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
h ttpinsideb ell
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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