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USC Computer Science Technical Reports, no. 560 (1993)

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USC Computer Science Technical Reports, no. 560 (1993)

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Content The T radeos of Multicast T rees and Algorithms
Liming W ei Deb orah Estrin
Computer Science Departmen t Computer Science Departmen tISI
Univ ersit y of Southern California Univ ersit y of Southern California
Los Angeles CA Los Angeles CA lw eiuscedu estrinuscedu
Abstract
Multicast trees can b e shared across sources shared trees
or ma y b e sourcesp ecic shortest path trees Inspired b y
recentin terests in using shared trees for in terdomain m ul
ticasting w ein v estigate the tradeos among shared tree
t yp es and source sp ecic shortest path trees b y comparing
p erformance o v er b oth individual m ulticast group and the
whole net w ork The p erformance is ev aluated in terms of
path length link cost and trac concen tration
W e presen t sim ulation results o v er a real net w ork as w ell
as random net w orks under dieren t circumstances One
practically signican t conclusion is that mem b er or sender
cen tered trees ha v e go o d dela y and cost prop erties on a v
erage but they exhibit hea vier trac concen tration whic h
mak es them inappropriate as the univ ersal form of trees for
all t yp es of applications
Keyw ords Multicast Routing Scalabilit y Cen ter Place
men t Strategy
Intro duction
Multimedia comm unication is often m ultip oin t and has con
tributed to the demand for m ulticast supp ort moreo v er
m ultimedia applications are often high bandwidth and dela y
sensitiv e Therefore it is essen tial to consider the eciency
and qualityof m ulticast distribution trees The problem
of computing the optimal m ulticast path in the shap e of a
tree or a group of trees has man y p oten tial solutions T radi
tional IP m ulticast proto cols ha v e solely used source ro oted
shortest path trees not b ecause it is the b est bandwidth
sa ving strategy but based on the fact that to da ys m ulticast
applications are primarily small scale and lo cal area F actors
suc h as proto col simplicit y and con v enience of dev elopmen t
ha v e dominated proto col design activities In the con text of
largescale wide area net w orking where resources are not as
plen tiful as in the lo cal area net w ork the previouslyignored
dierences b et w een tree t yp es ma y lead to signican tdier ences in cost and p erformance of m ulticast services Ho w
ev er to date there ha v e not b een systematic comparisons
among the dieren t solutions
In this pap er w e judge the qualit y of a tree according to
the follo wing three dimensions
Lo w dela y The dela yofam ulticast tree is ev aluated
in terms of the endtoend dela ybet w een a source and
This w ork has b een supp orted bya gran t from Sun Microsystems
inc Moun tain View California and b y NSF grantCD A receiv er relativ e to the shortest unicastpath dela ybe t w een the same source and receiv er
Lo w cost There are t w o dieren t costs asso ciated
with a m ulticast tree
a Cost of total bandwidth consumption b Cost of tree state information
In this pap er w e only deal with a and lea veb to
future analysis of proto col o v erhead
Ligh t trac concen trationWhen a m ulticast group
establishes its deliv ery trees across the net w ork traf
c from dieren t sources ma y share links that are not
shared when eac h source uses its o wn shortest path
tree W e compare the maxim um n um ber of o ws
on
a unidirectional link as a simple measure of the degree
of trac concen tration
When using the ab o veev aluation criteria w ebearin
mind the restrictions of real w orld large scale net w orks eg
limited kno wledge of global top ology at eachnet w ork no de
and need for algorithms that w ork ecien tly in a distributed
manner
The t yp es of trees can b e roughly divided in to source
sp ecic shortest path trees and group shared trees In the
former tree t yp e a shortest path tree ro oted at eac h sending
source needs to b e established in the later t yp e only one tree
shared b y all sources within the m ulticast group needs to b e
main tained In the past the v ast ma jorit y of researchon
tree t yp es ha v e b een ab out shared trees One common class
of shared trees under frequentin v estigation are the Steiner
minimal tree and its sub optimal appro ximations and
v ariances dealing with constrained dela ys The main ob
jectiv e of these algorithms w as to ac hiev e optimal cost The
other class of shared trees is c enterb ase d tree These w ere
mainly in tro duced in resp onse to other proto col design re
quiremen ts suc h as reduction of setup state information and
the need for proto col rendezv ous mec hanisms In this
pap er wein v estigate the dela y cost and trac concen tration
c haracteristics of v arious tree t yp es Our w ork encompasses
t w o elds the theoretical algorithms eld where most group
shared tree algorithms researc hes ha v e b een carried out and
the net w ork proto col design eld where real w orld proto cols
ha v e b een designed and studied A review of the related
bac kgrounds is necessary Wecalla streamofpac k ets on a link originated from a particular
source a o w
To appear in proceedings of the International Conference on Computer Communications and Net w orks
Background
As bac kground in this section w e discuss shortest path tree
Steiner minimal tree and trac concen tration
Sho rtest P ath T rees
A Shortest P ath T ree SPT ro oted at the source is com
p osed of the shortest paths b et w een the source and eachof
the receiversinthe m ulticast group Multicasting eliminates
duplicate data pac k et copies that w ould otherwise tra v erse
those links that are common to t w o or more of the source
toreceiv er shortest paths Ho w ev er a SPT algorithm do es
not attempt to minimize the total cost of distribution
Sourcero oted shortest path trees are easy to compute
and can b e implemen ted in a distributed fashion ecien tly In net w orks consisting of symmetric links or paths rev erse
path forw arding RPF algorithms can b e used to deriv e
shortest path trees from the unicast routing mec hanisms When asymmetric paths exist RPF will pro vide r e
verse shortest paths
or distributed link state proto cols suc h
as MOSPF can b e used to compute shortest path trees
Although not oering minimal cost paths proto cols based
on shortest path trees ha v e b een adopted most widely This is due to the fact that when compared with m ulti
ple unicast transmissions SPTbased m ulticast already pro
vides substan tial sa vings in link cost and it helps to a v oid
fanout problems at sources F or a virtual net w ork suc h
as the MBONE with relativ ely few globallyactiv em ul
ticast groups SPTs are satisfactory This is b ecause the
net w ork is not ric h in connectivit y and therefore dieren t
t yp es of trees w ould b e mapp ed to the same routes an yw a y P erhaps most imp ortan t is that to date the con trol asp ect
of proto cols instead of the t yp e of distribution trees dom
inates the eciency Ho w ev er to supp ort increasing usage
and large scale applications there is a need for proto col de
signers to explore prop erties of sharedtree t yp es The next
subsection discusses the minimal cost shared tree t yp e
Steiner Minimal T ree
A Steiner Minimal T ree SMT is dened to b e the minimal
cost subgraph tree spanning a giv en subset of no des in
a graph Since the SMT for all sources within a
m ulticast group is the same irresp ectiv e of the role of sender
or receiv er the n um b er of state en tries needed to main tain
the tree is only p er group Th us it scales w ell for big
m ulticast groups with large n um b ers of senders The Steiner
minimal tree problem has b een studied in tensiv ely in the
area of algorithms for the past half cen tury It is w ell kno wn that computation of a SMT is NP
complete and is not exp ected to ha v e p olynomial time so
lutions This computational complexit y prohibits on
demand computation o v er a reasonablysized graph Karp
has suggested sev eral tec hniques to reduce the problem size
for SMT computations Ho w ev er these graph reduction
tec hniques are highly graph dep enden t F or computer net
w orks as complex as the In ternet where the a v erage no de
degree of routers is higher than graph reduction will not
b e eectiv e enough in reducing the computational demand
to a practical range
Rev erse shortest paths will ha v e higher dela y than forw ard short
est paths b ecause the data follo ws the shortest path fr om the receiv er
instead of the shortest path to the receiv er
XY
Z
X
Y
Z
X
Y
Z
XY
Z

(b) Shared Tree
(c) Source Specific Tree
(a) Sample Network
Figure T rac concen tration example
Because of the diculties in obtaining SMT in larger
graphs it is often deemed acceptable to use near optimal
trees instead of SMTs V arious near optimal algorithms ex
ist that pro duce go o d appro ximations to SMT As will b e
discussed later no existing SMT algorithms can b e easily ap
plied in practical m ulticast proto cols designed for large scale
net w orks but SMT itself do es pro vide the absolute limit on
the minim um link cost and serv es as a go o d reference to
measure the costoptimalit y of other alternativetree t yp es
suc h as shortest path trees
In addition to computational o v erhead the w orstcase
maxim um endtoend path length of a SMT can b e as long
as the longest acyclic path within the graph Although the
w orst case scenario is unlik ely in net w orks with the ric h con
nectivityt ypical of to da ys net w orks it is imp ortan ttokno w
in the aver age case ho w go o d or bad a path length along a
SMT can b e In section w e will presen t sim ulation re
sults of a near optimal SMT algorithm o v er random graphs
T rac Concentration
Although trac concen tration determines the eectiv enet w ork capacit y for m ulticast applications the problem has
receiv ed less atten tion than the dela ycost tradeos in the
area of m ulticast routing A trac concen tration example
is illustrated in gure The net w ork is a simple threeno de
fully connected graph where all no de pairs are connected b y
symmetric links in opp osite directions There is a m ulticast
group consisting of receiv ers on no des X Y and Z and
t w o sources X and Z sending trac at unit rate Fig b
sho ws a shared tree used b y all senders of the group Fig c sho ws sourcesp ecic shortest path trees In g b link
Z Y has a load of o ws while in g c all links ha v e
a maxim um load of o w In Section w e presen t sim
ulation studies of trac concen tration in large graphs with
man y groups
In summary minimal cost and minimal dela y cannot
b oth b e ac hiev ed with an y single t yp e of tree With re
sp ect to dela y and cost shortest path trees SPT whic h
are source ro oted pro vide minimal dela y at the exp ense
of cost whereas Steiner minimal trees SMT shared p er
group minimize cost at the exp ense of dela y Bet w een them
are a sp ectrum of dieren tt yp es of trees oering dieren t
tradeos In addition dieren t algorithms use dieren t
strategies to place the routes and ma y result in dieren t
degrees of trac concen tration
The rest of this pap er is organized as follo ws Section presen ts a few t ypical candidate p olynomial time shared tree
algorithms Section describ es the net w ork top ology and
random net w ork mo del w e used in our study and Section
presen ts and analyzes our sim ulation results
P olynomial Time Algo rithms fo r Group Sha red T rees
Group shared trees are used in t w o prop osed mec hanisms
for scalable m ulticast routing Here w e briey discuss
related w ork in p olynomial time sharedtree computation
algorithms W e will presen t comparisons based on sim ula
tions of the relev antsc hemes in section W e divide these
algorithms in to t w o ma jor categories Pseudo Steiner T ree
algorithms and Cen ter Based T ree algorithms
App ro ximations of Steiner Minimal T ree
As men tioned b efore the Steiner Minimal tree is a NP
complete problem As a reference example w ec ho ose the
algorithm in v en ted b y Kou Mark o wsky and Berman re
ferred to as the KMB algorithm to appro ximate SMTs
It has b een estimated that the cost of a tree generated with
the KMB algorithm a v erages more than the cost of a
SMT Ho w ev er the KMB algorithm in its original form
needs the complete net w ork top ology and therefore is not
practical for large wide area in ternets
There exist distributed v ersions of KMB suchas the
one prop osed in where eac h no de only needs partial
kno wledge of the net w ork top ologyHo w ev er the required
message complexityw ould signican tly complicate proto col
design This issue in addition to con v ergence and stabil
it y problems mak es it impractical for to da ys in ternet w ork
en vironmen t
Doar Leslies Naiv e algorithm for construction of routes
for dynamic m ulticast groups computes the m ulticast route
b y com bining the shortest paths across initial m ulticast group
mem b ers then joining new mem b ers to the nearest attac h
men t p oin t on the existing tree Their sim ulation result
sho w ed that the cost of the naiv e trees is within times
that of the KMB trees and their maxim um path length is
around of the KMB trees
Although this particular algorithm requires complete kno wl
edge of the net w ork top ologyit ma y b e p ossible to mo dify it
for use without global kno wledge Ho w ev er b efore consid
ering it as a candidate for real net w orks one question that
needs to b e answ ered is ho w it compares with the widely
used shortest path trees The answ er can b e deriv ed from
the comparison of KMB trees and shortest path trees whic h
w e presen t in Section CenterBased T rees
T o cop e with the un b ounded dela y problems of the near
optimal Steiner trees W all prop osed sev eral cen terbased
tree algorithms The simplicit y of this class of algorithms
mak es it desirable for the design of practical proto cols and
w as used as the basis for the Core Based T ree in terdomain
m ulticast routing proto col as w ell as for the shared tree
mo de of another in terdomain m ulticast prop osal called PIM
Cen terBased T ree as the name indicates uses a short
est path tree ro oted at a no de in the cen ter of the net w ork
W all prop osed t w o ma jor strategies for lo cating cen
ters Cho ose the optimal net w ork no de so that the result
ing cen ter tree can ha v e minimal maxim umdela yor a v erage
dela y among all group mem b ers and senders Optimally
c ho ose a mem b er or sender of the group as the cen ter so
that the tree has minimal maxim umdela yora v eragedela y
among all mem b ersender cen tered trees W e denote the for
Figure The logical top ology of the early ARP Anet
mer category dela yoptimal CBT
the later dela yoptimal
MSPT for Mem b ersenderro oted Shortest P ath T ree
In this pap er w e also consider a third cen ter placemen t
strategy the minimal cost cen ter placemen t Suc h a tree
has the minimal sum of treelink costs W e call this cost
optimal CBT or if the cen ter is at a mem b er or sender
costoptimal MSPT
It is pro v en in that the maxim um delayof anoptimal
maxim umdela y cen ter based tree is b ounded at times that
of the maxim um dela y in shortest path trees It is also sho wn
that for optimal maxim umdela y MSPT the a v erage of the
maxim um dela ys o v er all senders turns out to b e less than
times the a v erage of maxim um dela ys along the shortest
paths
The fact that there exists dela y b ounds for these kind of
trees is encouraging Ho w ev er for practical purp oses what
w e really w anttokno w is not only the w orst case b ound
but also the aver age case maxim um delayof suc h a tree and
the distribution of suchdela ys Another unkno wn factor is
the cost of suc h trees are they c heap er than source ro oted
shortest path trees for pac k et deliv ery
Note that it is dicult to use CBTs in their original form
in real m ulticast proto cols b ecause nding the cen ter
for a group is an NPcomplete problem and it requires
kno wledge of of the whole top ology Alternativ e practical
forms can b e based on heuristic cen ter placemen t strategies
or use MSPT instead The sim ulation results in section
will sho w whether MSPT is a viable alternativ e
Finally to consider an y algorithm for practical applica
tion w eneedto knowhowev enly it distributes the routes
ie are there serious trac concen tration problems that re
duce net w ork utilizations Again sim ulation will b e used
to answ er these questions
Net w o rk T op ologies
W e rst sim ulate the SPT KMB and dieren tcen ter based
trees using the top ology of the early ARP Anet sho wn in
gure This net w ork has no des The a v erage no de
degree is relativ ely lo w b ecause of the scarcityof
link resources As the net w ork connectivityimpro v es the
a v erage no de degree will b e higher
W e then use sim ulations o v er dieren t classes of random
graphs to capture the comprehensivec haracteristics of the
algorithms and trees W e adopted the random graph mo del
in tro duced in whic h can generate a v ariet y of dieren t
graphs with classiable features connectivit y degrees dif
feren t edge distributions One adv an tage of this mo del o v er
F or formal denitions of these tree t yp es see app endix
N=20 ALPHA=0.10 BETA=2.800 Seed=11 NodeDegree=3.9 N=20 ALPHA=1.00 BETA=0.310 Seed=11 NodeDegree=3.9
a ! b ! Figure Random graphs of dieren t s
a purely random mo del is that it can more easily b e corre
lated to real w orld net w orks
In this graph mo del graphs are generated in t w o steps
no de co ordinates assignmen ts and edge additions The n
v ertices are randomly distributed o v er a rectangular co ordi
nate grid and are assigned in teger co ordinates Edges are
in tro duced according to the edge probabilit y function whic h
tak es a pair of no des u v as its v ariables
P u v ! e
d uv L
where d u v is the distance from no de u to v L is the
maxim um shortest path distance b et w een an y pair of no des
in the net w ork often called net w ork diameter
A larger v alue of increases the ratio of the
n um b er of long edges vs short edges and a bigger results
in a larger a v erage no de degree of the whole graph Figure illustrates the eect of using dieren tv alues of it sho ws
t w o no de graphs of the same a v erage no de degree and
under the same no de placemen t
W e assign the dela y of a link to b e the distance b et w een
the t w o end no des in the sim ulations
Exp erimental design
Because SPTs ha v e already ac hiev ed m uc h success in prac
tical proto col designs w e use the ratio of eac h measuremen t
on a certain tree vs the measuremento v er corresp onding
SPTs as the metric for comparisons among dieren talgo rithms In the follo wing presen tation w e use MaxD to de
note the maxim um dela y exp erienced from an y source to an y
receiv er along a sp ecic tree and R MaxD to denote the ra
tio b et w een MaxD of that particular tree and the MaxD of
corresp onding SPTs Av eD denotes the a v erage of the max
im um dela ys of all sources and R Av eD is the resp ectiv e
Av eD ratio to SPTs Cost ratio denotes the ratio b et w een
the sum of link cost of a particular tree and the sum of link
cost of the corresp onding SPT The app endix giv es formal
denitions of these terms
According to results from when the size of random graphs
is big enough most graph prop erties will hold when the graph size
gro ws ev en bigger W e rep eated our exp erimen ts with t w o dieren t
graph sizes and found the measured results are consisten t
The original mo del has restriction of W e found that
larger v alues of bey ond when com bined with appropriate small
v alues also generate graphs that sub jectiv ely at least app eared
to b e of practical signicance This observ ation w as made through
a Xwindo w based Random T op ology GeneratorPreview er that w e
dev elop ed to help visualize dieren t graphs
0 102030
Group Size
1.0
1.2
1.4
1.6
1.8
Ratio
KMB/SPT delay
Optimal-cost CBT/SPT delay
MSPT/SPT delay
Optimal-delay CBT/SPT delay
Figure Dela y ratios of KMB CBT and MSPT in
ARP Anet top ology
0 102030
Group Size
0.70
0.75
0.80
0.85
0.90
0.95
1.00
1.05
1.10
Cost Ratio
KMB/SPT cost
Optimal-cost CBT/SPT cost
MSPT Cost/SPT
Optimal-delay CBT/SPT cost
Figure Cost ratios of KMB CBT and MSPT in ARP Anet
top ology
W e constructed sets of exp erimen ts one to to measure
the costs and dela ys of trees the other to measure the trac
concen trations with dieren t tree t yp es
In the rst set of exp erimen ts w e compare KMB trees
with SPTs then cen ter based trees with SPTs Comparison
of SPT and KMB trees can rev eal howm uc h more ro om
for link cost sa vings wemayha vebey ond SPTs and ho w
m uc h more link dela ys w e should exp ect if suc h cost sa vings
are ac hiev ed using KMB trees Comparisons b et w een SPT
and cen ter based trees illustrate howm uc h more delaywill
b e incurred b y cen ter based trees on a v erage and whether
they use more or less bandwidth resource than the shortest
path trees
Since cen ter based trees are the only practical candi
date for constructing group shared trees in real w orld proto
cols w e only compare trac concen trations of cen ter
based trees with SPTs
The parameters that could aect the p erformance of dif
feren t distribution tree t yp es are Reasonableness of
graphs ie the prop ortion of short links vs long links
whichw as susp ected to ha veinuence o v er the delayand
cost of trees
graph no de degree m ulticast group size
n um b er of receiv er mem b ers in curren t group n um ber
of sources sending to the group Distribution of sources
and receiv er mem b ers and graph size
Wesa y a graph is more reasonable if there is a higher probabil
it y for a no de to b e connected to a near neigh b or than to a distan t
neigh bor
0 102030
Group Size
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
Ratio (KMB/SPT)
(a)
R
R
Cost_ratio
MaxD
AveD
0 102030
Group Size
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
Ratio (CBT/SPT)
R
R
Cost_
ratio
MaxD
AveD
(b)
0 102030
Group Size
0.5
1.0
1.5
2.0
Ratio (MSPT/SPT)
(c)
R
R
Cost_
ratio
MaxD
AveD
Figure Eect of group size on dela ys and costs in no de
random graphs a KMB b CBT c MSPT
Simulation Results
Most exp erimen ts in v olving random graphs w ere run o v er
graphs of t w o dieren t sizes no des and no des In
the dela y and cost comparisons for eac h random graph gen
erated a randomly selected m ulticast group w as put in the
graph then the shortest path trees the cen ter based trees
CBT and MSPT and the KMB trees w ere computed W e
assume eachm ulticast group mem b er can tak e the role of
b oth sender and receiv er Figure through g are the re
sults for the dela y and cost comparisons In these pictures
there are runs at eac h data p oin t Figure and g are the results for trac concen tration exp erimen ts There
w e also in v estigate situations where the n um b ers of senders
and receiv ers are not equal
Dela y and Cost SPT KMB CBT and MSPT
W e rst v ary sizes of the groups and see ho w the p erfor
mance c hanges Figures and sho w the sample means of
dela y ratios and cost ratios of KMBSPT CBTSPT and
optimaldela y MSPTSPT in the ARP Anet top ology
The
group size is v aried from to F or eac h group size w eran domly generate groups and compute the dieren tt yp es
of trees The error bars represen t the condence in ter
v als of the means In gure the a v erage dela ys of KMB is
signican tly larger than the other t yp es of trees esp ecially
when group size gets larger The a v erage dela ys of optimal
cost CBT are ab o v e the optimaldela y MSPT whic hin turn are ab o v e the optimaldela y CBT In gure KMB has the
minimal a v erage cost the optimalcost CBT is the second
lo w est Optimaldela y CBT and optimaldela y MSPT b eing
close to eac h other ha v e the highest a v erage cost in this com
parison Since it is dicult to adapt optimalcost cen ter tree
algorithms to an ecien t distributed algorithm real w orld
proto cols w ould c ho ose a form closer to optimaldela yCBT
or MSPT Therefore w e will omit results ab out optimal
cost cen ter trees and only sho w results ab out optimaldela y
W e omitted the a v erage dela ys of all receiv ers here The relation
ship among dieren t trees a v erage dela ys are similar to that of the
plotted a v eragemaxim um dela ys
0 102030
Normalized Count (%)
0.5
1.0
1.5
2.0
2.5
Ratio (KMB/SPT)
0 102030
0.5
1.0
1.5
2.0
2.5
Ratio (KMB/SPT)
R_MaxD Histogram
0 102030
Normalized Count (%)
0.5
1.0
1.5
2.0
2.5
0 102030
0.5
1.0
1.5
2.0
2.5
R_AveD Histogram
0 102030
Normalized Count (%)
0.5
1.0
1.5
2.0
2.5
0 102030
0.5
1.0
1.5
2.0
2.5
Cost_ratio Histogram
(a)
(b)
Figure Histograms of dela ys and costs for KMB trees
a Histograms for group size of b Histograms for group
size of cen ter trees Unless otherwise sp ecied CBT and MSPT
refer to optimaldela y CBT and MSPT resp ectiv ely W e then rep eated the same exp erimento v er a large n um
b er of random graphs Figure sho ws the eect of group
p opulation c hanges on the KMB and cen ter tree dela ys and
costs The results w ere tak en from no de graphs where
a v erage no de degree is restricted to plus or min us In
this gure and all subsequen t gures the error bars rep
resen t the standard deviations of the data set to giv ean
indication ho w wide the measured data is distributed
Histograms are plotted in gure to sho w the distribu
tions of the three parameters MaxD Av eD and C ost r atio
at group sizes of gure a and gure b F or
space reasons w e only include the KMB histograms here
The bin size is xed at Note that the horizon tal axis
of the histograms are normalized Eg a bar of length with ratio ! means of the measured data v alues are
at ratio of to " It can b e observ ed from gure that the dela ys of KMB
trees tend to gro w larger with larger groups while their costs
in comparison with shortest path trees tend to b e lo w er The
dela y and cost curv es of cen ter trees ho w ev er are rather at
With larger groups cen ter trees tend to ha v e shorter dela ys
than KMB trees but the p eaks of their cost ratio histograms
are higher than that of the KMB trees not sho wn here
KMB trees in general ha v e bigger v ariations in dela y than
cen ter trees The tails of R MaxD R Av eD in g a and b
extend to ab out while in corresp onding histograms for
CBTs and MSPTs the tails are b elo w and larger groups
ha v e sligh tly shorter tails The ab o v e observ ation exhibits
an in teresting asp ect of the fatesharing nature of cen ter
trees They are not optimal for ev ery one and o v erall they
are also not v ery bad for ev ery one either
W e rep eated the ab o v e exp erimen t in no de random
graphs with group sizes ranging from to The results
ha v e similar trends KMB dela y curv es are higher the a v
erage ratio of dela ys is around The KMB to SPT cost
ratio curveislo w er b y ab out CBT and MSPT dela y
curv es though sligh tly higher are not signican tly dieren t
than in no de graphs
02 46 8 10
Node Degree
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
Ratio (MSPT/SPT)
02 46 8 10
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
Ratio (CBT/SPT)
02 46 8 10
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
Ratio (KMB/SPT)
R_MaxD
02 46 8 10
Node Degree
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
02 46 8 10
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
02 46 8 10
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
R_AveD
02 46 8 10
Node Degree
0.5
0.6
0.7
0.8
0.9
1.0
1.1
02 46 8 10
0.5
0.6
0.7
0.8
0.9
1.0
1.1
02 46 8 10
0.5
0.6
0.7
0.8
0.9
1.0
1.1
Cost_ratio
(a)
(b)
(c)
Figure Comparisons of dela y and cost in no de graphs
a KMB tree vs SPT b cen ter tree vs SPT c MSPT vs
SPT
Noww e try to v ary the no de de gr e e of a graph and mea
sure the p erformance Figure sho ws the eects of dieren t
no de degrees on the dela y and cost prop erties All graphs
ha v e no des all m ulticast groups ha v e mem bers When
pro cessing sim ulation data the a v erage no de degrees are
rounded to the nearest in teger The v ertical axe are the a v
erages of R M axD R Av eD and C ost r atio resp ectiv ely Solid
lines represen t graphs with ! dotted lines represen t
graphs with ! A few observ ations that can b e made
from this picture are
All algorithms are relativ ely insensitiv e to the reason
ableness of graphs
The maxim um dela y within a KMB tree tends to b e
larger than that of a CBT tree The maxim um dela y
of a KMB tree increases faster when the a v erage no de
degree increases Note that MSPT R MaxD and R Av eD
curv es are quite close to those of CBTs
The KMB Cost r atio curv e decreases faster MSPT
C ost r atio is close to CBT Cost ratio Observ ations and ab o v e suggest that in the range of
graphs exp erimen ted MSPT will on the a v erage b e almost
as go o d as the optimal CBT
The ab o v e exp erimen ts w ere rerun o v er no de graphs
also with mem b er groups The R MaxD R Av eD and C ost ratio
all ha v e similar trends KMB R MaxD and R Av eD are sligh tly
higher b y and KMB Cost ratio curveissligh tly at
ter The c hanges in MSPT and CBT curv es are v ery small
Therefore at higher no de degrees the dierences b et w een
234567 89
Node Degree
5000
6000
7000
8000
9000
TCmax
TCmax , 300 groups per graph
234567 8 9
Node Degree
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
Ratio
Ratio of TCmax (CBT/SPT, MSPT/SPT)
SPT
CBT
MSPT
CBT/SPT
MSPT/SPT
a b
Figure T rac Concen tration in no de graphs with mem b er groups senders p er group
KMB R MaxD and CBT R MaxD and the dierences b et w een
KMB R Av eD and CBT R Av eD are sligh tly smaller than in
no de graphs
T rac Concentration SPT CBT and MSPT
In the trac concen tration exp erimen ts weuse a n um ber of
xed size m ulticast groups randomly placed in a graph and
part of the group mem b ers are assigned the roles of senders
randomly Assuming eac h source of the group generates
trac at constan t unit rate the total n umber ofunittrac
o ws that tra versealinkis coun ted
F or the same graph
and groups cen ter based trees and shortest path trees are
computed and link loads coun ted
The maxim um link loads for SPT CBT and MSPT are
sho wn in g a Figure a sho ws that as the a v er
age no de degree of the random graphs gro ws from to the SPTs maxim um link load decreases This is b ecause
there are more redundan t links when a v erage no de degree
is higher and there exist alternate paths b et w een man y
pairs of no des Ho w ev er cen ter based trees maxim um link
loads hardly c hange when the a v erage no de degree increases
whic h is not surprising considering the fact they are cen ter
or singlesourcero oted shortest path trees Fig b sho ws
the ratio of maxim um link loads of CBT and MSPT vs that
of SPTs W eha v e run the same exp erimen t under iden tical
congurations but with senders p er group the results are
comparable
This exp erimen t suggests that in net w orks with lo w con
nectivit y degree the link utilization pattern of cen terbased
trees will b e v ery close to that of shortest path trees Ho w
ev er when the a v erage no de degree increases cen terbased
trees main tain almost at maxim um link loads whereas the
maxim um link load of shortest path trees drop signican tly Often senders within the same m ulticast group ha v e similar send
ing rates but senders for dieren t groups mayha v e dieren t sending
rates The purp ose here is to sho w tendencies of route distributions
th us suc h uniform sending rate assumption w ould suce
Eachph ysical link is treated as t w o unidirectional links connect
ing the same pair of no des in opp osite directions
0 2000 4000 6000
Link Load
0
20
40
60
80
100
Count
0 2000 4000 6000
Link Load
0
20
40
60
80
100
0 2000 4000 6000
Link Load
0
20
40
60
80
100
a SPT b CBT c MSPT
Figure Distribution of link loads in the same graph
This indicates that added alternativ e paths are not b eing
sucien tly utilized b y cen ter based trees
T o see what prop ortions of links are highly loaded under
dieren t algorithms w ecoun ted the n um b er of links under
dieren t loads Figure sho ws the distribution of link loads
within one sp ecic no de graph The a v erage no de degree
is there are activ e groups all ha ving mem bers There are senders within eac h group When the n um ber
of senders increases from to with all three t yp es of trees
the distributions of maxim um link load roughly retain the
same shap e except that the lengths of the tails extend to
ab out ten times those in sender cases Only results from
sender exp erimen ts are sho wn in gure In gure the SPT histogram prole drops smo othly
with larger link loads more links are ligh tly loaded and
smaller n um b er of links ha v e high loads The CBT and
MSPT histograms b oth ha v e a narrowhighpeakatnear represen ting links underutilized The p eak is follo w ed b y
a long tail with a rather signican t p ortion at the end of
the tail If w e drawa v ertical line in the t w o righ t pictures
in g at the p osition equal to the maxim um link load
of the corresp onding SPT link load histogram as sho wn in
g b c in dotted lines the area to the righ tofthe line
represen ts the n um b er of links that needs higher link capac
ities to service the same conguration of m ulticast groups
as in their SPT coun terpart The link load distribution is
closely related to the distribution of memberlocationsofthe
m ulticast groups Changing the set of the m ulticast groups
mayc hange some details of the histograms but the proles
sho wn in g remain relativ ely constan t under random
distribution of m ulticast groups
This suggests that cen ter based trees ma y not b e ideal
for high bandwidth applications whic h after the m ultiplying
eect w ould create hot sp ots and reduce the eectiv en um ber
of trac o ws that can b e admitted in to the net w ork
Conclusions
SPT and shared tree SMT cen ter based approac hes are
complimen tary in terms of algorithm complexit y link costs
dela ys and trac concen tration densities SPT oers mini
mal dela y and is the simplest to compute SMT oers mini
mal cost and is the most dicult to compute Cen ter based
tree falls in b et w een these t w o extreme cases
SMT dela ys are not b ounded in theorysim ulation of
KMB trees resulted in widely distributed values Itisnot
lik ely that algorithms deriv ed from pseudooptimal SMT al
gorithms suc h as KMB can b e used in real m ulticast proto
cols
Cen ter based tree dela ys are not adequately b ounded
but are mostly fa v orably distributed MSPT dela ys and
costs are ab out the same as cen ter based trees with dela y
optimal cen ter placemen t MSPT and dela yoptimal CBT
also ha v e similar trac concen tration c haracteristics If the
use of dela yoptimal cen ter based trees is justied in a prac
tical proto col it will b e sucien t to use MSPT whichis
signican tly easier to compute than optimal cen ter based
trees
Weha vesho wn that the p erformance of these dieren t
trees is sensitiv e to group p opulation a v erage no dedegree
and lo cations of the group mem b ers It is relativ ely insen
sitiv e to the reasonableness of the edge distributions of a
graph
Our sim ulations sho w ed that dieren t algorithms indeed
lead to dieren t degrees of trac concen trations Hence se
lection of algorithms should not b e based purely on p erfor
mance for individual groups When there exists hea vy trac
concen tration the hea vily loaded links b ecome b ottlenec ks
Although source sp ecic SPTs consume more link band
width for eac h individual m ulticast group their demands
on bandwidth are more ev enly distributed than the cen ter
based trees esp ecially in net w orks with high connectivityde gree Hence a net w ork ma y supp ort more high bandwidth
m ulticast groups if SPTs are used instead of cen ter based
trees or MSPTs
Ackno wledgments
Wew ould lik e to thank Rob ert Braden Stephen Casner
Rob ert F elderman Y ak o v Rekter An thon y Li the p eople on
the idmr mailing list and man y other ISI division mem bers
for commen ts and suggestions on v arious sub jects of this
pap er
References
R M Karp R e ducibility among c ombinatorial pr ob
lemsPlenn um Press New Y ork P a w ei Win ter Steiner problem in net w orks A surv ey Networks # G P olyzos V Komp ella J P asquale Multicasting for
m ultimedia applications In Pr o c e e dings of the IEEE
Info c om A J Ballardie P FF rancis and J Cro w croft Core
based trees In Pr o c e e dings of the A CM SIGCOMM San F rancisco S Deering D Estrin D F arinacci V Jacobson C Liu
and L W ei An arc hitecture for widearea m ulticast
routing In Pr o c e e dings of the A CM SIGCOMM London Septem b er Y K Dalal and R M Metcalfe Rev erse path forw ard
ing of broadcast pac k ets Communic ations of the A CM #
Stev e Deering Sc alable Multic ast R outing Pr oto c ol PhD thesis Stanford Univ ersit y S Deering and D Cheriton Multicast routing in data
gram in ternet w orks and extended lans A CM T r ansac
tions on Computer Systems pages # Ma y J Mo y Multicast extensions to ospf RFC Marc h
Ron F rederic k Ietf audio video cast Internet So ciety
News Stev e Casner Second ietf in ternet audio cast Internet
So ciety News E N Gilb ert and H O P ollak Steiner minimal trees
SIAM Journal on Applie d Mathematics #
Jan uary S Deering D Estrin D F arinacci V Jacobson C Liu
and L W ei Proto col indep endentm ulticast pim
sparse mo de proto col Sp ecication Working Dr aft Marc h L Kou G Mark o wsky and L Berman A fast algo
rithm for steiner trees A cta Informatic a #
Matthew Doar and Ian Leslie Ho w bad is naivem ul
ticast routing In Pr o c e e dings of the IEEE Info c om Da vid W all Me chanisms for Br o adc ast and Sele ctive
Br o adc ast PhD thesis Stanford Univ ersit y June T ec hnical Rep ort N Bernard M W axman Routing of m ultip oin t connec
tions IEEE Journal on Sele ctedA r e as in Communic a
tions Decem b er Bela Bollobas R andom Gr aphs Academic Press Inc
Orlanndo Florida F ormal Denitions
W e use graph theory notations to dene the p erformance
measures of a m ulticast tree Let G ! V E C b e a directed
graph where V is a set of no des E ! f u v ju v V g is
a set of edges C ! fc u v j u v E g is a set of edge
costs Let M V b e the set of m ulticast mem b ers S V
b e the set of senders for M T M u! V M E M C M suc h
that T M u G beam ulticast tree for M whic h source u
uses for pac k et deliv ery T M b e the set of a sp ecic kind of
m ulticast trees for all sources in S for Mand d u v T M u
b e the path length from u to v via tree T M u W e dene
the p erformance measures as follo ws
Maxim um and a v erage dela y measures

First the
maxim um dela y for source u along tree T M uis maxD T M u ! MAX fd u v T M u j
for all v M g
the maxim um dela y for all sources in S for group M M axD T M ! MAX fmaxD T M u j
for all u S g
Despite the form this denition is the same as those used b yW all
in and the a v erage of the maxim um dela y for m ulticast
group M Av eD T M !
n
X
u S
maxD T M u n ! jS j F or con v enience of comparison w e normalize the de
la ys across dieren t graphs and use R MaxD the ratio
of maxim um dela ys and R Av eD the ratio of a v erage
maxim um dela ys
R MaxD T M !
MaxD T M M axD SP T M where SP T M is the set of shortest path trees for group
MAnd R Av eD T M !
Av eD T M Av eD SP T M An optimal maxim umdela y CBT can b e dened as
acen ter tree whose MaxD T M is minimal among all
cen ter trees
Link cost for trac from source u along tree T M Cost T M u !
X
ij E
M
u c i j F or shared group trees if the links are symmetric cij
! cj i and all sources are receiv ers themselv es C ost T M u will b e the same for all sources u An optimal cost CBT can b e dened as a cen ter tree
whose Cost T M u is minimal among all cen ter trees
F or nonshortestpathtree T Mthe ratiooflinkcost
for trac from source u is
Cost ratio T M u !
C ost T M u Cost SP T M u where Cost SP T M u is the link cost of a shortest
path tree ro oted at u extending to all mem bers of
group M When there are m ultiple shortest path trees
wepic k one at random
T rac Concen tration Let num flow i j b e the n um
ber of o ws passing link i j The maxim um link load
in a graph G when there are n activ e groups is
T C max G n ! MAX fnum flow i j j
for all i j E g The distribution function of all link loads of graph G
under n activ e groups is
D ist G i ! n um b er of links with i o ws

Note that w e could ha v e dened this ratio as
R M axD T M MaxD T M maxD SP T M u where u S and
maxD T M u M axD T M But this denition ma y not capture the c hange of dela ys in a mean
ingful w a y A source that has the largest maxim um delayin T M ma y
not ha v e the largest maxim um dela y in a SPT The alternativ ede nition R M axD T M giv es the ratio of the maxim um path length for
an individual source Denition presen ts the c hange in maxim um
dela y for a whole group The do wn side of denition is that it do es
not sho w the c hange of fate among the group mem b ers in dieren t
t yp es of trees

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Description

Liming Wei and Deborah Estrin. "The trade-offs of multicast trees and algorithms." Computer Science Technical Reports (Los Angeles, California, USA: University of Southern California. Department of Computer Science) no. 560 (1993).

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Creator Estrin, Deborah (author), Wei, Liming (author)

Core Title USC Computer Science Technical Reports, no. 560 (1993)

Alternative Title The trade-offs of multicast trees and algorithms (title)

Publisher Department of Computer Science,USC Viterbi School of Engineering, University of Southern California, 3650 McClintock Avenue, Los Angeles, California, 90089, USA (publisher)

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Format 8 pages (extent), technical reports (aat)

Language English

Unique identifier UC16270970

Identifier 93-560 The Trade-offs of Multicast Trees and Algorithms (filename)

Legacy Identifier usc-cstr-93-560

Format 8 pages (extent),technical reports (aat)

Rights Department of Computer Science (University of Southern California) and the author(s).

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Source 20180426-rozan-cstechreports-shoaf (batch), Computer Science Technical Report Archive (collection), University of Southern California. Department of Computer Science. Technical Reports (series)

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Repository Name USC Viterbi School of Engineering Department of Computer Science

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Title Computer Science Technical Report Archive

Coverage Temporal 1991/2017

Repository Email csdept@usc.edu

Repository Name USC Viterbi School of Engineering Department of Computer Science

Repository Location Department of Computer Science. USC Viterbi School of Engineering. Los Angeles\, CA\, 90089

Publisher Department of Computer Science,USC Viterbi School of Engineering, University of Southern California, 3650 McClintock Avenue, Los Angeles, California, 90089, USA (publisher)