Breadth-First

Search

L0A

L1

L2

Breadth-First Search

© 2004

drich, Tamassia

B C D

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Breadth-First

Search

(§ 12.3.3)Breadth-first

searchBFS

on a graph

ith n(BFS) is a

general

vertices and

m edgesniqu

efor traversing

s O(n + m )

timegrap

h BFS

can

be furtherBF

Straversal extended to

solveothergrap

hG

graph

problems

Visits all the

rtices

and Find and report

aedges of G

�with the minimum

Determines whether

G isnumber of edges

connected between two

givenComputes the connected vertice

scomponents

of GComputes a

spanning

Find a simple cycle, ifforest of

Gthere is one

Breadth-First Search

© 2004

drich, Tamassia

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BFS

Algorithm Algorithm BFS(G, s)Thealgorithm uses

a L0 ← new empty sequencemechanism for setting and L0.insertLast(s)getting

“labels”

of vertices bel(s, VISITED)an

dedges i ← 0

Algorithm BFS(G) while ¬Li.isEmpty()Input graph G Li +1 ← new empty sequenceOutput labeling of the edges for all v ∈ Li.elements()

and tition of the for e ∈ G.incidentEdges(v)vertices of G if getLabel(e) = UNEXPLORED

← opposite(v,e)for all u ∈if getLabel(w) = UNEXPLOREDsetLabel(u, UNEXPLORED)

setLabel(e, DISCOVERY)for all e ∈ G.edges() setLabel(w, VISITED)

setLabel(e, UNEXPLORED)Li +1.insertLast(w)

for all v ∈ G.vertices()

elseif getLabel(v) = UNEXPLORED

setLabel(e, CROSS)BFS(G, v) i ← i

+1Breadth-First Search

© 2004

drich, Tamassia

Lof L

se tLa bel (s,

Li

∈par all e

=G.v erti ces () w ←

∈w,

Li∈

e,i

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Example L0

Aunexplored vertex

A

L1visited vertex

AB C D

unexplored edgescovery

edgeE F

cross

edge

L0 L0A A

L1 L1B C D B C D

E F E F

Breadth-First Search

© 2004

drich, Tamassia

L

di

L L

Goo

4

Example

(cont.)L0

L1 L1

L0A A

L1

L2 L2

Breadth-First Search

© 2004

drich, Tamassia

L0

A A

B C D B C D

L2E F E F

L0

L1

B C D B C D

E F E F

Goo

5

Example

(cont.)L0 L0

A

L1 L1B C D

L2

E F E F

L0A

L1B C D

L2

E F

Breadth-First Search

© 2004

drich, Tamassia

LA

B C D

L L2

L

L

Goo

6

PropertiesNotatio

n Gs: connected component of sPropert

yB C D

s) visits

all the

vertices and

BFS(G,edges of Gs

E FProperty The discovery

edgeseled bys) form

aspanning

tree TsBFS(G, L

Property L1ac

hvertex

v in Li B C DThe h of Ts from s to v has

iedgesEvery path from

s to v in Gs hasi edges

Breadth-First Search

© 2004

drich, Tamassia

A

1s)

2lab

s)of Gs

0 A

3For e v

� pat

h toL2

� s in at E Fleast i

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Analysis Setting/getting a

vertex/edgelabel

takes

O(1) timeEach vertex is labeled

twiceonce

UNEXPLOREDonc

eVISITEDEach edge is

labeledtwiceonc

eUNEXPLOREDonc

eDISCOVERY or CROSSEach vertex is

insertedonce into a

sequence Li

Method incidentEdges is called once

for each

vertexBFS

runs+ m) time provided the graph

Orepresented by the adjacency

list structureRecall that Σv deg(v) = 2m�

Breadth-First Search

© 2004

drich, Tamassia

O

� as

� as

� as

� as or

in O (n + is

Goo

8

Applications

Using the template

method pattern,

we canspecialize the BFS traversal of a

graphG to

solve

the

following

problems

in O(n + m)

timeCompute the connected

components ofG

Compute a spanning forest of GFind a simple cycle

in G, or report that

G is a

�forestGiven

two

vertices of

G, find a

path in G betweenthe

mthe

nimum number

edges, orreport that no

suchpath exists

Breadth-First Search

© 2004

drich, Tamassia

�

�

� Gwith mi of

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9

DFS

vs.

BFSApplicatio

nsDFS

BFSSpanning forest,

connected√

components, paths, cyclesShortest pathsBiconnected

components

√

L0A A

L1

L2

DFS

BFSBreadth-First

Search10©

2004drich, Tamassia

√

√

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DFS

vs.

BFS

(cont.)Bac

kedge (v,w)

s edge

,w)ancestor of v

in the same level asthe

treeof discover

ythe next level inedge

sthe tree

of discoveryedge

sL0

L1

L2

DFS

BFS

Breadth-First Search

11© 2004

drich, Tamassia

Cross (v ,w

� w is an � w is inv or in

A A

B C D B C D

E F E F

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