BreadthFirstSearch

(1)

(2)

Applications

•

Find shortest path between vertices

•

Find if/where an element is in a graph

•

Several algorithms build upon BFS

–

Djikstra’s algorithm

–

Primm’s algorithm

–

Ford-Fulkersons algorithm

(3)

Basic Algorithm

•

Start at some vertex

•

Search for another vertex

•

Search by looking at closest vertices

•

Continue search by looking at the next closest

vertices

(4)

Algorithm Details

•

Graph can be represented as an adjacency matrix

•

In addition each vertex has the following

–

Parent vertex

–

Distance from source

–

State

•

Undiscovered

•

Discovered_A

– with at least one undiscovered adjacent vertex

•

Discovered_N

– with no undiscovered adjacent vertices

(5)

Algorithm Details

•

Maintain a FIFO queue of visited vertices

•

Add the source vertex to the queue

•

While the queue is not empty or desired vertex

is not found

–

Remove first vertex (U) in the queue

–

For each adjacent vertex (V) to vertex U

•

If V has not been discovered

–

Set state to Discovered_A

–

Set distance to U.distance + 1

(6)

Example Notes

•

Vertex State

–

Undiscovered = White

–

Discovered_A = Yellow

–

Discovered_N = Black

•

Edge State

–

Unvisited = Black

–

Visited = Yellow

(7)

Example 1 Notes

•

Source vertex = A

•

Searching to find shortest path to all other

(8)

Example 1

8

B

A

D

C

E

F

G

H

J

I

K

L

(9)

Example 1

Vertex

Parent

Vertex

From Source

Distance

State

A NULL NULL Undiscovered

B NULL NULL Undiscovered

C NULL NULL Undiscovered

D NULL NULL Undiscovered

E NULL NULL Undiscovered

F NULL NULL Undiscovered

G NULL NULL Undiscovered

H NULL NULL Undiscovered

I NULL NULL Undiscovered

J NULL NULL Undiscovered

K NULL NULL Undiscovered

(10)

Example 1

10

B

A

D

C

E

F

G

H

J

I

K

L

Queue

A

• Set state of A to Discovered_A • Set parent of A to null

(11)

Example 1

Vertex

Parent

Vertex

From Source

Distance

State

A NULL 0 Discovered_A

B NULL NULL Undiscovered

(12)

Example 1

12

B

A

D

C

E

F

G

H

J

I

K

L

Queue

(13)

Example 1

B

A

D

C

E

F

G

H

J

I

K

L

Queue

B

• Visit vertex B

• Add vertex B to the queue • Set parent of vertex B to A

(14)

Example 1

14

Vertex

Parent

Vertex

From Source

Distance

State

B A 1 Discovered_A

(15)

Example 1

B

A

D

C

E

F

G

H

J

I

K

L

Queue

C B

• Visit vertex C

• Add vertex C to the queue • Set parent of vertex C to A

(16)

Example 1

16

Vertex

Parent

Vertex

From Source

Distance

State

B A 1 Discovered_A

C A 1 Discovered_A

(17)

Example 1

B

A

D

C

E

F

G

H

J

I

K

L

Queue

D C B

• Visit vertex D

• Add vertex D to the queue • Set parent of vertex D to A

(18)

Example 1

18

Vertex

Parent

Vertex

From Source

Distance

State

B A 1 Discovered_A

C A 1 Discovered_A

D A 1 Discovered_A

(19)

Example 1

B

A

D

C

E

F

G

H

J

I

K

L

Queue

D C B

(20)

Example 1

20

Vertex

Parent

Vertex

From Source

Distance

State

A NULL 0 Discovered_N

B A 1 Discovered_A

C A 1 Discovered_A

D A 1 Discovered_A

(21)

Example 1

B

A

D

C

E

F

G

H

J

I

K

L

Queue

D C

(22)

Example 1

22

B

A

D

C

E

F

G

H

J

I

K

L

Queue

E D C

• Visit vertex E

• Add vertex E to the queue • Set parent of vertex E to B

• Set distance of vertex E to distance of B + 1

(23)

Example 1

Vertex

Parent

Vertex

From Source

Distance

State

B A 1 Discovered_A

C A 1 Discovered_A

D A 1 Discovered_A

E B 2 Discovered_A

(24)

Example 1

24

B

A

D

C

E

F

G

H

J

I

K

L

Queue

F E D C

• Visit vertex F

• Add vertex F to the queue • Set parent of vertex F to B

• Set distance of vertex F to distance of B + 1

(25)

Example 1

Vertex

Parent

Vertex

From Source

Distance

State

B A 1 Discovered_A

C A 1 Discovered_A

D A 1 Discovered_A

E B 2 Discovered_A

F B 2 Discovered_A

(26)

Example 1

26

B

A

D

C

E

F

G

H

J

I

K

L

Queue

F E D C

(27)

Example 1

Vertex

Parent

Vertex

From Source

Distance

State

B A 1 Discovered_N

C A 1 Discovered_A

D A 1 Discovered_A

E B 2 Discovered_A

F B 2 Discovered_A

(28)

Example 1

28

B

A

D

C

E

F

G

H

J

I

K

L

Queue

F E D

(29)

Example 1

B

A

D

C

E

F

G

H

J

I

K

L

Queue

F E D

• Visit vertex F

(30)

Example 1

30

B

A

D

C

E

F

G

H

J

I

K

L

Queue

F E D

(31)

Example 1

Vertex

Parent

Vertex

From Source

Distance

State

B A 1 Discovered_N

C A 1 Discovered_N

D A 1 Discovered_A

E B 2 Discovered_A

F B 2 Discovered_A

(32)

Example 1

32

B

A

D

C

E

F

G

H

J

I

K

L

Queue

F E

(33)

Example 1

B

A

D

C

E

F

G

H

J

I

K

L

Queue

G F E

• Visit vertex G

• Add vertex G to the queue • Set parent of vertex G to D

(34)

Example 1

34

Vertex

Parent

Vertex

From Source

Distance

State

B A 1 Discovered_N

C A 1 Discovered_N

D A 1 Discovered_A

E B 2 Discovered_A

F B 2 Discovered_A

G D 2 Discovered_A

(35)

Example 1

B

A

D

C

E

F

G

H

J

I

K

L

Queue

H G F E

• Visit vertex H

• Add vertex H to the queue • Set parent of vertex H to D

(36)

Example 1

36

Vertex

Parent

Vertex

From Source

Distance

State

B A 1 Discovered_N

C A 1 Discovered_N

D A 1 Discovered_A

E B 2 Discovered_A

F B 2 Discovered_A

G D 2 Discovered_A

H D 2 Discovered_A

(37)

Example 1

B

A

D

C

E

F

G

H

J

I

K

L

Queue

H G F E

(38)

Example 1

38

Vertex

Parent

Vertex

From Source

Distance

State

B A 1 Discovered_N

C A 1 Discovered_N

D A 1 Discovered_N

E B 2 Discovered_A

F B 2 Discovered_A

G D 2 Discovered_A

H D 2 Discovered_A

(39)

Example 1

B

A

D

C

E

F

G

H

J

I

K

L

Queue

H G F

(40)

Example 1

40

B

A

D

C

E

F

G

H

J

I

K

L

Queue

H G F

(41)

Example 1

Vertex

Parent

Vertex

From Source

Distance

State

B A 1 Discovered_N

C A 1 Discovered_N

D A 1 Discovered_N

E B 2 Discovered_N

F B 2 Discovered_A

G D 2 Discovered_A

H D 2 Discovered_A

(42)

Example 1

42

B

A

D

C

E

F

G

H

J

I

K

L

Queue

H G

(43)

Example 1

B

A

D

C

E

F

G

H

J

I

K

L

Queue

I H G

• Visit vertex I

• Add vertex I to the queue • Set parent of vertex I to F

(44)

Example 1

44

Vertex

Parent

Vertex

From Source

Distance

State

B A 1 Discovered_N

C A 1 Discovered_N

D A 1 Discovered_N

E B 2 Discovered_N

F B 2 Discovered_A

G D 2 Discovered_A

H D 2 Discovered_A

I F 3 Discovered_A

(45)

Example 1

B

A

D

C

E

F

G

H

J

I

K

L

Queue

I H G

(46)

Example 1

46

Vertex

Parent

Vertex

From Source

Distance

State

B A 1 Discovered_N

C A 1 Discovered_N

D A 1 Discovered_N

E B 2 Discovered_N

F B 2 Discovered_N

G D 2 Discovered_A

H D 2 Discovered_A

I I 3 Discovered_A

(47)

Example 1

B

A

D

C

E

F

G

H

J

I

K

L

Queue

I H

(48)

Example 1

48

B

A

D

C

E

F

G

H

J

I

K

L

Queue

I H

• Visit vertex I

(49)

Example 1

B

A

D

C

E

F

G

H

J

I

K

L

Queue

I H

(50)

Example 1

50

Vertex

Parent

Vertex

From Source

Distance

State

B A 1 Discovered_N

C A 1 Discovered_N

D A 1 Discovered_N

E B 2 Discovered_N

F B 2 Discovered_N

G D 2 Discovered_N

H D 2 Discovered_A

I F 3 Discovered_A

(51)

Example 1

B

A

D

C

E

F

G

H

J

I

K

L

Queue

I

(52)

Example 1

52

B

A

D

C

E

F

G

H

J

I

K

L

Queue

I

(53)

Example 1

Vertex

Parent

Vertex

From Source

Distance

State

B A 1 Discovered_N

C A 1 Discovered_N

D A 1 Discovered_N

E B 2 Discovered_N

F B 2 Discovered_N

G D 2 Discovered_N

H D 2 Discovered_N

I F 3 Discovered_A

(54)

Example 1

54

B

A

D

C

E

F

G

H

J

I

K

L

Queue

(55)

Example 1

B

A

D

C

E

F

G

H

J

I

K

L

Queue

J

• Visit vertex J

• Add vertex J to the queue • Set parent of vertex J to I

(56)

Example 1

56

Vertex

Parent

Vertex

From Source

Distance

State

B A 1 Discovered_N

C A 1 Discovered_N

D A 1 Discovered_N

E B 2 Discovered_N

F B 2 Discovered_N

G D 2 Discovered_N

H D 2 Discovered_N

I F 3 Discovered_A

J I 4 Discovered_A

(57)

Example 1

B

A

D

C

E

F

G

H

J

I

K

L

Queue

K J

• Visit vertex K

• Add vertex K to the queue • Set parent of vertex K to I

(58)

Example 1

58

Vertex

Parent

Vertex

From Source

Distance

State

B A 1 Discovered_N

C A 1 Discovered_N

D A 1 Discovered_N

E B 2 Discovered_N

F B 2 Discovered_N

G D 2 Discovered_N

H D 2 Discovered_N

I F 3 Discovered_A

J I 4 Discovered_A

K I 4 Discovered_A

(59)

Example 1

B

A

D

C

E

F

G

H

J

I

K

L

Queue

K J

(60)

Example 1

60

Vertex

Parent

Vertex

From Source

Distance

State

B A 1 Discovered_N

C A 1 Discovered_N

D A 1 Discovered_N

E B 2 Discovered_N

F B 2 Discovered_N

G D 2 Discovered_N

H D 2 Discovered_N

I F 3 Discovered_N

J I 4 Discovered_A

K I 4 Discovered_A

(61)

Example 1

B

A

D

C

E

F

G

H

J

I

K

L

Queue

K

(62)

Example 1

62

B

A

D

C

E

F

G

H

J

I

K

L

Queue

K

(63)

Example 1

Vertex

Parent

Vertex

From Source

Distance

State

B A 1 Discovered_N

C A 1 Discovered_N

D A 1 Discovered_N

E B 2 Discovered_N

F B 2 Discovered_N

G D 2 Discovered_N

H D 2 Discovered_N

I F 3 Discovered_N

J I 4 Discovered_N

K I 4 Discovered_A

(64)

Example 1

64

B

A

D

C

E

F

G

H

J

I

K

L

Queue

(65)

Example 1

B

A

D

C

E

F

G

H

J

I

K

L

Queue

L

• Visit vertex L

• Add vertex L to the queue • Set parent of vertex L to K

(66)

Example 1

66

Vertex

Parent

Vertex

From Source

Distance

State

B A 1 Discovered_N

C A 1 Discovered_N

D A 1 Discovered_N

E B 2 Discovered_N

F B 2 Discovered_N

G D 2 Discovered_N

H D 2 Discovered_N

I F 3 Discovered_N

J I 4 Discovered_N

K I 4 Discovered_A

(67)

Example 1

B

A

D

C

E

F

G

H

J

I

K

L

Queue

L

(68)

Example 1

68

Vertex

Parent

Vertex

From Source

Distance

State

B A 1 Discovered_N

C A 1 Discovered_N

D A 1 Discovered_N

E B 2 Discovered_N

F B 2 Discovered_N

G D 2 Discovered_N

H D 2 Discovered_N

I F 3 Discovered_N

J I 4 Discovered_N

K I 4 Discovered_N

(69)

Example 1

B

A

D

C

E

F

G

H

J

I

K

L

Queue

(70)

Example 1

70

B

A

D

C

E

F

G

H

J

I

K

L

Queue

(71)

Example 1

Vertex

Parent

Vertex

From Source

Distance

State

B A 1 Discovered_N

C A 1 Discovered_N

D A 1 Discovered_N

E B 2 Discovered_N

F B 2 Discovered_N

G D 2 Discovered_N

H D 2 Discovered_N

I F 3 Discovered_N

J I 4 Discovered_N

K I 4 Discovered_N

(72)

Example 1

72

B

A

D

C

E

F

G

H

J

I

K

L

Queue

(73)

Time Analysis

•

O

(V + E)

–

E = edges

–

V = vertices

–

Visit all vertices and edges exactly once

–

Initial setup =

O

(V)

–

Dequeue =

O

(E)

(74)

Space Analysis

•

O

(V)

–

All vertices in queue in the worst case

(75)

Complete

•

Complete = the algorithm will always find the

solution if the solution exists

•

BFS is complete

(76)

Optimal

•

Optimal = always finds the optimal solution

•

BFS is optimal

–

See proof in book

–

Short answer = when vertex is first visited there is

no shorter path to that vertex

•

Parent vertex is saved