Dis

trib

uted

Str

uctu

res f

or M

ulti-

Hop

N

etw

orks

Raj

moh

an R

ajar

aman

Nor

thea

stern

Uni

vers

ity

Partl

y ba

sed

on a

tuto

rial,

join

t with

Tor

sten

Sue

l, at

the

DIM

AC

S Su

mm

er

Scho

ol o

n Fo

unda

tions

of W

irele

ss N

etwo

rks a

nd A

pplic

atio

ns, A

ugus

t 200

0

Sept

embe

r 10,

200

2

ro

utin

g ta

bles

sp

anni

ng su

bgra

phs

sp

anni

ng tr

ees,

broa

dcas

t tre

es

cl

uste

rs, d

omin

atin

g se

ts

hi

erar

chic

al n

etw

ork

deco

mpo

sitio

n

Focu

s of t

his T

utor

ial

We

are

inte

rest

ed in

com

putin

g an

d m

aint

aini

ngva

rious

sort

s of g

loba

l/loc

al st

ruct

ures

indy

nam

ic d

istri

bute

d/m

ulti-

hop/

wire

less

net

work

s

Wha

t is M

issi

ng?

Sp

ecifi

c ad

hoc

net

wor

k ro

utin

g pr

otoc

ols

A

d H

oc N

etw

orki

ng [P

erki

ns 0

1]

Tuto

rial b

y N

itin

Vai

dya

http

://w

ww

.crh

c.ui

uc.e

du/~

nhv/

pres

enta

tions

.htm

l

Ph

ysic

al a

nd M

AC

laye

r iss

ues

C

apac

ity o

f wire

less

net

wor

ks [G

upta

-Kum

ar

00, G

ross

glau

ser-

Tse

01]

Fa

ult-t

oler

ance

and

wire

less

secu

rity

In

trodu

ctio

n (

netw

ork

mod

el, p

robl

ems,

perf

orm

ance

mea

sure

s)

Pa

rt I:

-bas

ics a

nd e

xam

ples

-rou

ting

& r

outin

g ta

bles

-t

opol

ogy

cont

rol

Pa

rt II

: -s

pann

ing

tree

s-d

omin

atin

g se

ts &

clu

ster

ing

-hie

rarc

hica

l clu

ster

ingOve

rvie

w

Mul

ti-H

op N

etw

ork

Mod

el

dy

nam

ic n

etw

ork

un

dire

cted

so

rt-o

f-al

mos

t pla

nar?

Wha

t is a

Hop

?

B

road

cast

with

in a

cer

tain

rang

e

Var

iabl

e ra

nge

depe

ndin

g on

pow

er c

ontro

l cap

abili

ties

In

terf

eren

ce a

mon

g co

nten

ding

tran

smis

sion

s

MA

C la

yer c

onte

ntio

n re

solu

tion

prot

ocol

s, e.

g., I

EEE

802.

11, B

luet

ooth

Pa

cket

radi

o ne

twor

k m

odel

(PR

N)

M

odel

eac

h ho

p as

a

broa

dcas

t hop

an

d co

nsid

er

inte

rfer

ence

in a

naly

sis

M

ultih

op n

etw

ork

mod

el

Ass

ume

an u

nder

lyin

g M

AC

laye

r pro

toco

l

The

netw

ork

is a

dyn

amic

inte

rcon

nect

ion

netw

ork

In

pra

ctic

e, b

oth

view

s im

porta

nt

W

irele

ss N

etw

orki

ng w

ork

-oft

en h

euri

stic

in n

atur

e-f

ew p

rova

ble

boun

ds-e

xper

imen

tal e

valu

atio

ns in

(rea

listic

) set

tings

D

istri

bute

d C

ompu

ting

wor

k-p

rova

ble

boun

ds-o

ften

wor

st-c

ase

assu

mpt

ions

and

gen

eral

gra

phs

-oft

en c

ompl

icat

ed a

lgor

ithm

s-a

ssum

ptio

ns n

ot a

lway

s app

licab

le to

wir

eles

s

Lite

ratu

re

Perf

orm

ance

Mea

sure

s

Ti

me

C

omm

unic

atio

n

M

emor

y re

quire

men

ts

A

dapt

abili

ty

Ener

gy c

onsu

mpt

ion

O

ther

QoS

mea

sure

s

path

leng

th

num

ber

of m

essa

ges

corr

elat

ion

St

atic

Li

mite

d m

obili

ty-a

few

nod

es m

ay fa

il, re

cove

r, or

be

mov

ed (s

enso

r net

wor

ks)

-tou

gh e

xam

ple:

thro

w a

mill

ion

node

s out

of a

n ai

rpla

ne

H

ighl

y ad

aptiv

e/m

obile

-tou

gh e

xam

ple:

a hu

ndre

d ai

rpla

nes/v

ehic

les m

ovin

g at

hig

h sp

eed

-im

poss

ible

(?):

a m

illio

n m

osqu

itoes

with

wir

eles

s lin

ks

N

omad

ic/v

iral

mod

el:

-dis

conn

ecte

d ne

twor

k of

hig

hly

mob

ile u

sers

-exa

mpl

e:

vi

rus t

rans

miss

ion

in a

pop

ulat

ion

of b

luet

ooth

use

rs

Deg

rees

of M

obili

ty/A

dapt

abili

ty

Mai

n Pr

oble

ms C

onsi

dere

d

ch

angi

ng, a

rbitr

ary

topo

logy

ne

ed ro

utin

g ta

bles

to fi

nd p

ath

to d

estin

atio

n

rela

ted

prob

lem

: fin

ding

clo

sest

item

of c

erta

in ty

pe

Rou

ting:

sour

ce

desti

natio

n

Top

olog

y C

ontr

ol:

G

iven

a c

olle

ctio

n of

nod

es o

n th

e pl

ane,

and

tran

smis

sion

ca

pabi

litie

s of t

he n

odes

, det

erm

ine

a to

polo

gy th

at is

:

conn

ecte

d

low

-deg

ree

a

span

ner:

dis

tanc

e be

twee

n tw

o no

des i

n th

e to

polo

gy is

cl

ose

to th

at in

the

trans

mis

sion

gra

ph

an e

nerg

y-sp

anne

r: it

has e

nerg

y-ef

ficie

nt p

aths

ad

apta

ble:

one

can

mai

ntai

n th

e ab

ove

prop

ertie

s ef

ficie

ntly

whe

n no

des m

ove

Span

ning

Tre

es:

K-D

omin

atin

g Se

ts:

us

eful

for r

outin

g

sing

le p

oint

of f

ailu

re

non-

min

imal

rout

es

man

y va

riant

s

de

fines

par

titio

n of

the

netw

ork

into

zon

es1-

dom

inat

ing

set

Clu

ster

ing:

Hie

rarc

hica

l Clu

sterin

g

di

sjoi

nt o

r ove

rlapp

ing

fla

t or h

iera

rchi

cal

in

tern

al a

nd b

orde

r nod

es a

nd e

dges

Flat

Clu

sterin

g

Bas

ic R

outin

g Sc

hem

es

Pr

oact

ive

Rou

ting:

-

keep

rout

ing

info

rmat

ion

curr

ent a

t all

times

-g

ood

for s

tatic

net

wor

ks-e

xam

ples

: dis

tanc

e ve

ctor

(DV

), lin

k st

ate

(LS)

alg

orith

ms

R

eact

ive

Rou

ting:

-fin

d a

rout

e to

the

desti

natio

n on

ly a

fter a

requ

est c

omes

in

-goo

d fo

r mor

e dy

nam

ic n

etw

orks

-exa

mpl

es: A

OD

V, d

ynam

ic so

urce

rout

ing

(DSR

), TO

RA

H

ybri

d Sc

hem

es:

-ke

ep so

me

info

rmat

ion

curr

ent

-exa

mpl

e: Z

one

Rou

ting

Prot

ocol

(ZR

P)

-exa

mpl

e: U

se sp

anni

ng tr

ees f

or n

on-o

ptim

al ro

utin

g

Proa

ctiv

e R

outin

g (D

ista

nce

Vec

tor)

Eac

h no

de m

aint

ains

dis

tanc

e to

eve

ry o

ther

nod

eU

pdat

ed b

etw

een

neig

hbor

s usi

ng B

ellm

an-F

ord

bi

ts sp

ace

requ

irem

ent

Sin

gle

edge

/nod

e fa

ilure

may

requ

ire m

ost n

odes

to

cha

nge

mos

t of t

heir

entri

esS

low

upd

ates

Tem

pora

ry lo

ops

half

ofth

e no

des

half

ofth

e no

des

)lo

g(

2n

nO

Rea

ctiv

e R

outin

g-

Ad-

Hoc

On

Dem

and

Dis

tanc

e V

ecto

r (A

OD

V) [

Perk

ins-

Roy

er 9

9]-

Dyn

amic

Sou

rce

Rou

ting

(DSR

) [Jo

hnso

n-M

altz

96]

-Te

mpo

rally

Ord

ered

Rou

ting

Alg

orith

m [

Park

-Cor

son

97]

If

sour

ce d

oes n

ot k

now

pat

h to

des

tinat

ion,

issu

es d

isco

very

requ

est

D

SR c

ache

s rou

te to

des

tinat

ion

Ea

sier

to a

void

rout

ing

loop

s

sour

ce

desti

natio

n

Hyb

rid

Sche

mes

-Zo

ne R

outin

g [H

aas9

7]

ev

ery

node

kno

ws a

zone

of ra

dius

r ar

ound

it

node

s at d

ista

nce

exac

tly r

are

calle

d pe

riph

eral

bo

rder

cast

ing:

se

ndin

g a

mes

sage

to a

ll pe

riphe

ral n

odes

glob

al ro

ute

sear

ch;

bord

erca

stin

gre

duce

s sea

rch

spac

e

radi

us d

eter

min

es tr

ade-

off

m

aint

ain

up-to

-dat

e ro

utes

in z

one

and

cach

e ro

utes

to e

xter

nal n

odes

r

Rou

ting

usin

g Sp

anni

ng T

ree

Se

nd p

acke

t fro

m so

urce

to ro

ot, t

hen

to d

estin

atio

n

O(n

log

n) to

tal,

and

at th

e ro

ot

sour

ce

root

desti

natio

n

N

on-o

ptim

al, a

nd b

ottle

neck

at r

oot

N

eed

to o

nly

mai

ntai

n sp

anni

ng tr

ee

Rou

ting

by C

lust

erin

g

G

atew

ay n

odes

mai

ntai

n ro

utes

with

in c

lust

er

Rou

ting

amon

g ga

tew

ay n

odes

alo

ng a

span

ning

tree

or u

sing

DV

/LS

algo

rithm

s

Hie

rarc

hica

l clu

ster

ing

(e.g

., [L

auer

86,

Ram

anat

han-

Stee

nstru

p 98

])

Rout

ing

by O

ne-L

evel

Clu

sterin

g[B

aker

-Eph

rem

edis

81]

Hie

rarc

hica

l Rou

ting

Th

e no

des o

rgan

ize

them

selv

es in

to a

hie

rarc

hy

The

hier

arch

y im

pose

s a n

atur

al a

ddre

ssin

g sc

hem

e

Qua

si-h

iera

rchi

cal r

outin

g: E

ach

node

mai

ntai

ns

next

hop

nod

e on

a p

ath

to e

very

oth

er le

vel-j

clu

ster

w

ithin

its l

evel

-(j+

1) a

nces

tral c

lust

er

St

rict-h

iera

rchi

cal r

outin

g: E

ach

node

mai

ntai

ns

next

leve

l-j c

lust

er o

n a

path

to e

very

oth

er le

vel-j

cl

uste

r with

in it

s lev

el-(

j+1)

anc

estra

l clu

ster

bo

unda

ry l

evel

-j cl

uste

rs in

its l

evel

-(j+

1) c

lust

ers a

nd

thei

r nei

ghbo

ring

clus

ters

Exa

mpl

e: S

tric

t-H

iera

rchi

cal R

outin

g

Ea

ch n

ode

mai

ntai

ns:

N

ext h

op n

ode

on a

min

-cos

t pat

h to

eve

ry o

ther

nod

e in

clu

ster

C

lust

er b

ound

ary

node

on

a m

in-c

ost p

ath

to n

eigh

borin

g cl

uste

r

Nex

t hop

clu

ster

on

the

min

-cos

t pat

h to

any

oth

er c

lust

er in

supe

rclu

ster

Th

e cl

uste

r lea

der p

artic

ipat

es in

com

putin

g th

is in

form

atio

n an

d di

strib

utin

g it

to n

odes

in it

s clu

ster

Spac

e R

equi

rem

ents

and

Ada

ptab

ility

Ea

ch n

ode

has

entri

es

is th

e nu

mbe

r of l

evel

s

is th

e m

axim

um, o

ver a

ll j,

of th

e nu

mbe

r of l

evel

-j cl

uste

rs in

a le

vel-(

j+1)

clu

ster

If

the

clus

terin

g is

regu

lar,

num

ber o

f ent

ries p

er

node

is

R

estru

ctur

ing

the

hier

arch

y:

Clu

ster

lead

ers s

plit/

mer

ge c

lust

ers w

hile

mai

ntai

ning

si

ze b

ound

s (O

(1) g

ap b

etw

een

uppe

r and

low

er b

ound

s)

Som

etim

es n

eed

to g

ener

ate

new

add

ress

es

Nee

d lo

catio

n m

anag

emen

t (na

me-

to-a

ddre

ss m

ap)

)(m

CO

m C

)(

/1m

mn

O

Spac

e R

equi

rem

ents

for

Rou

ting

D

ista

nce

Vec

tor:

O(n

log

n) b

its p

er n

ode,

O(n

^2 lo

g n)

tota

l

Rou

ting

via

span

ning

tree

: O(n

log

n) to

tal,

very

non

-opt

imal

O

ptim

al (i

.e.,

shor

test

pat

h) ro

utin

g re

quire

s The

ta(n

^2)

bits

tota

l on

alm

ost a

ll gr

aphs

[B

uhrm

an-H

oepm

an-V

itany

i 00]

A

lmos

t opt

imal

rout

ing

(with

stre

tch

< 3)

requ

ires T

heta

(n^2

)on

som

e gr

aphs

[Fra

igni

aud-

Gav

oille

95, G

avoi

lle-G

engl

er97

, Gav

oille

-Per

enne

s 96]

Tr

adeo

ff b

etw

een

stre

tch

and

spac

e: [

Pele

g-U

pfal

89]

-upp

er b

ound

: O

(n

) m

emor

y w

ith st

retc

h O

(k)

-low

er b

ound

: The

ta(n

)

bits

with

stre

tch

O(k

)

-abo

ut O

(n

) w

ith st

retc

h 5

[E

ilam

-Gav

oille

-Pel

eg 0

0]

1+1/

k 1+1/

(2k+

4)

3/2

R

ecal

l cor

rela

tion

mem

ory/

adap

tabi

lity

ada

ptab

ility

shou

ld re

quire

long

er p

aths

How

ever

, not

muc

h kn

own

form

ally

O

nly

sing

le-m

essa

ge ro

utin

g (n

o at

tem

pt to

avo

id b

ottle

neck

s)

R

esul

ts fo

r gen

eral

gra

phs.

For s

peci

al c

lass

es, b

ette

r res

ults

:-t

rees

, mes

hes,

rings

etc

.

-out

erpl

anar

and

deco

mpo

sabl

e gr

aphs

[F

rede

ricks

on-J

anar

dan

86]

-pla

nar g

raph

s:

O(n

) w

ith st

retc

h 7

[Fr

eder

icks

on/J

anar

dan

86]

Not

e:

1+ep

s

Loc

atio

n M

anag

emen

t

A

nam

e-to

-add

ress

map

ping

serv

ice

C

entra

lized

app

roac

h: U

se re

dund

ant l

ocat

ion

man

ager

s tha

t sto

re m

ap

Upd

atin

g co

sts i

s hig

h

Sear

chin

g co

st is

rela

tivel

y lo

w

Clu

ster

-bas

ed a

ppro

ach:

Use

hie

rarc

hica

l cl

uste

ring

to o

rgan

ize

loca

tion

info

rmat

ion

Lo

catio

n m

anag

er in

a c

lust

er st

ores

add

ress

map

ping

s fo

r nod

es w

ithin

the

clus

ter

M

appi

ng re

ques

t pro

gres

sive

ly m

oves

up

the

clus

ter

until

add

ress

reso

lved

C

omm

on is

sues

with

dat

a lo

catio

n in

P2P

syst

ems

Con

tent

-and

Loc

atio

n-A

ddre

ssab

le R

outin

g

ho

w d

o w

e id

entif

y no

des?

-

ever

y no

de h

as a

n ID

ar

e th

e ID

s fix

ed o

r can

they

be

chan

ged?

W

hy w

ould

a n

ode

wan

t to

send

a m

essa

ge to

nod

e 01

0654

1 ?

(inst

ead

of se

ndin

g to

a n

ode

cont

aini

ng a

giv

en it

em o

r a n

ode

in a

give

n ar

ea)

sour

ce

desti

natio

n01

0564

1de

stina

tion

(3,3

)

Geo

grap

hica

l Rou

ting

U

se o

f geo

grap

hy to

ach

ieve

scal

abili

ty

Proa

ctiv

e al

gorit

hms n

eed

to m

aint

ain

stat

e pr

opor

tiona

l to

num

ber o

f nod

es

Rea

ctiv

e al

gorit

hms,

with

agg

ress

ive

cach

ing,

als

o st

ores

larg

e st

ate

info

rmat

ion

at so

me

node

s

Nod

es o

nly

mai

ntai

n in

form

atio

n ab

out l

ocal

ne

ighb

orho

ods

R

equi

res r

easo

nabl

y ac

cura

te g

eogr

aphi

c po

sitio

ning

sy

stem

s (G

PS)

A

ssum

e bi

dire

ctio

nalr

adio

reac

habi

lity

Ex

ampl

e pr

otoc

ols:

Lo

catio

n-A

ided

Rou

ting

[Ko-

Vai

dya

98],

Rou

ting

in

the

Plan

e [H

assi

n-Pe

leg

96],

GPS

R [K

arp-

Kun

g 00

]

Gre

edy

Peri

met

er S

tate

less

Rou

ting

G

PSR

[Kar

p-K

ung

00]

G

reed

y fo

rwar

ding

Fo

rwar

d to

nei

ghbo

r clo

sest

to d

estin

atio

n

Nee

d to

kno

w th

e po

sitio

n of

the

desti

natio

n

DS

GPS

R: P

erim

eter

For

war

ding

G

reed

y fo

rwar

ding

doe

s not

alw

ays w

ork

Th

e pa

cket

cou

ld g

et st

uck

at a

lo

cal m

axim

um

Pe

rimet

er fo

rwar

ding

atte

mpt

s to

forw

ard

the

pack

et a

roun

d th

e v

oid

D xU

se ri

ght-h

and

rule

to e

nsur

e pr

ogre

ssO

nly

wor

ks fo

r pla

nar g

raph

sN

eed

to re

stric

t the

set o

f edg

es u

sed

Prox

imity

Gra

phs

Rel

ativ

e N

eigh

borh

ood

Gra

ph(R

NG

): Th

ere

is a

n ed

ge b

etw

een

u an

d v

only

if

ther

e is

no

verte

x w

such

that

d(u

,w) a

nd

d(v,

w) a

re b

oth

less

than

d(u

,v)

Gab

riel

Gra

ph(G

G):

Ther

e is

an

edge

bet

wee

n u

and

v if

ther

e is

no

verte

x w

in th

e ci

rcle

with

di

amet

er c

hord

(u,v

)

Prox

imity

Gra

phs a

nd G

PSR

U

se g

reed

y fo

rwar

ding

on

the

entir

e gr

aph

W

hen

gree

dy fo

rwar

ding

reac

hes a

loca

l m

axim

um, s

witc

h to

per

imet

er fo

rwar

ding

Ope

rate

on

plan

ar su

bgra

ph (R

NG

or G

G, f

or e

xam

ple)

Forw

ard

it al

ong

a fa

ce in

ters

ectin

g lin

e to

des

tinat

ion

C

an sw

itch

to g

reed

y fo

rwar

ding

afte

r rec

over

ing

from

lo

cal m

axim

um

Dis

tanc

e an

d nu

mbe

r of h

ops t

rave

rsed

co

uld

be m

uch

mor

e th

an o

ptim

al

Span

ners

and

Str

etch

St

retc

h of

asu

bgra

phH

is th

e m

axim

um ra

tio o

f the

di

stan

ce b

etw

een

two

node

s in

H to

that

bet

wee

n th

em

in G Ex

tens

ivel

y st

udie

d in

the

grap

h al

gorit

hms a

nd g

raph

theo

ry

liter

atur

e [E

ppst

ein

96]

D

ista

nce

stre

tch

and

topo

logi

cal s

tretc

h

A sp

anne

r is a

subg

raph

that

has

con

stan

t stre

tch

N

eith

er R

NG

nor

GG

is a

span

ner

Th

e D

elau

nay

trian

gula

tion

yiel

ds a

pla

nar d

ista

nce-

span

ner

Th

e Y

ao-g

raph

[Yao

82]i

s als

o a

sim

ple

dist

ance

-spa

nner

Ene

rgy

Con

sum

ptio

n &

Pow

er C

ontr

ol

C

omm

only

ado

pted

pow

er a

ttenu

atio

n m

odel

:

is b

etw

een

2 an

d 4

A

ssum

ing

unifo

rm th

resh

old

for r

ecep

tion

pow

er a

nd

inte

rfer

ence

/noi

se le

vels

, ene

rgy

cons

umed

for t

rans

mitt

ing

from

to

ne

eds t

o be

pro

porti

onal

to

Po

wer

con

trol:

Rad

ios h

ave

the

capa

bilit

y to

adj

ust t

heir

pow

er le

vels

so a

s to

reac

h de

stin

atio

n w

ith d

esire

d fid

elity

En

ergy

con

sum

ed a

long

a p

ath

is si

mpl

y th

e su

m o

f the

tra

nsm

issi

on e

nerg

ies a

long

the

path

link

s

Def

ine

ener

gy-s

tretc

h an

alog

ous t

o di

stan

ce-s

tretc

h

αdi

stan

cepow

erTr

ansm

it

Pow

er

R

ecei

ved

∝

α

uv

α ),

(v

ud

Ene

rgy-

Aw

are

Rou

ting

A

pat

h w

ith m

any

shor

t hop

s con

sum

es le

ss e

nerg

y th

an a

pa

th w

ith a

few

larg

e ho

ps

Whi

ch e

dges

to u

se?

(Con

side

red

in to

polo

gy c

ontro

l)

Can

mai

ntai

n e

nerg

y co

st

info

rmat

ion

to fi

nd m

inim

um-e

nerg

y pa

ths [

Rod

oplu

-Men

g 98

]

Rou

ting

to m

axim

ize

netw

ork

lifet

ime

[Cha

ng-T

assi

ulas

99

]

Form

ulat

e th

e se

lect

ion

of p

aths

and

pow

er le

vels

as a

n op

timiz

atio

n pr

oble

m

Sugg

ests

the

use

of m

ultip

le ro

utes

bet

wee

n a

give

n so

urce

-de

stin

atio

n pa

ir to

bal

ance

ene

rgy

cons

umpt

ion

En

ergy

con

sum

ptio

n al

so d

epen

ds o

n tra

nsm

issi

on ra

te

Sche

dule

tran

smis

sion

s laz

ily [P

rabh

akar

et a

l 200

1]

Can

split

traf

fic a

mon

g m

ultip

le ro

utes

at r

educ

ed ra

te [S

hah-

Rab

aey

02]

Top

olog

y C

ontr

ol

G

iven

:

A c

olle

ctio

n of

nod

es in

the

plan

e

Tran

smis

sion

rang

e of

the

node

s (as

sum

ed

equa

l)

Goa

l: To

det

erm

ine

a su

bgra

phof

the

trans

mis

sion

gra

ph G

that

is

Con

nect

ed

Lo

w-d

egre

e

Smal

l stre

tch,

hop

-stre

tch,

and

pow

er-s

tretc

h

The

Yao

Gra

ph

D

ivid

e th

e sp

ace

arou

nd e

ach

node

into

sect

ors (

cone

s)

of a

ngle

Each

nod

e ha

s an

edge

to n

eare

st n

ode

in e

ach

sect

or

Num

ber o

f edg

es is

θ

For

any

edg

e (u

,v) i

n tra

nsm

issi

on g

raph

Th

ere

exis

ts e

dge

(u,w

) in

sam

e se

ctor

such

th

at w

is c

lose

r to

v th

an u

isH

as st

retc

h ))2/

sin(

21/(1

θ−

)(n

O

u

wv

Var

iant

s of t

heY

aoG

raph

Li

near

num

ber o

f edg

es, y

et n

ot c

onst

ant-d

egre

e

Can

der

ive

a co

nsta

nt-d

egre

e su

bgra

ph b

y a

phas

e of

edg

e re

mov

al [W

atte

nhof

er e

t al 0

0, L

i et a

l 01]

In

crea

ses s

tretc

h by

a c

onst

ant f

acto

r

Nee

d to

pro

cess

edg

es in

a c

oord

inat

ed o

rder

Y

Y g

raph

[Wan

g-Li

01]

M

ark

near

est n

eigh

bors

as b

efor

e

Edge

(u,v

) add

ed if

u is

nea

rest

nod

e in

sect

or su

ch th

at

u m

arke

d v

H

as O

(1) e

nerg

y-st

retc

h [J

ia-R

-Sch

eide

ler0

2]

Is th

e Y

Y g

raph

als

o a

dist

ance

-spa

nner

?

Res

tric

ted

Del

auna

y G

raph

R

DG

[Gao

et a

l 01]

U

se su

bset

of e

dges

from

the

Del

auna

ytri

angu

latio

n

Span

ner (

O(1

) dis

tanc

e-st

retc

h); c

onst

ruct

ible

loca

lly

Not

con

stan

t-deg

ree,

but

pla

nar a

nd li

near

# e

dges

U

sed

RD

G o

n cl

uste

rhea

ds to

redu

ce d

egre

e

Span

ners

and

Geo

grap

hic

Rou

ting

Sp

anne

rs g

uara

ntee

exi

stenc

eof

shor

t or e

nerg

y-ef

ficie

nt p

aths

Fo

r som

e gr

aphs

(e.g

.,Y

aogr

aph)

eas

y to

con

stru

ct

C

an u

se g

reed

y an

d pe

rimet

er fo

rwar

ding

(GPS

R)

Sh

orte

st-p

ath

rout

ing

on sp

anne

r sub

grap

h

Pr

oper

ties o

f gre

edy

and

perim

eter

forw

ardi

ng

[Gao

et a

l 01]

for g

raph

s with

co

nsta

nt d

ensit

y

If g

reed

y fo

rwar

ding

doe

s not

reac

h lo

cal m

axim

um,

then

-

hop

path

foun

d, w

here

is

opt

imal

If

ther

e is

a

perim

eter

pat

h o

f ho

ps, t

hen

-h

op

path

foun

d

l)

(2 l

O)

(2 l

Ol

Dyn

amic

Mai

nten

ance

of T

opol

ogy

Ed

ges o

f pro

xim

ity g

raph

s eas

y to

mai

ntai

n

A n

ode

mov

emen

t onl

y af

fect

s nei

ghbo

ring

node

s

Fo

r Yao

gra

ph a

nd R

DG

, cos

t of u

pdat

e pr

opor

tiona

l to

size

of n

eigh

borh

ood

Fo

r spe

cial

ized

subg

raph

s of t

he Y

aogr

aph

(suc

h as

the

YY

gra

ph),

upda

te c

ost c

ould

be

high

er

A c

asca

ding

eff

ect c

ould

cau

se n

on-lo

cal c

hang

es

Perh

aps,

can

avoi

d m

aint

aini

ng e

xact

pro

perti

es a

nd

have

low

am

ortiz

ed c

ost

Use

ful S

truc

ture

s for

Mul

ti-ho

p N

etw

orks

G

loba

l stru

ctur

es:

M

inim

um sp

anni

ng tr

ees &

min

imum

bro

adca

st tr

ees

Lo

cal s

truct

ures

:

D

omin

atin

g se

ts: d

istri

bute

d al

gorit

hms a

nd tr

adeo

ffs

H

iera

rchi

cal s

truct

ures

:

Sp

arse

nei

ghbo

rhoo

d co

vers

Mod

el A

ssum

ptio

ns

G

iven

an

arbi

trary

mul

tihop

net

wor

k, re

pres

ente

d by

an

undi

rect

ed g

raph

A

sync

hron

ous c

ontro

l; ru

nnin

g tim

e bo

unds

as

sum

e sy

nchr

onou

s com

mun

icat

ion

N

odes

are

ass

umed

to b

e sta

tiona

ry d

urin

g th

e co

nstru

ctio

n ph

ases

D

ynam

ic m

aint

enan

ce: A

naly

ze th

e ef

fect

of

indi

vidu

al n

ode

mov

emen

ts

MA

C a

nd p

hysi

cal l

ayer

con

sider

atio

ns a

re

orth

ogon

al

App

licat

ions

of S

pann

ing

Tre

es

Fo

rms a

bac

kbon

e fo

r rou

ting

Fo

rms t

he b

asis

for c

erta

in n

etw

ork

parti

tioni

ng

tech

niqu

es

Subt

rees

of a

span

ning

tree

may

be

usef

ul d

urin

g th

e co

nstru

ctio

n of

loca

l stru

ctur

es

Prov

ides

a c

omm

unic

atio

n fr

amew

ork

for g

loba

l co

mpu

tatio

n an

d br

oadc

asts

Arb

itrar

y Sp

anni

ng T

rees

A

des

igna

ted

node

star

ts th

e f

lood

ing

pr

oces

s

Whe

n a

node

rece

ives

a m

essa

ge, i

t fo

rwar

ds it

to it

s nei

ghbo

rs th

e fir

st ti

me

M

aint

ain

sequ

ence

num

bers

to d

iffer

entia

te

betw

een

diff

eren

t ST

com

puta

tions

N

odes

can

ope

rate

asy

nchr

onou

sly

N

umbe

r of m

essa

ges i

s

;

wor

st-c

ase

time,

for s

ynch

rono

us c

ontro

l, is

)

(mO

))(

Dia

m(

GO

Min

imum

Spa

nnin

g T

rees

Th

e ba

sic

algo

rithm

[Gal

lagh

er-H

umbl

et-S

pira

83]

m

essa

ges a

nd

ti

me

Im

prov

ed ti

me

and/

or m

essa

ge c

ompl

exity

[Chi

n-Ti

ng 8

5, G

afni

86,

Aw

erbu

ch 8

7]

Firs

t sub

-line

ar ti

me

algo

rithm

[Gar

ay-K

utte

n-Pe

leg

93]:

Im

prov

ed to

Taxo

nom

y an

d ex

perim

enta

l ana

lysi

s [Fa

lout

sos-

Mol

le96

]

low

er b

ound

[Rab

inov

ich-

Pele

g 00

]

)lo

g(

nn

mO

+)

log

(n

nO

)lo

gD(

*61.0

nn

O+

)lo

g/

(n

nD

+Ω

)lo

g(

*n

nD

O+

The

Bas

ic A

lgor

ithm

D

istri

bute

d im

plem

enta

tion

of B

orou

vka

sal

gorit

hm [B

orou

vka

26]

Ea

ch n

ode

is in

itial

ly a

frag

men

t

Frag

men

t r

epea

tedl

y fin

ds a

min

-wei

ght e

dge

leav

ing

it an

d at

tem

pts t

o m

erge

with

the

neig

hbor

ing

frag

men

t, sa

y

If fr

agm

ent

a

lso

choo

ses t

he sa

me

edge

, the

n m

erge

O

ther

wis

e, w

e ha

ve a

sequ

ence

of f

ragm

ents

, whi

ch

toge

ther

form

a fr

agm

ent

1F

2F2F

Subt

letie

s in

the

Bas

ic A

lgor

ithm

A

ll no

des o

pera

te a

sync

hron

ousl

y

Whe

n tw

o fr

agm

ents

are

mer

ged,

we

shou

ld

rel

abel

th

e sm

alle

r fra

gmen

t.

Mai

ntai

n a

leve

l for

eac

h fr

agm

ent a

nd e

nsur

e th

at

frag

men

t with

smal

ler l

evel

is re

labe

led:

W

hen

frag

men

ts o

f sam

e le

vel m

erge

, lev

el in

crea

ses;

ot

herw

ise,

leve

l equ

als l

arge

r of t

he tw

o le

vels

In

effic

ienc

y: A

larg

e fr

agm

ent o

f sm

all l

evel

may

m

erge

with

man

y sm

all f

ragm

ents

of l

arge

r lev

els

Asy

mpt

otic

Impr

ovem

ents

to th

e B

asic

A

lgor

ithm

Th

e fr

agm

ent l

evel

is se

t to

log

of th

e fr

agm

ent

size

[Chi

n-Ti

ng 8

5,G

afni

85]

R

educ

es ru

nnin

g tim

e to

Im

prov

ed b

y en

surin

g th

at c

ompu

tatio

n in

leve

l fr

agm

ent i

s blo

cked

for

tim

e

Red

uces

runn

ing

time

to

)lo

g(

*n

nO

)(n

O

l)

2(l

O

Leve

l 1Le

vel 1Le

vel 2

ASu

blin

ear

Tim

e D

istr

ibut

ed

Alg

orith

m

A

ll pr

evio

us a

lgor

ithm

s per

form

com

puta

tion

over

fr

agm

ents

of M

ST, w

hich

may

hav

e di

amet

er

Tw

o ph

ase

appr

oach

[GK

P 93

, KP

98]

C

ontro

lled

exec

utio

n of

the

basi

c al

gorit

hm, s

topp

ing

whe

n fr

agm

ent d

iam

eter

reac

hes a

cer

tain

size

Ex

ecut

e an

edg

e el

imin

atio

n pr

oces

s tha

t req

uire

s pr

oces

sing

at t

he c

entra

l nod

e of

a B

FS tr

ee

R

unni

ng ti

me

is

R

equi

res a

fair

amou

nt o

f syn

chro

niza

tion)

log

)(

Dia

m(

*n

nG

O+

)(n

Ω

Min

imum

Ene

rgy

Bro

adca

st R

outin

g

G

iven

a se

t of n

odes

in th

e pl

ane,

nee

d to

bro

adca

st

from

a so

urce

to o

ther

nod

es

In a

sing

le st

ep, a

nod

e m

ay b

road

cast

with

in a

ra

nge

by a

ppro

pria

tely

adj

ustin

g tra

nsm

it po

wer

En

ergy

con

sum

ed b

y a

broa

dcas

t ove

r ran

ge

is

prop

ortio

nal t

o

Prob

lem

: Com

pute

the

sequ

ence

of b

road

cast

step

s th

at c

onsu

me

min

imum

tota

l ene

rgy

O

ptim

um st

ruct

ure

is a

dire

cted

tree

root

ed a

t the

sour

ce

α rr

Ene

rgy-

Eff

icie

nt B

road

cast

Tre

es

NP-

hard

for g

ener

al g

raph

s, co

mpl

exity

for t

he

plan

e st

ill o

pen

G

reed

y he

uris

tics p

ropo

sed

[Wie

selth

iere

t al 0

0]

Min

imum

span

ning

tree

with

edg

e w

eigh

ts e

qual

to

ener

gy re

quire

d to

tran

smit

over

the

edge

Sh

orte

st p

ath

tree

with

sam

e w

eigh

ts

Bou

nded

Incr

emen

tal P

ower

(BIP

): A

dd n

ext n

ode

into

br

oadc

ast t

ree,

that

requ

ires m

inim

um e

xtra

pow

er

M

ST a

nd B

IP h

ave

cons

tant

-fac

tor a

ppro

xim

atio

n ra

tios,

whi

le S

PT h

as ra

tio

[Wan

et a

l 01]

If

wei

ghts

are

squa

re o

f Euc

lidea

n di

stan

ces,

then

MST

fo

r any

poi

nt se

t in

unit

disk

is a

t mos

t 12

)(n

Ω

A

dom

inat

ing

set

of

is a

subs

et o

f

such

that

fo

r eac

h

,

eith

er

, or

th

ere

exis

ts

,

s.t.

.

A

-dom

inat

ing

set i

s a su

bset

su

ch th

at e

ach

node

is

with

in

hops

of a

nod

e in

.

Dom

inat

ing

Sets

),

(E

VG

=D

Dv∈

Du∈

Ev

u∈)

,(

VV

v∈

kD

kD

App

licat

ions

Fa

cilit

y lo

catio

n

A se

t of

-dom

inat

ing

cent

ers c

an b

e se

lect

ed to

loca

te

serv

ers o

r cop

ies o

f a d

istri

bute

d di

rect

ory

D

omin

atin

g se

ts c

an se

rve

as lo

catio

n da

taba

se fo

r st

orin

g ro

utin

g in

form

atio

n in

ad

hoc

netw

orks

[Lia

ng

Haa

s00]

U

sed

in d

istri

bute

d co

nstru

ctio

n of

min

imum

sp

anni

ng tr

ee [K

utte

n-Pe

leg

98]

k

An

Ada

ptiv

e D

iam

eter

-2 C

lust

erin

g

A

par

titio

ning

of t

he n

etw

ork

into

clu

ster

s of

diam

eter

at m

ost 2

[Lin

-Ger

la97

]

Prop

osed

for s

uppo

rting

spat

ial b

andw

idth

reus

e

Sim

ple

algo

rithm

in w

hich

eac

h no

de se

nds a

t m

ost o

ne m

essa

ge

The

Clu

ster

ing

Alg

orith

m

Ea

ch n

ode

has a

uni

que

ID a

nd k

now

s nei

ghbo

r ids

Ea

ch n

ode

deci

des i

ts c

lust

er le

ader

imm

edia

tely

af

ter i

t has

hea

rd fr

om a

ll ne

ighb

ors o

f sm

alle

r id

If

any

of t

hese

nei

ghbo

rs is

a c

lust

er le

ader

, it p

icks

one

O

ther

wis

e, it

pic

ks it

self

as a

clu

ster

lead

er

Bro

adca

sts i

ts id

and

clu

ster

lead

er id

to n

eigh

bors

12

34

56

78

Prop

ertie

s of t

he C

lust

erin

g

Ea

ch n

ode

send

s at m

ost o

ne m

essa

ge

A n

ode

u se

nds a

mes

sage

onl

y w

hen

it ha

s dec

ided

its

clus

ter l

eade

r

The

runn

ing

time

of th

e al

gorit

hm is

O(D

iam

(G))

Th

e cl

uste

r cen

ters

toge

ther

form

a 2

-dom

inat

ing

set

Th

e be

st u

pper

bou

nd o

n th

e nu

mbe

r of c

lust

ers i

s O

(V)

Dyn

amic

Mai

nten

ance

Heu

rist

ic

Ea

ch n

ode

mai

ntai

ns th

e id

s of n

odes

in it

s clu

ster

W

hen

a no

de u

mov

es, e

ach

node

v in

the

clus

ter

does

the

follo

win

g:

If u

has t

he h

ighe

st c

onne

ctiv

ity in

the

clus

ter,

then

v

chan

ges c

lust

er b

y fo

rmin

g a

new

one

or m

ergi

ng w

ith a

ne

ighb

orin

g on

e

Oth

erw

ise,

v re

mai

ns in

its o

ld c

lust

er

A

imed

tow

ard

mai

ntai

ning

low

dia

met

er

The

Min

imum

Dom

inat

ing

Set

Prob

lem

N

P-ha

rd fo

r gen

eral

gra

phs

A

dmits

a P

TAS

for p

lana

r gra

phs [

Bake

r 94]

R

educ

es to

the

min

imum

set c

over

pro

blem

Th

e be

st p

ossi

ble

poly

-tim

e ap

prox

imat

ion

ratio

(to

with

in a

low

er o

rder

add

itive

term

) for

MSC

an

d M

DS,

unl

ess N

P ha

s

-tim

e de

term

inis

tic a

lgor

ithm

s [Fe

ige

96]

A

sim

ple

gree

dy a

lgor

ithm

ach

ieve

s ap

prox

imat

ion

ratio

,

is 1

plu

s the

max

imum

de

gree

[Joh

nson

74,

Chv

atal

79]

)(lo

g∆=

∆O

H∆

)lo

g(lo

gn

O n

A

n Ex

ampl

eGre

edy

Alg

orith

m

Dis

trib

uted

Gre

edy

Impl

emen

tatio

n

[L

iang

-Haa

s00]

A

chie

ves t

he sa

me

appr

oxim

atio

n ra

tio a

s the

ce

ntra

lized

gre

edy

algo

rithm

.

Alg

orith

m p

roce

eds i

n ro

unds

C

alcu

late

the

span

for e

ach

node

,

whi

ch is

the

num

ber o

f un

cove

red

node

s tha

t

cove

rs.

C

ompa

re sp

ans b

etw

een

node

s with

in d

ista

nce

2 of

eac

h ot

her.

A

ny n

ode

sele

cts i

tsel

f as a

dom

inat

or, b

reak

ing

tie b

y no

de

ID ,

if it

has t

he m

axim

um sp

an w

ithin

dis

tanc

e 2.

uu

Dis

trib

uted

Gre

edy

Span

Cal

cula

tion

R

ound

1

2 2

5

5

3

3

4

43

3

Dis

trib

uted

Gre

edy

Can

dida

te se

lect

ion

R

ound

1

2 2

5

5

3

3

4

43

3

Dis

trib

uted

Gre

edy

Dom

inat

or se

lect

ion

R

ound

1

Dis

trib

uted

Gre

edy

Span

cal

cula

tion

R

ound

2

2

2

3

43

3

Dis

trib

uted

Gre

edy

Can

dida

te se

lect

ion

R

ound

2

2

2

3

43

3

Dis

trib

uted

Gre

edy

Dom

inat

or se

lect

ion

R

ound

2

Dis

trib

uted

Gre

edy

Span

cal

cula

tion

R

ound

3

1

11

Dis

trib

uted

Gre

edy

Can

dida

te se

lect

ion

R

ound

3

1

11

Dis

trib

uted

Gre

edy

Dom

inat

or se

lect

ion

R

ound

3

Low

er B

ound

on

Run

ning

Tim

e of

D

istr

ibut

ed G

reed

y

Run

ning

tim

e is

for t

he

cate

rpill

ar

grap

h, w

hich

has

a

chai

n of

nod

es w

ith

decr

easi

ng sp

an.

)(

nΩ

Sim

ply

rou

ndin

g up

sp

an is

a

cure

for t

he c

ater

pilla

r gra

ph,

but p

robl

em st

ill e

xist

s as i

n th

e rig

ht g

raph

, whi

ch ta

kes

runn

ing

time

. )

( nΩ

Fast

er A

lgor

ithm

s

-d

omin

atin

g se

t alg

orith

m [K

utte

n-Pe

leg

98]

Run

ning

tim

e is

o

n an

y ne

twor

k.B

ound

on

DS

is a

n ab

solu

te b

ound

, not

rela

tive

to th

e op

timal

resu

lt.

-app

roxi

mat

ion

in w

orst

cas

e.

U

ses d

istri

bute

d co

nstru

ctio

n of

MIS

and

span

ning

fo

rest

s

A lo

cal r

ando

miz

ed g

reed

y al

gorit

hm, L

RG

[Jia

-R-

Suel

01]

C

ompu

tes a

n

siz

e D

S in

tim

e w

ith

high

pro

babi

lity

G

ener

aliz

es to

wei

ghte

d ca

se a

nd m

ultip

le c

over

age

k)

(log*

nO

)(n

Ω

)(lo

gnO

)(lo

g2n

O

Loc

al R

ando

miz

ed G

reed

y -L

RG

Each

roun

d of

LRG

con

sist

s of t

hese

step

s.

Rou

nded

span

cal

cula

tion

: Eac

h no

de

cal

cula

tes i

ts

span

, the

num

ber o

f yet

unc

over

ed n

odes

that

c

over

s;

it ro

unds

up

its sp

an to

the

near

est p

ower

of b

ase

, eg

2.

Can

dida

te se

lect

ion

: A n

ode

anno

unce

s its

elf a

s a

cand

idat

e if

it ha

s the

max

imum

roun

ded

span

am

ong

all n

odes

with

in d

istan

ce 2

.

Supp

ort c

alcu

latio

n: E

ach

unco

vere

d no

de

cal

cula

tes

its su

ppor

t num

ber

, whi

ch is

the

num

ber o

f ca

ndid

ates

that

cov

ers

.

Dom

inat

or se

lect

ion:

Eac

h ca

ndid

ate

se

lect

s its

elf a

do

min

ator

with

pro

babi

lity

, w

here

is

th

e m

edia

n su

ppor

t of a

ll th

e un

cove

red

node

s tha

t

cove

rs.

u

vv

b

)(us

)(

/1v

med

)(v

med

uv

Perf

orm

ance

Cha

ract

eris

tics o

f LR

G

Ter

min

ates

in

roun

ds w

hpA

ppro

xim

atio

n ra

tio is

in

exp

ecta

tion

and

whp

R

unni

ng ti

me

is in

depe

nden

t of d

iam

eter

and

ap

prox

imat

ion

ratio

is a

sym

ptot

ical

ly o

ptim

alT

rade

off

betw

een

appr

oxim

atio

n ra

tio a

nd ru

nnin

g tim

e Ter

min

ates

in

rou

nds w

hpA

ppro

xim

atio

n ra

tio is

in

exp

ecta

tion

In

expe

rimen

ts, f

or a

rand

om la

yout

on

the

plan

e:

Dis

tribu

ted

gree

dy p

erfo

rms s

light

ly b

ette

r

)lo

g(lo

g∆

nO

)(lo

g∆O

)(lo

gnO

)lo

g(lo

g∆

nO

∆+

H)1(

ε

Hie

rarc

hica

l Net

wor

k D

ecom

posi

tion

Sp

arse

nei

ghbo

rhoo

d co

vers

[Aw

erbu

ch-P

eleg

89,

Li

nial

-Sak

s 92]

A

pplic

atio

ns in

loca

tion

man

agem

ent,

repl

icat

ed d

ata

man

agem

ent,

rout

ing

Pr

ovab

le g

uara

ntee

s, th

ough

diff

icul

t to

adap

t to

a dy

nam

ic

envi

ronm

ent

R

outin

g sc

hem

e us

ing

hier

arch

ical

par

titio

ning

[D

olev

et a

l 95]

A

dapt

ive

to to

polo

gy c

hang

es

Wee

k gu

aran

tees

in te

rms o

f stre

tch

and

mem

ory

per n

ode

Spar

se N

eigh

borh

ood

Cov

ers

A

n r-

neig

hbor

hood

cov

er is

a se

t of o

verla

ppin

g cl

uste

rs su

ch th

at th

e r-

zone

of a

ny n

ode

is in

one

of

the

clus

ters

A

im: H

ave

cove

rs th

at a

re lo

w d

iam

eter

and

hav

e sm

all o

verla

p

Trad

eoff

bet

wee

n di

amet

er a

nd o

verla

p

Set o

f r-z

ones

: Hav

e di

amet

er r

but o

verla

p n

Th

e en

tire

netw

ork:

Ove

rlap

1 bu

t dia

met

er c

ould

be

n

Spar

se r-

neig

hbor

hood

with

O(r

log(

n)) d

iam

eter

cl

uste

rs a

nd O

(log(

n)) o

verla

p [P

eleg

89, A

wer

buch

-Pe

leg

90]

Spar

se N

eigh

borh

ood

Cov

ers

Se

t of s

pars

e ne

ighb

orho

od c

over

s

-nei

ghbo

rhoo

d co

ver:

For e

ach

node

:

For a

ny

, the

-z

one

is c

onta

ined

with

in a

clu

ster

of

dia

met

er

Th

e no

de is

in

clus

ters

A

pplic

atio

ns:

Tr

acki

ng m

obile

use

rs

Dis

tribu

ted

dire

ctor

ies f

or re

plic

ated

obj

ects

r)

log

(n

rO

)(lo

g2n

O

ni

log

0≤

≤

r

i 2

Onl

ine

Tra

ckin

g of

Mob

ile U

sers

G

iven

a fi

xed

netw

ork

with

mob

ile u

sers

N

eed

to su

ppor

t loc

atio

n qu

ery

oper

atio

ns

Hom

e lo

catio

n re

gist

er (H

LR) a

ppro

ach:

W

hene

ver a

use

r mov

es, c

orre

spon

ding

HLR

is u

pdat

ed

Inef

ficie

nt if

use

r is n

ear t

he se

eker

, yet

HLR

is fa

r

Perf

orm

ance

issu

es:

C

ost o

f que

ry:

ratio

with

di

stan

ce

betw

een

sour

ce a

nd

dest

inat

ion

C

ost o

f upd

atin

g th

e da

ta st

ruct

ure

whe

n a

user

mov

es

Mob

ile U

ser

Tra

ckin

g: In

itial

Set

up

Th

e sp

arse

-

neig

hbor

hood

cov

er fo

rms a

re

gion

al d

irect

ory

at le

vel

A

t lev

el ,

eac

h no

de u

sele

cts a

hom

e cl

uste

r th

at c

onta

ins t

he

-zon

e of

u

Each

clu

ster

has

a le

ader

nod

e.

Initi

ally

, eac

h us

er re

gist

ers i

ts lo

catio

n w

ith

the

hom

e cl

uste

r lea

der a

t eac

h of

the

leve

ls

i 2i

)(lo

gnO

i 2i

The

Loc

atio

n U

pdat

e O

pera

tion

W

hen

a us

er X

mov

es, X

leav

es a

forw

ardi

ng

poin

ter a

t the

pre

viou

s hos

t.

Use

r X u

pdat

es it

s loc

atio

n at

onl

y a

subs

et o

f ho

me

clus

ter l

eade

rs

For e

very

sequ

ence

of m

oves

that

add

up

to a

di

stan

ce o

f at l

east

,

X u

pdat

es it

s loc

atio

n w

ith

the

lead

er a

t lev

el

A

mor

tized

cos

t of a

n up

date

is

for a

se

quen

ce o

f mov

es to

talin

g di

stan

ce

i 2 i)

log

(n

dO

d

The

Loc

atio

n Q

uery

Ope

ratio

n

To

loca

te u

ser X

, go

thro

ugh

the

le

vels

star

ting

from

0 u

ntil

the

user

is lo

cate

d

At l

evel

, q

uery

eac

h of

the

clus

ters

u b

elon

gs

to in

the

-n

eigh

borh

ood

cove

r

Follo

w th

e fo

rwar

ding

poi

nter

s, if

nece

ssar

y

Cos

t of q

uery

:

, if

is th

e di

stan

ce

betw

een

the

quer

ying

nod

e an

d th

e cu

rren

t lo

catio

n of

the

user

i 2i

)lo

g(

nd

Od

)(lo

gnO

Com

men

ts o

n th

e T

rack

ing

Sche

me

D

istri

bute

d co

nstru

ctio

n of

spar

se c

over

s in

time

[Aw

erbu

chet

al 9

3]

The

stor

age

load

for l

eade

r nod

es m

ay b

e ex

cess

ive;

us

e ha

shin

g to

dis

tribu

te th

e le

ader

ship

role

(per

us

er) o

ver t

he c

lust

er n

odes

Dis

tribu

ted

dire

ctor

ies f

or a

cces

sing

repl

icat

ed

obje

cts [

Aw

erbu

ch-B

arta

l-Fia

t 96]

A

llow

s rea

ds a

nd w

rites

on

repl

icat

ed o

bjec

ts

An

-c

ompe

titiv

e al

gorit

hm a

ssum

ing

each

nod

e ha

s tim

es m

ore

mem

ory

than

the

optim

al

Unc

lear

how

to m

aint

ain

spar

se n

eigh

borh

ood

cove

rs

in a

dyn

amic

net

wor

k)lo

glo

g(

2n

nn

mO

+

)(lo

gnO

)(lo

gnO

Bub

bles

Rou

ting

and

Part

ition

ing

Sche

me

A

dapt

ive

sche

me

by [D

olev

et a

l 95]

H

iera

rchi

cal P

artit

ioni

ng o

f a sp

anni

ng tr

ee st

ruct

ure

Pr

ovab

le b

ound

s on

effic

ienc

y fo

r upd

ates

2-le

vel p

artit

ioni

ngof

a sp

anni

ng tr

ee

root

Bub

bles

(co

nt.)

Si

ze o

f clu

ster

s at e

ach

leve

l is b

ound

ed

Clu

ster

size

gro

ws e

xpon

entia

lly

#

of le

vels

equ

al to

# o

f rou

ting

hops

Tr

adeo

ff b

etw

een

num

ber o

f rou

ting

hops

and

upd

ate

cost

s

Ea

ch c

lust

er h

as a

lead

er w

ho h

as r

outin

g in

form

atio

n

Gen

eral

idea

: -r

oute

up

the

tree

unt

il in

the

sam

e cl

uste

r as d

estin

atio

n,

-the

n ro

ute

down

-mai

ntai

n by

rebu

ildin

g/fix

ing

thin

gs lo

cally

insid

e su

btre

es

Bub

bles

Alg

orith

m

A

par

titio

n is

an

[x,y

]-pa

rtitio

n if

all i

ts c

lust

ers a

re o

f siz

ebe

twee

n x

and

y

A

par

titio

n P

is a

refin

emen

t of a

noth

er p

artit

ion

Pif

each

clus

ter i

n P

is c

onta

ined

in so

me

clus

ter o

f P.

A

n (x

_1, x

_2,

, x_k

)-hie

rarc

hica

l par

titio

ning

is a

sequ

ence

of p

artit

ions

P_1

, P_2

, ..,

P_k

such

that

-P_i

is a

n [x

_i, d

x_i

] par

titio

ning

(

dis

the

degr

ee)

-P_i

is a

refin

emen

t of P

_(i-1

)

C

hoos

e x

_(k+

1) =

1 a

nd x

_i =

x_(

i+1)

n1/

k

Clu

ster

ing

Con

stru

ctio

n

B

uild

a sp

anni

ng tr

ee, s

ay, u

sing

BFS

Le

t P_1

be

the

clus

ter c

onsi

stin

g of

the

entir

e tre

e

Pa

rtitio

n P_

1 in

to c

lust

ers,

resu

lting

in P

_2

R

ecur

sive

ly p

artit

ion

each

clu

ster

M

aint

enan

ce ru

les:

-whe

n a

new

nod

e is

add

ed, t

ry to

incl

ude

in e

xist

ing

clus

ter,

else

split

clu

ster

-whe

n a

node

is re

mov

ed, i

f nec

essa

ry c

ombi

ne c

lust

ers

m

emor

y re

quire

men

t

ad

apta

bilit

y

k

hops

dur

ing

rout

ing

m

atch

ing

low

er b

ound

for b

ound

ed d

egre

e gr

aphs

N

ote:

Bub

bles

doe

s not

pro

vide

a n

on-tr

ivia

l upp

er b

ound

on st

retc

h in

the

non-

hop

mod

el

Perf

orm

ance

Bou

nds

kk

nd

/12

3n

kdn

klo

g/1

1+