

c b,a,e 2 a black

e

s

a

d

cb


1

1

2

2

a s,b,c,e 1 s black

b s,a,c,e 1 s black

3

e a,b,c,d 2 a black

c b,a,e 2 a black

Q=[d]

Q=[]

Q=[e]

d e inf nil white

d e 3 e grey

d e 3 e black

d e inf nil white

s a,b 0 nil black

s a,b 0 nil black

s a,b 0 nil black

s a,b 0 nil black


8/20

BFS and DFS 8

BFS Summary

There are several things we should ask about BFS:

Why does BFS discover every vertex

reachable from ?

Why does BFS give the distance from

to

,for every reachable vertex ?

Why/How does BFS give a shortest path from

to , for every reachable vertex ?

How do we construct a breadth-first tree after

we have executed BFS?

What is the complexity of BFS?

It is not hard to convince yourself that the first 3

are true, although proving them is a little harder.

It is not too difficult to see that the complexity isY

.

We wont prove any of these things explicitly.


9/20

BFS and DFS 9

Depth-First Search Let

be a graph.

A depth-first search of

will

Find every vertex in

.

Timestamp every vertex withI

the first time it was discovered, andI

the last time it is examined.

Construct a depth-first forest. That is, a

partition of

into depth-first trees.

The timestamps are used by other graph

algorithms.

Depth-first search is so called because it continues

along a path until it finds no new vertices, at

which point it backtracks.

Unlike BFS, DFS has no source vertex.

DFS starts searching from every vertex of the

graph eventually. We will see what this means.


10/20

BFS and DFS 10

How DFS Works

For each node in the graph we will need to store

the list of adjacent vertices,

the discover timestamp of (

).

the final timestamp of ( a

).

predecessor of

(

), and the color of (

).

The color of a vertex indicates the status of a

vertex:

A white vertex has not been discovered.

A gray vertex has been discovered, but it mayhave undiscovered neighbors.

A black vertex has been discovered, and it has

no undiscovered neighbors.

The algorithm starts with

! # $ %

, and

b ' (

) 1 for all

.

Since both

and a

are guaranteed to be set

in the algorithm, we need not initialize them.


11/20

BFS and DFS 11

More of How DFS Works

Unlike BFS, DFS is recursive.

When DFS visits a white node

, it does

the following:

Color gray (

5 6 8 @

).

Set discover time of

to the current time

(

) f 1 ).

Increment current time ( ) f 1

) f 1 ).

For all@

adjacent to I

If@

is white,P Set to be the predecessor of

@

(

@

).P Recursively visit node

@

.

Color

black (

p q 8

s).

Set finish time of to the current time

( a

) f 1 ).

Increment current time ( ) f 1

) f 1 ).

Notice that ) f 1 has to be a global variable.

We will now see the algorithm in its entirety.


12/20

BFS and DFS 12

DFS Algorithm

Let

be a graph.

The depth-first algorithm is as follows:

DFS(G)

ForAll u in V

c(u)=white

p(u)=NILtime=0

ForAll u in V

If c(u)=white

DFS-Visit(u)

DFS-Visitis as follows:

DFS-Visit(u)c(u)=gray

d(u)=time++

ForAll v adjacent to u

If c(v)=white

p(v)=u

DFS-Visit(v)

c(u)=blackf(u)=time++

An example should help make things clearer.


13/20

BFS and DFS 13

DFS Example

e

f g

NIL NIL NIL NIL

W W W W

v

p

c

d

f

a b c d e f g ha

b c d

h

1/

NIL

2/

2

G G

1 3

4/3/

e

f g

W

NIL NIL NIL NIL NIL

W W W W

vp

c

d

f

a b c d e f g h

a

b c d

h

1/NIL

2/

2

a

G

b

1 3

3/

4a b c

e

f g

W W

NILNIL NIL NIL NIL NIL

W W W W

v

p

c

d

f

a b c d e f g ha

b c d

h

1/

NIL

1

2/

2

a

e

f g

time=1

W W W

NIL NILNIL NIL NIL NIL NIL

W W W W

v

p

c

d

f

a b c d e f g ha

b c d

h

1/

NIL

1

G

time=2

time=3

time=4

G

G

G

G

G

G


14/20

BFS and DFS 14

e

f g

NIL NIL NIL NIL

W W W W

v

p

c

df

a b c d e f g ha

b c d

h

1/

NIL

2/

2

G G

1 3

4/3/

4

a b c

time=4

G G

e

f g

NIL NIL NIL

W W W

v

p

c

d

f

a b c d e f g ha

b c d

h

1/

NIL

2/

2

G G

1 3

4/3/

4

a b c

G

5/

time=5 5

d

G G

e

f g

NIL NIL NIL

W W W

v

p

c

d

f

a b c d e f g ha

b c d

h

1/

NIL

2/

2

G G

1 3

4/3/

4

a b c

G

5

d

5/6

6

G

time=6

B

e

f g

NIL NIL NIL

W W W

v

p

c

d

f

a b c d e f g ha

b c d

h

1/

NIL

2/

2

G G

1 3

3/

4

a b c

G

5

d

5/6

6

B4/7

time=7

7

B

e

f g

NIL NIL

W W

v

p

c

d

f

a b c d e f g ha

b c d

h

1/

NIL

2/

2

G G

1 3

3/

4a b c

G

5d

5/6

6

B

7

B4/7

8/ time=8

c

8

G


15/20

BFS and DFS 15

e

f g

NIL NIL

W W

v

p

c

d

f

a b c d e f g ha

b c d

h

1/

NIL

2/

2

G G

1 3

3/

4

a b c

G

5

d

5/6

6

B

7

B4/7

8/ time=8

c

8

G

e

f g

NIL NIL

W W

v

p

c

d

f

a b c d e f g ha

b c d

h

1/

NIL

2/

2

G G

1 3

3/

4

a b c

G

5

d

5/6

6

B

7

B4/7

c

88/9 time=99

B

e

f g

NIL NIL

W W

v

p

c

d

f

a b c d e f g ha

b c d

h

1/

NIL

2/

21 3 4

a b c

5

d

5/6

67

4/7

c

88/99

time=10

3/10 B

10

B B BGG

e

f g

NIL NIL

W W

v

p

c

df

a b c d e f g ha

b c d

h

1/

NIL

2/

2

G G

1 3

3/

4

a b c

G

5

d

5/6

6

B

7

B4/7

8/ time=8

c

8

G

e

f g

NIL NIL

W W

v

p

c

d

f

a b c d e f g ha

b c d

h

1/

NIL

21 3 4a b c

5d

5/6

67

4/7

c

88/99

3/10 B

10

B B BG

time=11

2/11 B

11


16/20

BFS and DFS 16

e

f g

NIL NIL

W W

v

p

c

df

a b c d e f g ha

b c d

h

1/

NIL

21 3 4

a b c

5

d

5/6

67

4/7

c

88/99

3/10 B

10

B B BG

time=11

2/11 B

11

e

f g

NIL NIL

W W

v

p

c

d

f

a b c d e f g ha

b c d

hNIL

21 3 4

a b c

5

d

5/6

67

4/7

c

88/99

3/10 B

10

B B BB

11time=12

B2/11

1/12

12

f g

NIL NIL

W

v

p

c

d

f

a b c d e f g ha

b c d

hNIL

21 3 4

a b c

5

d

5/6

67

4/7

c

88/99

3/10 B

10

B B BB

11

B2/11

1/12

12

13e

time=1313/

G

f g

NIL

v

p

c

d

f

a b c d e f g ha

b c d

hNIL

21 3 4a b c

5d

5/6

67

4/7

c

88/99

3/10 B

10

B B BB

11

B2/11

1/12

12

13e

13/

G

14time=15

14/15

B

15

f g

NIL

v

p

c

d

f

a b c d e f g ha

b c d

hNIL

21 3 4

a b c

5

d

5/6

67

4/7

c

88/99

3/10 B

10

B B BB

11

B2/11

1/12

12

13e

13/

14/

G G

14time=14

e

e


17/20

BFS and DFS 17

DFS Example Finish

f g

NIL

v

p

c

d

f

a b c d e f g ha

b c d

hNIL

21 3 4

a b c

5

d

5/6

67

4/7

c

88/99

3/10 B

10

B B BB

11

B2/11

1/12

12

13e

14

14/15

B

15

time=1616

13/16

f g

NIL

v

p

c

d

f

a b c d e f g ha

b c d

hNIL

21 3 4

a b c

5

d

5/6

67

4/7

c

88/99

3/10 B

10

B B BB

11

B2/11

1/12

12

13e

13/

G

14time=15

14/15

B

15

e

e

B

The depth-first forest for this example is

f ga

b c d

h e


18/20

BFS and DFS 18

DFS Summary

The running time of DFS isY

.

The edges

are the edges in the

depth-first forest.

As stated previously, the timestamps are useful in

other algorithms.

There are several other things that can be saidabout depth-first searching, but we wont.

Instead, we will discuss one application, the

Topological Sort.


19/20

BFS and DFS 19

Topological Sort

Let

be a directed acyclic graph.

A topological sort of

is a linear ordering of the

vertices of

such that if F is before u , then

u

F

is not an edge.

Example: Consider the following Graph:

A B C D

E F G H

I

Here is one possible topological sort:

C EA F H IG D B

Notice that no edges go from right to left.


20/20

BFS and DFS 20

Topological Sort Algorithm

Let

be a directed acyclic graph.

To topologically sort

, we use a slightly

modified version of DFS.

At the end ofDFS-Visit( F ), place F at the head of

a linked list.

When DFS is done, the linked list contains thevertices of

topologically sorted.

Example:

1/16

2/15 3/14

17/18

7/10

8/9

4/5

17/18

C

1/16

A

2/15

E

3/14

F

7/10

H

8/9

I

4/5

B

A B C D

I

GFE H6/13

11/12

11/126/13

G D

bfsanddfs (1) bfs dfs

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