Hierarc hical Net w orks and the

LSA NSquared Problem in OSPF Routing

Alfred V Aho

Da vid Lee

Bell Lab oratories Lucen t T ec hnologies

Murra y Hill New Jersey

Abstract

With N routers in a net w ork running the

OSPF routing proto col a net w ork top ol

ogy up date can generate on the order

of N

LSA pac k ets This phenomenon

kno wn as the LSA NSquared Problem

sev erely degrades net w ork p erformance

and scalabilit y Hierarc hical OSPF net

w ork arc hitectures ha v e b een prop osed to

reduce the n um b er of LSAs that are gen

erated b y a net w ork top ology up date W e

sho w that equalsize areas minimize the

n um b er of LSAs Then w e deriv e the opti

mal n um b er of areas and the size of areas

to create the net w ork pro ducing the min

imal n um b er of LSAs Finally w e sho w

that the optimal net w ork arc hitecture re

duces the n um b er of LSAs from O N

to

O

p

N N

In tro duction

As the n um b er of routers in an Op en

Shortest P ath First OSPF net w ork in

creases the consumption of resources

within the routers suc h as CPU mem

ory and bandwidth do es not gro w lin

early but at a m uc h faster rate Within

an y net w ork there are limits to these

resources and these limits signican tly

hamp er net w ork scalabilit y F or instance

as the net w ork size gro ws the band

width and CPU consumption for pro

cessing Link State Adv ertisemen ts LSA

from the OSPF routing proto col increases

rapidly leading to routing instabilit y

Sp ecically for a fully connected net

w ork con taining N routers an y net w ork

top ology c hange caused b y a link or in ter

face going do wn or coming up can generate

O N

LSAs creating what is kno wn as

the LSA Nsquar e d pr oblem F or net w orks

with large n um b ers of routers this

o o d

of LSAs can signican tly degrade net w ork

p erformance and in tro duce routing insta

bilities

Sev eral tec hniques ha v e b een prop osed

to cop e with the LSA Nsquared problem

in OSPF net w orks A spanningtree net

w ork arc hitecture has b een prop osed

but this solution requires a proto col mec h

anism so that all the routers can agree on

a spanning tree F urthermore an y tree

link going do wn requires reestablishing a

spanning tree whic h slo ws do wn the net

w ork up dating pro cess and adds to the

complexit y of the proto col T o cop e with

the problems arising from a spanningtree

arc hitecture a congurationinformation

approac h w as prop osed Ho w ev er this

solution also requires an additional proto

col mec hanism and an incorrect congu

ration w ould in tro duce routing errors

Our goal is to reduce the n um b er of

LSAs resulting from net w ork top ology up

dates b y making OSPF net w orks scalable

and at the same time main taining the

OSPF routing proto col as it is curren tly

dened W e build on the tec hnique pro

p osed in taking adv an tage of OSPF

as a lev el hierarc hical routing proto col

Supp ose that there are N routers in an

OSPF net w ork W e partition them in to n

areas of m N n routers eac h F or ro

bustness the routers in eac h area are fully

connected ie eac h router has a link to

ev ery router in the same area In addi

tion eac h area has one or more Area Bor

der Routers ABRs to connect the areas

The ABRs from all the areas in the net

w ork are fully connected with one another

and routers in dieren t areas comm unicate

via their corresp onding ABRs

In tuitiv ely the hierarc hical net w ork ar

c hitecture reduces the n um b er of LSAs

from net w ork up dates b ecause there is no

need for a router to send LSAs directly

to the routers in the other areas In ter

area LSAs are only sen t and receiv ed b y

ABRs Y et t w o basic questions remain

unansw ered

What is the optimal size of areas so

that the n um b er of LSAs exc hanged

from eac h net w ork top ology up date is

minimized

What is the corresp onding n um b er

of LSAs exc hanged in a net w ork in

whic h the areas ha v e the optimal size

W e rst sho w that for a xed n um

b er of areas equalsize areas minimize the

n um b er of LSAs from eac h net w ork top ol

ogy up date W e then deriv e the optimal

area size that minimizes the the n um b er

of LSAs The total n um b er of LSAs for a

net w ork top ology up date is reduced from

O N

for a

at net w ork to O N

for the

optimal net w ork F or clarit y w e initially

consider the case in whic h eac h area has

only one ABR w e discuss the general case

in the conclusion

EqualSize Areas Mini

mize LSAs

Supp ose that there are N routers in a net

w ork with n areas In this section w e sho w

that the n um b er of LSAs exc hanged from

eac h net w ork up date is minimized if all ar

eas ha v e the same size

Prop osition A net w ork arc hitecture in

whic h areas are of equal size minimizes the

n um b er of LSAs exc hanged arising from

eac h net w ork up date

Pr o of Supp ose that a hierarc hical net w ork

of n areas has a minimal n um b er of LSAs

exc hanged from eac h net w ork top ology up

date but supp ose that in this net w ork

there are t w o areas A and A

of m and

m

routers resp ectiv ely with m

m

W e

rst consider the case that a linkin terface

go es do wnup in an area other than these

t w o The ABR in area A receiv es LSAs

from other ABRs and then

o o ds the

routers in A with LSAs Since the n um

b er of LSAs from the other ABRs only de

p ends on the n um b er of areas and is inde

p enden t of the size of A w e initially con

sider the LSAs generated within A from

eac h LSA receiv ed b y its ABR

The ABR in A sends m LSAs to all

the routers in the area except for itself In

resp onse eac h router other than the ABR

sends m LSAs to all the routers except

for itself and the ABR The total n um b er

of the LSAs in area A is m m

m m

Similarly the total

n um b er of the LSAs in area A

is m

Hence the total n um b er of the LSAs in the

t w o areas is m

m

Consider a dieren t net w ork with n ar

eas in whic h eac h area has the same n um

b er of routers as the previous one except

for areas A and A

whic h no w b oth ha v e

m

routers where m

m m

A similar analysis sho ws that in this

net w ork eac h linkin terface that go es

do wnup generates a total of m

LSAs Since m

m

it can b e easily

sho wn that m

m

m

Therefore the original net w ork ar

c hitecture do es not generate the minimal

n um b er of LSAs as w e had assumed a

con tradiction

The case that a net w ork up date o ccurs

in area A or A

can b e handled similarly

F rom no w on w e only consider the case

in whic h all the areas ha v e the same size

LSAs from Net w ork Up

dates

Supp ose that w e ha v e a net w ork with

N routers in whic h there are n areas

of m N n routers eac h W e w an t

to compute an optimal partition of the

routers in to areas so that the n um b er of

LSAs exc hanged from eac h net w ork up

date is minimized W e rst coun t the

n um b er of LSAs generated from eac h net

w ork up date There are t w o cases dep end

ing on whether a lo w erlev el in traarea

or upp erlev el in terarea linkin terface is

in v olv ed

Case A lo w erlev el linkin terface in an

area go es do wnup

a Consider the area where

the linkin terface go es do wnup Eac h

of the t w o adjacen t routers sends m

LSAs to all the routers in the area ex

cept for itself and the other router adja

cen t to the link In resp onse eac h of the

m routers sends m LSAs to all

the routers in the area except for itself

and the router from whic h it has receiv ed

the LSA The resulting n um b er of LSAs is

If N n is not an in teger some areas will ha v e

one more router than the others The optimalit y

argumen t can b e easily extended to this situation

m m m m m

The total n um b er of the LSAs from b oth

adjacen t routers is m m

b In eac h of the other n areas up on

receiving an LSA an ABR sends m

LSAs to all the routers in its area except

for itself In resp onse eac h of the m

routers sends m LSAs to all the routers

in the area except for itself and the ABR

for that area The total n um b er of LSAs

in the area is m m m

m

Since the ABR resp onds to the

LSAs from b oth routers adjacen t to the

up dated linkin terface the total n um b er

of the LSAs is m

c Finally let us consider LSAs among

the ABRs F or eac h LSA from a router

adjacen t to the up dated linkin terface the

originating ABR sends n LSAs to all

the ABRs except for itself Eac h of the n

ABRs sends n LSAs to all the ABRs

except for itself and the originating ABR

The resulting n um b er of LSAs is n

n n n

Since the ABR

resp onds to the LSAs from b oth routers

adjacen t to the up dated linkin terface the

total n um b er of the LSAs is n

W e therefore ha v e the follo wing result

Lemma The total n um b er of LSAs in

Case is m m n m

n

Case An upp erlev el linkin terface b e

t w een t w o ABRs go es do wnup

a F or LSAs among ABRs the argumen t

is similar to Case a The total n um b er

of LSAs is n n

b F or LSAs in eac h of the n areas the ar

gumen t is similar to Case b The total

n um b er of LSAs is m

In summary

Lemma The total n um b er of LSAs in

Case is n n n m

Optimization

W e no w determine the quan tit y n

the

optimal n um b er of areas that minimizes

the n um b er of LSAs from eac h net w ork

up date W e consider the t w o cases sep

arately

Case A lo w erlev el linkin terface in an

area go es do wnup

F rom Lemma the total n um b er of

LSAs is m m n m

n

W e w an t to minimize the

n um b er of LSAs sub ject to the constrain t

mn N W e apply Lagrange m ultipliers

and obtain the result

n

n

N N

Solving the equation w e see that

n

s

r

s

r

where

N N

T aking an

appro ximation w e see that

n

s

N

In summary

Lemma F or Case the optimal n um

b er of areas n

and the corresp onding

n um b er of routers in an area m

are

n

s

N

m

p

N

and the total n um b er of the LSAs is ap

pro ximately

p

N N

Case An upp erlev el linkin terface b e

t w een t w o ABRs go es do wnup

F rom Lemma the total n um b er of

LSAs is n n n m

W e

w an t to minimize the LSAs sub ject to the

constrain t mn N Again applying La

grange m ultipliers w e obtain

n

n

N

Solving the equation w e get

n

s

r

s

r

where

N

T aking an appro ximation w e see that

n

s

N

m

p

N

and the total n um b er of the LSAs is ap

pro ximately

p

N N

Lemma F or Case the optimal n um

b er of areas n

and the corresp onding

n um b er of routers in an area m

are

n

s

N

m

p

N

and the total n um b er of LSAs is appro xi

mately

p

N N

F rom Lemma and w e obtain our

main result

Theorem T o minimize the the n um b er

of LSAs generated from net w ork up dates

the optimal n um b er of areas and routers

in eac h area are

n

s

N

m

p

N

and the total n um b er of the LSAs from

eac h net w ork up date is appro ximately

p

N N

Figure compares the n um b ers of LSAs

from net w ork up dates b et w een the opti

mal hierarc hical net w ork and

at net w ork

It clearly sho ws ho w the LSA Nsquared

problem hamp ers the scalabilit y of IP net

w orks while the optimal hierarc hical net

w ork arc hitecture is more practical

Giv en a n um b er of routers in a net w ork

engineers ma y w an t to kno w exactly what

is the n um b er of areas and what is the

n um b er of routers in eac h area so that the

n um b er of LSAs from net w ork up dates is

minimized The reader is referred to

Conclusion

Going from a

at net w ork to the optimal

hierarc hical arc hitecture signican tly re

duces the n um b er of LSAs resulting from

net w ork up dates from O N

to O

p

N

N F or instance if N then a

at net w ork will require on the order of

LSAs for eac h net w ork up date whic h

is unacceptable The optimal hierarc hical

net w ork has areas with eac h area con

taining routers The n um b er of LSAs

from eac h net w ork up date is no w on the

order of

t w o orders of magnitude

smaller than that in the

at net w ork

F or clarit y w e considered the case in

whic h eac h area has only one ABR An op

timal hierarc hical net w ork has n areas and

hence ABRs with appro ximately n

LSAs

exc hanged for eac h net w ork up date Sup

p ose that the n areas ha v e k

i

i n

ABRs resp ectiv ely Then there are

P

n

k

i

ABRs If the n um b er of ABRs in the areas

is small

P

n

k

i

is still on the order of n and

the resulting n um b er of LSAs exc hanged

from a net w ork up date remains of a simi

lar order Ho w ev er if the k

i

s are close to

m ie a large p ortion of the routers in an

area are ABRs then

P

n

k

i

is on the order

of N and the resulting LSAs exc hanged

from a net w ork up date is of order N

that of a

at net w ork

Since the areas and routers are fully con

nected the net w ork is robust On the

other hand it ma y tak e hops for all the

routers to b e informed of a net w ork top ol

ogy c hange instead of F or high sp eed

routers the time for t w o additional hops

is negligible

Hierarc hical net w orks w ere rst studied

b y L Kleinro c k and F Kamoun

They w an ted to minimize the size of rout

ing tables and to reduce the consumption

of CPU memory and bandwidth for net

w ork up dates They studied hierarc hical

net w orks with an arbitrary n um b er of lev

els and sho w ed that the optimal n um b er of

lev els is ln N where ln is the natural loga

rithm On the other hand for a t w olev el

hierarc hical net w ork as in OSPF the opti

mal n um b er of areas degree of clustering

is

p

N with

p

N no des in eac h area

In this pap er w e fo cused on the mini

mization of the n um b er of LSAs exc hanged

from eac h linkin terface going do wnup

and obtained a dieren t result W e com

puted the n um b er of LSAs exc hanged af

ter eac h net w ork top ology up date F or a

series of net w ork top ology up dates if the

ABRs aggregate LSAs then the amor

tized n um b er of LSAs from all the up dates

is smaller

Ac kno wledgemen ts

Insigh tful discussions and commen ts from

R Hao are deeply appreciated

References

F Kamoun and L Kleinro c k

Sto c hastic P erformance Ev aluation of

Hierarc hical Routing for Lare Net

w orks Computer Networks V ol

pp

L Kleinro c k and F Kamoun

Hierarc hical Routing for Large Net

w orks P erformance Ev aluation and

Optimization Computer Networks

V ol pp

S Kini and R Dub e Redun

dan t Link State Adv ertisemen t Re

duction in OSPF T e ch Memo Bel l

L ab or atories

J T Mo y OSPF A natomy

of an Internet R outing Pr oto c ol

AddisonW esley

J T Mo y M K Glrish and P Ka vi

Cascades Approac h to w ards

Building Scalable Wide Area Net

w orks Memo Casc ade Communic a

tions Corp

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labscomuserleeda vidospflsa

0

2000

4000

6000

8000

10000

0

0.5

1

1.5

2

x 10

6

Number of Routers

Number of LSAs

Flat vs. Optimal Hierarchical Network

opt. hierarch. network

flat network

Figure A Comparison of LSAs

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