graphs: representation and elementary algorithms · inkscape xfig audit-libs-python policycoreutils...

Graphs: Representation and

Elementary Algorithms

Antonio Carzaniga

Faculty of InformaticsUniversità della Svizzera italiana

April 23, 2015

© 2007 Antonio Carzaniga

Outline

Graphs: definitions

Representations

Breadth-first search

Depth-first search


Example


Example

GE

VS

TI

VD

FR

BE

OW

NW

UR

GR

NE

LU

ZG

SZ

GL

SG

JU

SO

BS

BL

AG

SH

ZH

TG AR

AL


Same Example (Better Layout)

GE VS TI

VD FR BE

OW

NW

UR GR

NE LU

ZG

SZ

GL

SG

JU

SO

BS BL AG SH

ZH

TG

AR

AL


Many Examples

Social networks: who knows who

The Web graph: which page links to which

The Internet graph: which router links to which

Citation graphs: who references whose papers

Planar graphs: which country is next to which

Well-shaped meshes: pretty pictures with triangles

Geometric graphs: who is near who

Random graphs: whichever. . .

Examples and descriptions taken from Daniel A. Spielman’s course “Graphs and Networks.”


Definitions

A graph

G = (V , E)

V is the set of vertices (also called nodes)

E is the set of edges


Definitions

A graph

G = (V , E)



◮ E ⊆ V × V , i.e., E is a relation between vertices

◮ an edge e = (u, v) ∈ V is a pair of vertices u ∈ V and v ∈ V


Definitions

A graph

G = (V , E)



◮ E ⊆ V × V , i.e., E is a relation between vertices

◮ an edge e = (u, v) ∈ V is a pair of vertices u ∈ V and v ∈ V

An undirected graph is characterized by a symmetric relation

between vertices

◮ an edge is a set e = {u, v} of two vertices


Example (2)

0/0

15/0

0/15

30/0

0/30

40/0

0/40

15/15

30/15

15/30

30/30

40/15

15/40

40/30

30/40

game A

game B

deuce

adv. A

adv. B


Example (3)

perl-HTML-Tagset

perl-HTML-Parser

perl-libwww-perl

gd

libwmf

gcgc-devel

automake

libtool

cyrus-sasl-md5

gaim

xorg-x11-drv-magictouch

xorg-x11-drivers

giflibgiflib-devel

tk tix

ruby-libsruby

bluez-gnome

bluez-utils

xorg-x11-fonts-base

xorg-x11-server-Xorg

xosd

xorg-x11-drv-i810

xorg-x11-drv-i128

xorg-x11-drv-digitaledge

xorg-x11-drv-dummy

xorg-x11-drv-glint

xorg-x11-drv-cirrus xorg-x11-drv-summa

xorg-x11-drv-s3virge

xorg-x11-drv-mutouch

xorg-x11-drv-ast

xorg-x11-drv-calcomp

xorg-x11-drv-chips

xorg-x11-drv-void

xorg-x11-drv-vga

xorg-x11-drv-hyperpen

xorg-x11-drv-s3

xorg-x11-drv-jamstudio

xorg-x11-drv-voodoo

xorg-x11-drv-ati

xorg-x11-drv-vesa

xorg-x11-drv-i740

xorg-x11-drv-spaceorb

xorg-x11-drv-nsc

synaptics

xorg-x11-drv-palmax

xorg-x11-drv-savage

xorg-x11-drv-aiptek

xorg-x11-drv-apm

xorg-x11-drv-tdfx

xorg-x11-drv-keyboard

xorg-x11-drv-neomagic

xorg-x11-drv-cyrix

xorg-x11-drv-v4l

xorg-x11-drv-citron

xorg-x11-drv-vmware

xorg-x11-drv-mouse

xorg-x11-drv-trident

xorg-x11-drv-magellan

xorg-x11-drv-tseng

xorg-x11-drv-dynapro

xorg-x11-drv-elo2300

xorg-x11-drv-sis

xorg-x11-drv-ur98

xorg-x11-drv-joystick

xorg-x11-drv-fpit

xorg-x11-drv-acecad

xorg-x11-drv-sisusb

xorg-x11-drv-microtouch

xorg-x11-drv-ark

xorg-x11-drv-dmc

xorg-x11-drv-elographics

xorg-x11-drv-evdev

xorg-x11-drv-fbdev

xorg-x11-drv-mga

xorg-x11-drv-nv

xorg-x11-drv-penmount

xorg-x11-drv-rendition

xorg-x11-drv-siliconmotion

xorg-x11-drv-tek4957

xorg-x11-drv-via

xorg-x11-drv-vmmouse

xmms-xosd

mkisofsdvd+rw-tools

gtk2

AdobeReader_enu

gail

gconf-editor

gimp

gnome-icon-theme

gnome-keyring

gtk2-devel

gtk2-engines

gtkhtml2

libbonoboui

libgcj

libglade2

libgnomecanvas

libgnomeprintui22

libgnomeui

librsvg2

libwnck

openoffice.org-core

RealPlayer

wireshark-gnome

gimp-print-plugin gimp-data-extras

redhat-artwork

java-1.4.2-gcj-compat

gjdoc

evolution-data-server

openoffice.org-writer

openoffice.org-base openoffice.org-calc

openoffice.org-draw

openoffice.org-graphicfilter

openoffice.org-impress

openoffice.org-langpack-it

lame

libgnomecups

libgnomeprint22

gtk+

gtk+-devel

gtk-engines

imlib

usbview

gnutls

libsoup

basesystem

glibc

j2sdk

jdk

ppp

glibc-devel

bind

pam

openldap

sed

lockdev

glibc-headers

xorg-x11-drv-i810-devel

ghostscript

ghostscript-fonts

ghostscript-gtk

pstoedit

tetex-fontools

transfig

inkscape

xfig

audit-libs-python

policycoreutils

libXcursor-devel

ttmkfdir

xorg-x11-fonts-truetype

xorg-x11-xfs

initscripts

dhclient

dhcpv6_client

hal

ipsec-tools

kbd

mkinitrd

openssh

portmap

privoxy

vte

xinetd

gnome-mount

pm-utils

openssh-clients

openssh-server

libdvdreadlibdvdnav

bluez-libs

xalan-j2

bsf

bsh

libxslt

libgnome

scrollkeeper

xmlsec1

xmlsec1-nss

evince

galeon

gnumeric

libgnomedb

elfutils

rpm-build

libgpg-errorlibgpg-error-devel

kakasi

namazuperl-Text-Kakasi

boostboost-devel

wireshark

gawk

autoconf

ethtool

R

e2fsprogs-libs

e2fsprogs

firefox

zip

libstdc++-devel

gcc-c++

startup-notification

gnome-desktop

metacity

perl

docbook-dtds

git-core

latex2html

lm_sensors

netpbm-progs

perl-Compress-Zlib

perl-DateManip

perl-DBI

perl-Error

perl-Parse-Yappperl-XML-Parser

subversion

swish++

w3m

xscreensaver-base

xscreensaver-extras

git-cvs

git-arch

git-svn

git-email

perl-Git

tetex-latex

subversion-perl

w3m-img

tzdata

glibc-common

m4

bison

bash

bash-completion

gpm

info

sysklogd

tar

gdb

tetex-dvips

tetex-fonts

xboard

libICE

libICE-devel

libSM-devel

dbus-python

avahi

avahi-glib

cairo

cairo-devel

pango

pango-devel

gnome-vfs2

gnome-vfs2-smb

bzip2-libs

bzip2

man

tla

tetex-eurofont

tetex-font-cm-lgc

tetex-font-kerkis

tetex-unicode

atk

atk-devel

logrotate

unzip

xmms

gkrellmms

libgomp

gcc

xerces-j2

gcc32

glib-devel

alsa-lib

alsa-lib-devel

SDL-devel

cryptsetup-luks

aspell-en

aspell

aspell-it

cvsps

zlib-devel

freetype-devel

libmng-devel

libpng-devel

libxml2-devel

libXft-devel

fontconfig-devel

glib2

GConf2

glib2-devel

glib-java

libcroco

libgsf

libnotify

libwpd

nc

shared-mime-info

notification-daemon

hsqldb

libXfixes-devel

pciutils

binutils

tcl

bind-libs

bind-utils

torcs-data-tracks-road

torcs

torcs-data

torcs-data-cars-extra

torcs-data-cars-nascar

torcs-data-tracks-dirt

torcs-data-tracks-oval

torcs-robots

tmpwatch

tetex

lcdf-typetools

coreutils

darcs

dhcp

fontconfig

guile

hicolor-icon-theme

minicom

xorg-x11-xinit

qt

urw-fonts

dejavu-lgc-fonts

bitmap-fonts

fonts-chinese

fonts-hebrew

fonts-hindi

fonts-japanese

fonts-korean

libXft

xorg-x11-fonts-miscxorg-x11-fonts-syriac

xorg-x11-fonts-Type1

SysVinit

passwd

util-linux

usermode

udev

libSM

libXt-devel

python

PyXML

rhpl

rpm-python

tkinter

yum

python-imaging

gnome-mime-data

skype

nspr

nss

libjpeg

libjpeg-devel

xorg-x11-util-macros

libXmu-devel

libXaw-devel

ed

audit-libs

shadow-utils

rpm

tcpdump

gnupg2

libart_lgpl

vim-minimal

cvs

sudo

popplerpoppler-utils

python-elementtree

openjade

libofx

gsynaptics

paps

cups

yum-metadata-parser

openssl

curl

pstack

ImageMagick

ImageMagick-c++

lvm2

mesa-libGLUmesa-libGLU-devel

dictd

dict-web1913

dmraid

mesa-libGLwmesa-libGLw-devel

pkgconfig

libX11-devel

mesa-libGL-devel

libXext-devel

libXrender-devel

libXpm-devel

gstreamer-toolsgstreamer

gstreamer-plugins-good

gstreamer-plugins-base

graphviz

gkrellmgkrellm-freq

gkrellm-volume

gkrellm-wifi

timidity++

SDL_mixer

dbus

NetworkManager-glib

psutils

mesa-libGL

xorg-x11-proto-devel

jpackage-utils

antlr

gnu-crypto

beecrypt

rpm-libs

groff

ffmpeglibquicktime

libXdmcp-devel

libXi-devel

libXinerama-devel

libXrandr-devel

libXv-devel

libXxf86dga-devel

libXxf86vm-devel

dbus-glib

crontabsanacron

libXcursor

libmng

which

htmlview

libcap

ntp

gimp-print

mktemp

grub

gzip

MAKEDEV

mgetty

openssl097a

cracklib-dicts

xorg-x11-resutils

SDL

libdvlibdv-devel

mplayermplayer-fonts

tomcat5-jsp-2.0-api

desktop-file-utils

gqview

redhat-menus

gnome-menus

xorg-x11-utils

libbonobo

perl-NKF

libXfixes

libao

vorbis-tools

grip

ORBit2

findutils

popt

imake

libraw1394libiec61883

less

calc

iptablesiptables-ipv6

perl-File-MMagic

freetype

freetype-demos

cpio

libdaemon

db4

libXfont

patch

net-tools

libvorbis

setup

filesystem

libXxf86dga

libX11

libusb

hwdata

usbutils

device-mapper

kpartx

libpng

gnuplot

maxima

libsemanage

libgcrypt

libattrlibattr-devel libacl-devel

libvolume_id

xorg-x11-server-utils

libacl

libXrandr

cracklib

esound

ncurses

ncurses-devel

libXpm

chkfontpath

zlib

netpbm

libselinux

libselinux-python

cdparanoia-libscdparanoia

libxml2

libxml2-python

vcdimager

expat

samba-common

samba-client

chkconfig

at

cyrus-sasl

iputils

xmms-libs

xmms-cdread

xmms-mp3

pinentry

setarch

maxima-runtime-gcl

cups-libs

module-init-tools

libsepol

gslgsl-devel

libXmu

xml-common

fedora-release-notes

fedora-release

livna-release

mingetty

libpcap

libXdmcp

libXau-devel

cyrus-sasl-lib

cyrus-sasl-plain

rsync

xml-commons-resolver

psmisc

libXaw

tomcat5-servlet-2.4-api

libXau

xorg-x11-drv-via-devel

dmidecode

libXinerama

wget dillo

elfutils-libs

dialog

sgml-common

opensp

libtiff

libtiff-devel

libstdc++

libXxf86vm

rcs

glib

libXi

libXt

libXv

grep

bc

irda-utils

krb5-libs

tcsh

libXext

kernel-headers

tetex-afm

libgcc

libXrender

file

python-sqlite

elfutils-libelf

libogg

readline

diffutils

cpp

nash

audiofilearts


Graph Representation

How do we represent a graph G = (E ,V) in a computer?




Adjacency-list representation

V = {1,2, . . . |V |}

G consists of an array Adj

A vertex u ∈ V is represented by an element in the array Adj





V = {1,2, . . . |V |}

G consists of an array Adj

A vertex u ∈ V is represented by an element in the array Adj

Adj[u] is the adjacency list of vertex u

◮ the list of the vertices that are adjacent to u

◮ i.e., the list of all v such that (u, v) ∈ E


Example

1 2 3 4

5 6 7 8

9 10 11 12


Example

1 2 3 4

5 6 7 8

9 10 11 12Adj

1 5 6

2 3 7

3 4 7

4 7

5 9

6 2 7 9

7 8 10 11

8 11 12

9 10

10

11 12

12


Using the Adjacency-List

Adj

1 5 6

2 3 7

3 4 7

4 7

5 9

6 2 7 9

7 8 10 11

8 11 12

9 10

10

11 12

12



Accessing a vertex u? Adj

1 5 6

2 3 7

3 4 7

4 7

5 9

6 2 7 9

7 8 10 11

8 11 12

9 10

10

11 12

12



Accessing a vertex u? O(1)

◮ optimal

Adj

1 5 6

2 3 7

3 4 7

4 7

5 9

6 2 7 9

7 8 10 11

8 11 12

9 10

10

11 12

12




◮ optimal

Iteration through V ?

Adj

1 5 6

2 3 7

3 4 7

4 7

5 9

6 2 7 9

7 8 10 11

8 11 12

9 10

10

11 12

12




◮ optimal

Iteration through V ? Θ(|V |)◮ optimal

Adj

1 5 6

2 3 7

3 4 7

4 7

5 9

6 2 7 9

7 8 10 11

8 11 12

9 10

10

11 12

12




◮ optimal


Iteration through E?

Adj

1 5 6

2 3 7

3 4 7

4 7

5 9

6 2 7 9

7 8 10 11

8 11 12

9 10

10

11 12

12




◮ optimal


Iteration through E? Θ(|V | + |E|)◮ okay (not optimal)

Adj

1 5 6

2 3 7

3 4 7

4 7

5 9

6 2 7 9

7 8 10 11

8 11 12

9 10

10

11 12

12




◮ optimal



Checking (u, v) ∈ E?

Adj

1 5 6

2 3 7

3 4 7

4 7

5 9

6 2 7 9

7 8 10 11

8 11 12

9 10

10

11 12

12




◮ optimal



Checking (u, v) ∈ E? O(|V |)

Adj

1 5 6

2 3 7

3 4 7

4 7

5 9

6 2 7 9

7 8 10 11

8 11 12

9 10

10

11 12

12




◮ optimal



Checking (u, v) ∈ E? O(|V |)

◮ bad

Adj

1 5 6

2 3 7

3 4 7

4 7

5 9

6 2 7 9

7 8 10 11

8 11 12

9 10

10

11 12

12


Graph Representation (2)


Graph Representation (2)

Adjacency-matrix representation

V = {1,2, . . . |V |}

G consists of a |V | × |V | matrix A

A = (aij) such that

aij =

1 if (i, j) ∈ E

0 otherwise


Example

1 2 3 4

5 6 7 8

9 10 11 12


Example

1 2 3 4

5 6 7 8

9 10 11 12

1

2

3

4

5

6

7

8

9

10

11

12

1 2 3 4 5 6 7 8 9 10 11 12


Example

1 2 3 4

5 6 7 8

9 10 11 12

1

2

3

4

5

6

7

8

9

10

11

12

1 2 3 4 5 6 7 8 9 10 11 12

1 1

1 1

1 1

1

1

1 1 1

1 11

1 1

1

1


Using the Adjacency-Matrix

1

2

3

4

5

6

7

8

9

10

11

12

1 2 3 4 5 6 7 8 9 101112

1 1

1 1

1 1

1

1

1 1 1

1 11

1 1

1

1



Accessing a vertex u?

1

2

3

4

5

6

7

8

9

10

11

12

1 2 3 4 5 6 7 8 9 101112

1 1

1 1

1 1

1

1

1 1 1

1 11

1 1

1

1




◮ optimal 1

2

3

4

5

6

7

8

9

10

11

12

1 2 3 4 5 6 7 8 9 101112

1 1

1 1

1 1

1

1

1 1 1

1 11

1 1

1

1




◮ optimal

Iteration through V ?

1

2

3

4

5

6

7

8

9

10

11

12

1 2 3 4 5 6 7 8 9 101112

1 1

1 1

1 1

1

1

1 1 1

1 11

1 1

1

1




◮ optimal


1

2

3

4

5

6

7

8

9

10

11

12

1 2 3 4 5 6 7 8 9 101112

1 1

1 1

1 1

1

1

1 1 1

1 11

1 1

1

1




◮ optimal


Iteration through E?

1

2

3

4

5

6

7

8

9

10

11

12

1 2 3 4 5 6 7 8 9 101112

1 1

1 1

1 1

1

1

1 1 1

1 11

1 1

1

1




◮ optimal


Iteration through E? Θ(|V |2)◮ possibly very bad

1

2

3

4

5

6

7

8

9

10

11

12

1 2 3 4 5 6 7 8 9 101112

1 1

1 1

1 1

1

1

1 1 1

1 11

1 1

1

1




◮ optimal



Checking (u, v) ∈ E?

1

2

3

4

5

6

7

8

9

10

11

12

1 2 3 4 5 6 7 8 9 101112

1 1

1 1

1 1

1

1

1 1 1

1 11

1 1

1

1




◮ optimal



Checking (u, v) ∈ E? O(1)

1

2

3

4

5

6

7

8

9

10

11

12

1 2 3 4 5 6 7 8 9 101112

1 1

1 1

1 1

1

1

1 1 1

1 11

1 1

1

1




◮ optimal



Checking (u, v) ∈ E? O(1)

◮ optimal

1

2

3

4

5

6

7

8

9

10

11

12

1 2 3 4 5 6 7 8 9 101112

1 1

1 1

1 1

1

1

1 1 1

1 11

1 1

1

1


Space Complexity


Space Complexity


Θ(|V | + |E|)


Space Complexity


Θ(|V | + |E|)

optimal


Space Complexity


Θ(|V | + |E|)

optimal


Θ(|V |2)


Space Complexity


Θ(|V | + |E|)

optimal


Θ(|V |2)

possibly very bad


Space Complexity


Θ(|V | + |E|)

optimal


Θ(|V |2)

possibly very bad

When is the adjacency-matrix “very bad”?


Choosing a Graph Representation


◮ generally good, especially for its optimal space complexity

◮ bad for dense graphs and algorithms that require random

access to edges

◮ preferable for sparse graphs or graphs with low degree


Choosing a Graph Representation


◮ generally good, especially for its optimal space complexity

◮ bad for dense graphs and algorithms that require random

access to edges

◮ preferable for sparse graphs or graphs with low degree


◮ suffers from a bad space complexity

◮ good for algorithms that require random access to edges

◮ preferable for dense graphs


Breadth-First Search

One of the simplest but also a fundamental algorithm


Breadth-First Search

One of the simplest but also a fundamental algorithm

Input: G = (V , E) and a vertex s ∈ V

◮ explores the graph, touching all vertices that are reachable

from s

◮ iterates through the vertices at increasing distance (edge

distance)

◮ computes the distance of each vertex from s

◮ produces a breadth-first tree rooted at s

◮ works on both directed and undirected graphs


Example

1 2 3 4

5 6 7 8

9 10 11 12


BFS Algorithm

BFS(G, s)

1 for each vertex u ∈ V(G) \ {s}

2 color[u] = white

3 d[u] = ∞

4 π[u] = nil

5 color[s] = gray

6 d[s] = 0

7 π[s] = nil

8 Q = ∅

9 Enqueue(Q, s)

10 while Q 6= ∅

11 u = Dequeue(Q)

12 for each v ∈ Adj[u]

13 if color[v] == white

14 color[v] = gray

15 d[v] = d[u]+ 1

16 π[v] = u

17 Enqueue(Q, v)

18 color[u] = black

1 2 3 4

5 6 7 8

9 10 11 12


BFS Algorithm

BFS(G, s)


2 color[u] = white

3 d[u] = ∞

4 π[u] = nil

5 color[s] = gray

6 d[s] = 0

7 π[s] = nil

8 Q = ∅

9 Enqueue(Q, s)

10 while Q 6= ∅

11 u = Dequeue(Q)



14 color[v] = gray

15 d[v] = d[u]+ 1

16 π[v] = u

17 Enqueue(Q, v)

18 color[u] = black

1 2 3 4

5 6 7 8

9 10 11 12

u = 1

Q = ∅


BFS Algorithm

BFS(G, s)


2 color[u] = white

3 d[u] = ∞

4 π[u] = nil

5 color[s] = gray

6 d[s] = 0

7 π[s] = nil

8 Q = ∅

9 Enqueue(Q, s)

10 while Q 6= ∅

11 u = Dequeue(Q)



14 color[v] = gray

15 d[v] = d[u]+ 1

16 π[v] = u

17 Enqueue(Q, v)

18 color[u] = black

1 2 3 4

5 6 7 8

9 10 11 12

u = 1

Q = {5}


BFS Algorithm

BFS(G, s)


2 color[u] = white

3 d[u] = ∞

4 π[u] = nil

5 color[s] = gray

6 d[s] = 0

7 π[s] = nil

8 Q = ∅

9 Enqueue(Q, s)

10 while Q 6= ∅

11 u = Dequeue(Q)



14 color[v] = gray

15 d[v] = d[u]+ 1

16 π[v] = u

17 Enqueue(Q, v)

18 color[u] = black

1 2 3 4

5 6 7 8

9 10 11 12

u = 1

Q = {5,6}


BFS Algorithm

BFS(G, s)


2 color[u] = white

3 d[u] = ∞

4 π[u] = nil

5 color[s] = gray

6 d[s] = 0

7 π[s] = nil

8 Q = ∅

9 Enqueue(Q, s)

10 while Q 6= ∅

11 u = Dequeue(Q)



14 color[v] = gray

15 d[v] = d[u]+ 1

16 π[v] = u

17 Enqueue(Q, v)

18 color[u] = black

1 2 3 4

5 6 7 8

9 10 11 12

u = 5

Q = {6}


BFS Algorithm

BFS(G, s)


2 color[u] = white

3 d[u] = ∞

4 π[u] = nil

5 color[s] = gray

6 d[s] = 0

7 π[s] = nil

8 Q = ∅

9 Enqueue(Q, s)

10 while Q 6= ∅

11 u = Dequeue(Q)



14 color[v] = gray

15 d[v] = d[u]+ 1

16 π[v] = u

17 Enqueue(Q, v)

18 color[u] = black

1 2 3 4

5 6 7 8

9 10 11 12

u = 5

Q = {6,9}


BFS Algorithm

BFS(G, s)


2 color[u] = white

3 d[u] = ∞

4 π[u] = nil

5 color[s] = gray

6 d[s] = 0

7 π[s] = nil

8 Q = ∅

9 Enqueue(Q, s)

10 while Q 6= ∅

11 u = Dequeue(Q)



14 color[v] = gray

15 d[v] = d[u]+ 1

16 π[v] = u

17 Enqueue(Q, v)

18 color[u] = black

1 2 3 4

5 6 7 8

9 10 11 12

u = 6

Q = {9}


BFS Algorithm

BFS(G, s)


2 color[u] = white

3 d[u] = ∞

4 π[u] = nil

5 color[s] = gray

6 d[s] = 0

7 π[s] = nil

8 Q = ∅

9 Enqueue(Q, s)

10 while Q 6= ∅

11 u = Dequeue(Q)



14 color[v] = gray

15 d[v] = d[u]+ 1

16 π[v] = u

17 Enqueue(Q, v)

18 color[u] = black

1 2 3 4

5 6 7 8

9 10 11 12

u = 6

Q = {9,2,7}


BFS Algorithm

BFS(G, s)


2 color[u] = white

3 d[u] = ∞

4 π[u] = nil

5 color[s] = gray

6 d[s] = 0

7 π[s] = nil

8 Q = ∅

9 Enqueue(Q, s)

10 while Q 6= ∅

11 u = Dequeue(Q)



14 color[v] = gray

15 d[v] = d[u]+ 1

16 π[v] = u

17 Enqueue(Q, v)

18 color[u] = black

1 2 3 4

5 6 7 8

9 10 11 12

u = 9

Q = {2,7}


BFS Algorithm

BFS(G, s)


2 color[u] = white

3 d[u] = ∞

4 π[u] = nil

5 color[s] = gray

6 d[s] = 0

7 π[s] = nil

8 Q = ∅

9 Enqueue(Q, s)

10 while Q 6= ∅

11 u = Dequeue(Q)



14 color[v] = gray

15 d[v] = d[u]+ 1

16 π[v] = u

17 Enqueue(Q, v)

18 color[u] = black

1 2 3 4

5 6 7 8

9 10 11 12

u = 9

Q = {2,7,10}


BFS Algorithm

BFS(G, s)


2 color[u] = white

3 d[u] = ∞

4 π[u] = nil

5 color[s] = gray

6 d[s] = 0

7 π[s] = nil

8 Q = ∅

9 Enqueue(Q, s)

10 while Q 6= ∅

11 u = Dequeue(Q)



14 color[v] = gray

15 d[v] = d[u]+ 1

16 π[v] = u

17 Enqueue(Q, v)

18 color[u] = black

1 2 3 4

5 6 7 8

9 10 11 12

u = 2

Q = {7,10}


BFS Algorithm

BFS(G, s)


2 color[u] = white

3 d[u] = ∞

4 π[u] = nil

5 color[s] = gray

6 d[s] = 0

7 π[s] = nil

8 Q = ∅

9 Enqueue(Q, s)

10 while Q 6= ∅

11 u = Dequeue(Q)



14 color[v] = gray

15 d[v] = d[u]+ 1

16 π[v] = u

17 Enqueue(Q, v)

18 color[u] = black

1 2 3 4

5 6 7 8

9 10 11 12

u = 2

Q = {7,10,3}


BFS Algorithm

BFS(G, s)


2 color[u] = white

3 d[u] = ∞

4 π[u] = nil

5 color[s] = gray

6 d[s] = 0

7 π[s] = nil

8 Q = ∅

9 Enqueue(Q, s)

10 while Q 6= ∅

11 u = Dequeue(Q)



14 color[v] = gray

15 d[v] = d[u]+ 1

16 π[v] = u

17 Enqueue(Q, v)

18 color[u] = black

1 2 3 4

5 6 7 8

9 10 11 12

u = 7

Q = {10,3}


BFS Algorithm

BFS(G, s)


2 color[u] = white

3 d[u] = ∞

4 π[u] = nil

5 color[s] = gray

6 d[s] = 0

7 π[s] = nil

8 Q = ∅

9 Enqueue(Q, s)

10 while Q 6= ∅

11 u = Dequeue(Q)



14 color[v] = gray

15 d[v] = d[u]+ 1

16 π[v] = u

17 Enqueue(Q, v)

18 color[u] = black

1 2 3 4

5 6 7 8

9 10 11 12

u = 7

Q = {10,3,8}


BFS Algorithm

BFS(G, s)


2 color[u] = white

3 d[u] = ∞

4 π[u] = nil

5 color[s] = gray

6 d[s] = 0

7 π[s] = nil

8 Q = ∅

9 Enqueue(Q, s)

10 while Q 6= ∅

11 u = Dequeue(Q)



14 color[v] = gray

15 d[v] = d[u]+ 1

16 π[v] = u

17 Enqueue(Q, v)

18 color[u] = black

1 2 3 4

5 6 7 8

9 10 11 12

u = 7

Q = {10,3,8,11}


BFS Algorithm

BFS(G, s)


2 color[u] = white

3 d[u] = ∞

4 π[u] = nil

5 color[s] = gray

6 d[s] = 0

7 π[s] = nil

8 Q = ∅

9 Enqueue(Q, s)

10 while Q 6= ∅

11 u = Dequeue(Q)



14 color[v] = gray

15 d[v] = d[u]+ 1

16 π[v] = u

17 Enqueue(Q, v)

18 color[u] = black

1 2 3 4

5 6 7 8

9 10 11 12

u = 10

Q = {3,8,11}


BFS Algorithm

BFS(G, s)


2 color[u] = white

3 d[u] = ∞

4 π[u] = nil

5 color[s] = gray

6 d[s] = 0

7 π[s] = nil

8 Q = ∅

9 Enqueue(Q, s)

10 while Q 6= ∅

11 u = Dequeue(Q)



14 color[v] = gray

15 d[v] = d[u]+ 1

16 π[v] = u

17 Enqueue(Q, v)

18 color[u] = black

1 2 3 4

5 6 7 8

9 10 11 12

u = 3

Q = {8,11}


BFS Algorithm

BFS(G, s)


2 color[u] = white

3 d[u] = ∞

4 π[u] = nil

5 color[s] = gray

6 d[s] = 0

7 π[s] = nil

8 Q = ∅

9 Enqueue(Q, s)

10 while Q 6= ∅

11 u = Dequeue(Q)



14 color[v] = gray

15 d[v] = d[u]+ 1

16 π[v] = u

17 Enqueue(Q, v)

18 color[u] = black

1 2 3 4

5 6 7 8

9 10 11 12

u = 3

Q = {8,11,4}


BFS Algorithm

BFS(G, s)


2 color[u] = white

3 d[u] = ∞

4 π[u] = nil

5 color[s] = gray

6 d[s] = 0

7 π[s] = nil

8 Q = ∅

9 Enqueue(Q, s)

10 while Q 6= ∅

11 u = Dequeue(Q)



14 color[v] = gray

15 d[v] = d[u]+ 1

16 π[v] = u

17 Enqueue(Q, v)

18 color[u] = black

1 2 3 4

5 6 7 8

9 10 11 12

u = 8

Q = {11,4}


BFS Algorithm

BFS(G, s)


2 color[u] = white

3 d[u] = ∞

4 π[u] = nil

5 color[s] = gray

6 d[s] = 0

7 π[s] = nil

8 Q = ∅

9 Enqueue(Q, s)

10 while Q 6= ∅

11 u = Dequeue(Q)



14 color[v] = gray

15 d[v] = d[u]+ 1

16 π[v] = u

17 Enqueue(Q, v)

18 color[u] = black

1 2 3 4

5 6 7 8

9 10 11 12

u = 8

Q = {11,4,12}


BFS Algorithm

BFS(G, s)


2 color[u] = white

3 d[u] = ∞

4 π[u] = nil

5 color[s] = gray

6 d[s] = 0

7 π[s] = nil

8 Q = ∅

9 Enqueue(Q, s)

10 while Q 6= ∅

11 u = Dequeue(Q)



14 color[v] = gray

15 d[v] = d[u]+ 1

16 π[v] = u

17 Enqueue(Q, v)

18 color[u] = black

1 2 3 4

5 6 7 8

9 10 11 12

u = 11

Q = {4,12}


BFS Algorithm

BFS(G, s)


2 color[u] = white

3 d[u] = ∞

4 π[u] = nil

5 color[s] = gray

6 d[s] = 0

7 π[s] = nil

8 Q = ∅

9 Enqueue(Q, s)

10 while Q 6= ∅

11 u = Dequeue(Q)



14 color[v] = gray

15 d[v] = d[u]+ 1

16 π[v] = u

17 Enqueue(Q, v)

18 color[u] = black

1 2 3 4

5 6 7 8

9 10 11 12

u = 4

Q = {12}


BFS Algorithm

BFS(G, s)


2 color[u] = white

3 d[u] = ∞

4 π[u] = nil

5 color[s] = gray

6 d[s] = 0

7 π[s] = nil

8 Q = ∅

9 Enqueue(Q, s)

10 while Q 6= ∅

11 u = Dequeue(Q)



14 color[v] = gray

15 d[v] = d[u]+ 1

16 π[v] = u

17 Enqueue(Q, v)

18 color[u] = black

1 2 3 4

5 6 7 8

9 10 11 12

u = 12

Q = ∅


BFS Algorithm

BFS(G, s)


2 color[u] = white

3 d[u] = ∞

4 π[u] = nil

5 color[s] = gray

6 d[s] = 0

7 π[s] = nil

8 Q = ∅

9 Enqueue(Q, s)

10 while Q 6= ∅

11 u = Dequeue(Q)



14 color[v] = gray

15 d[v] = d[u]+ 1

16 π[v] = u

17 Enqueue(Q, v)

18 color[u] = black

1 2 3 4

5 6 7 8

9 10 11 12


Complexity of BFS

BFS(G, s)


2 color[u] = white

3 d[u] = ∞

4 π[u] = nil

5 color[s] = gray

6 d[s] = 0

7 π[s] = nil

8 Q = ∅

9 Enqueue(Q, s)

10 while Q 6= ∅

11 u = Dequeue(Q)



14 color[v] = gray

15 d[v] = d[u]+ 1

16 π[v] = u

17 Enqueue(Q, v)

18 color[u] = black


Complexity of BFS

BFS(G, s)


2 color[u] = white

3 d[u] = ∞

4 π[u] = nil

5 color[s] = gray

6 d[s] = 0

7 π[s] = nil

8 Q = ∅

9 Enqueue(Q, s)

10 while Q 6= ∅

11 u = Dequeue(Q)



14 color[v] = gray

15 d[v] = d[u]+ 1

16 π[v] = u

17 Enqueue(Q, v)

18 color[u] = black

We enqueue a vertex

only if it is white, and

we immediately color

it gray; thus, we

enqueue every vertex

at most once


Complexity of BFS

BFS(G, s)


2 color[u] = white

3 d[u] = ∞

4 π[u] = nil

5 color[s] = gray

6 d[s] = 0

7 π[s] = nil

8 Q = ∅

9 Enqueue(Q, s)

10 while Q 6= ∅

11 u = Dequeue(Q)



14 color[v] = gray

15 d[v] = d[u]+ 1

16 π[v] = u

17 Enqueue(Q, v)

18 color[u] = black

We enqueue a vertex



it gray; thus, we


at most once

So, the (dequeue)

while loop executes

O(|V |) times


Complexity of BFS

BFS(G, s)


2 color[u] = white

3 d[u] = ∞

4 π[u] = nil

5 color[s] = gray

6 d[s] = 0

7 π[s] = nil

8 Q = ∅

9 Enqueue(Q, s)

10 while Q 6= ∅

11 u = Dequeue(Q)



14 color[v] = gray

15 d[v] = d[u]+ 1

16 π[v] = u

17 Enqueue(Q, v)

18 color[u] = black

We enqueue a vertex



it gray; thus, we


at most once

So, the (dequeue)

while loop executes

O(|V |) times

For each vertex u, the

inner loop executes

Θ(|Eu|), for a total of

O(|E|) steps


Complexity of BFS

BFS(G, s)


2 color[u] = white

3 d[u] = ∞

4 π[u] = nil

5 color[s] = gray

6 d[s] = 0

7 π[s] = nil

8 Q = ∅

9 Enqueue(Q, s)

10 while Q 6= ∅

11 u = Dequeue(Q)



14 color[v] = gray

15 d[v] = d[u]+ 1

16 π[v] = u

17 Enqueue(Q, v)

18 color[u] = black

We enqueue a vertex



it gray; thus, we


at most once

So, the (dequeue)

while loop executes

O(|V |) times

For each vertex u, the

inner loop executes

Θ(|Eu|), for a total of

O(|E|) steps

So, O(|V | + |E|)


Depth-First Search


Depth-First Search

Immediately follow the links of the most recently-visited

vertex, then backtrack when you reach a dead-end

◮ i.e., backtrack when the current vertex has no more adjacent

vertices that have not yet been visited


Depth-First Search





Input: G = (V , E)

◮ explores the graph, touching all vertices


Depth-First Search





Input: G = (V , E)


◮ produces a depth-first forest, consisting of all the depth-first

trees defined by the DFS exploration


Depth-First Search





Input: G = (V , E)


◮ produces a depth-first forest, consisting of all the depth-first

trees defined by the DFS exploration

◮ associates two time-stamps to each vertex

◮ d[u] records when u is first discovered

◮ f [u] records when DFS finishes examining u’s edges, and

therefore backtracks from u


DFS Algorithm

DFS(G)

1 for each vertex u ∈ V(G)

2 color[u] = white

3 π[u] = nil

4 time = 0 // “global” variable

5 for each vertex u ∈ V(G)

6 if color[u] == white

7 DFS-Visit(u)

DFS-Visit(u)

1 color[u] = grey

2 time = time+ 1

3 d[u] = time



6 π[v] = u

7 DFS-Visit(v)

8 color[u] = black

9 time = time+ 1

10 f [u] = time


Complexity of DFS


Complexity of DFS

The loop in DFS-Visit(u) (lines 4–7) accounts for Θ(|Eu|)


Complexity of DFS


We call DFS-Visit(u) once for each vertex u

◮ either in DFS, or recursively in DFS-Visit

◮ because we call it only if color[u] = white, but then we

immediately set color[u] = grey


Complexity of DFS


We call DFS-Visit(u) once for each vertex u

◮ either in DFS, or recursively in DFS-Visit

◮ because we call it only if color[u] = white, but then we

immediately set color[u] = grey

So, the overall complexity is Θ(|V | + |E|)


Applications of DFS: Topological Sort



Given a directed acyclic graph (DAG)

◮ find an ordering of vertices such that you only end up with

forward links



Given a directed acyclic graph (DAG)

◮ find an ordering of vertices such that you only end up with

forward links

Example: dependencies in software packages

◮ find an installation order for a set of software packages

◮ such that every package is installed only after all the packages

it depends on


Topological Sort Algorithm



Topological-Sort(G)

1 DFS(G)

2 output V sorted in reverse order of f [·]



Topological-Sort(G)

1 DFS(G)

2 output V sorted in reverse order of f [·]

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pants

belt

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