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USC Computer Science Technical Reports, no. 614 (1995)

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USC Computer Science Technical Reports, no. 614 (1995)

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Content Sharing the Cost of Multicast T rees An Axiomatic Analysis
Shai Herzog
Scott Shenk er
and Deb orah Estrin
herzogisiedu shenk erparcxero xcom estrinuscedu
Abstract
Giv en the need to pro vide users with reasonable feedbac k ab out the costs their net w ork usage
incurs and the increasingly commercial nature of the In ternet w e b eliev e that the allo cation of cost
among users will pla y an imp ortan t role in future net w orks This pap er discusses cost allo cation in the
con text of m ulticast o ws The question w e discuss is this when a single data o w is shared among
man y receiv ers ho w do es one split the cost of that o w among the receiv ers Multicast routing reduces
net w ork usage b y using a single shared deliv ery tree W e address the issue of howthese sa vings are
allo cated among the v arious mem b ers of the m ulticast group W e rst consider an axiomatic approac h
to the problem analyzing the implications of dieren t distributiv e notions on the resulting allo cations
W e then consider a onepass mec hanism to implementsuc h allo cation sc hemes and in v estigate the family
of allo cation sc hemes suchmec hanisms can supp ort
In tro duction
Because of the long history of go v ernmen t subsidy in the United States and other coun tries and the
relativ ely co op erativeIn ternet user p opulation the data net w orking researc h comm unit y has paid little
atten tion to the issue of cost allo cation In the United States ho w ev er w e are no w approac hing an almost
completely commercial ie unsubsidized In ternet en vironmen t This means that issues of cost reco v ery
and prot incen tiv es for b etter or w orse b ecome m uc h more relev an t
W e b eliev e that cost allo cation
will b ecome a v ery imp ortan t issue in this en vironmen t By cost allo cation w e mean the assignmentto
v arious users of some measure of the cost of the net w ork resources they are consuming
It is imp ortan t to note that these costs are not necessarily nancial hence the quotes around the term in
the title nor are these costs necessarily extracted from the users in the form of pa ymen ts
When users are
co op erativ e they w antto mak e the righ t decision ab out the wise use of net w ork resources Suc h decisions
This researchw as supp orted b y the Adv anced Researc h Pro jects Agency under Ft Huac h uca con tract n um ber D ABT
C en titled Gigabit Net w ork Comm unications Researc h The views and conclusions con tained in this do cumen t
are those of the authors and should not b e in terpreted as represen ting the ocial p olicies either expressed or implied of the
Adv anced Researc h Pro jects Agency the Departmen t of the Arm y or the US Go v ernmen t
Computer Science Departmen t Univ ersit y of Southern CA LA CA USCInformation Sciences Institute Admiralt yW a y Marina DelRey CA XeroxP AR C Co y ote Hill Road P alo Alto CA In addition as argued in when data net w orks oer m ultiple qualities of service QoS whic h is fast approac hing
on the horizon one m ust consider user incen tiv es to ensure that these QoS features are used appropriately Appropriate use
leads to the ecien t use of net w ork resources where eciency is dened in These incen tiv e issues are also relev antto
con trolling usage ev en in a single QoS en vironmentsee W e should also p oin t out that ev en though costs are allo cated to individu al users the reco v ery of those costs migh t o ccur
at a m uc h higher lev el of gran ularit yF or instance USC migh t b e resp onsible for all of the costs allo cated to its studen ts and
sta

can only b e made if users are informed ab out the cost their usage imp oses on the net w ork F eedbac k
ab out these costs is imp ortantif w e wish to promote the ecien t use of net w ork resources This cost can
b e in terms of congestioninduced p erformance degradation imp osed on other users andor in terms of the
actual capital and main tenance cost of the net w ork facilities themselv es
This pap er discusses cost allo cation in the con text of m ulticast o ws The question w e discuss is this
when a single data o w is shared among man y receiv ers ho w do es one split the cost of that o w among
the receiv ers
The whole p ointofm ulticast is to reduce net w ork usage b y sharing a deliv ery tree W e
ask ho w are these sa vings allo cated among the v arious mem b ers of the m ulticast group W e should note
that w e are making the fundamen tal assumption that costs are assigned to the receiv ers of the m ulticast
group and not to the sources This is b ecause m ulticast mem b ership is t ypically receiv er initiated see
Some applications ma y imp ose an arrangemen t to share costs b et w een the sender and receiv er but
w e consider this to b e a higherla y er issue
W e use an axiomatic approac h to the problem of cost allo cation W e rst discuss sev eral desirable prop
erties that a cost allo cation sc heme should ha v e suc h as anon ymit y essen tially symmetry among group
mem b ers Using these prop erties as a basic set of axioms w e analyze the family of cost allo cation functions
that are consisten t with these axioms Th us this axiomatic approac h allo ws us to explore the implications
of v arious notions of fairness and equit y in cost allo cation Of course one m ust not consider only suc h
idealized prop erties but m ust also consider ho w to implementsuc h a cost allo cation strategyW e discuss
one approac h to implemen ting cost allo cation strategies called the onepass approac h and analyze to what
exten t this approac h can b e made consisten t with the basic axioms
T o fo cus atten tion on the basic issues our in v estigation is in the con text of a v ery simple net w ork mo del
w e consider generalizations in Section There is a single static distribution tree ie routing and group
mem b ership do es not c hange with a single source and m ultiple receiv ers The net w ork pro vides a single
qualit y of service to all mem b ers of the m ulticast group There are costs asso ciated with eac h link tra v ersed
b y the o w These costs could b e tied to usage or reserv ations or b oth for the purp oses of our mo del
it do esnt matter The problem is to fully allo cate the cost of the o w among the receiv ers W e do not
discuss issues of securityor of net w ork dynamics
This pap er has sections W e presen t the basic mo del in Section and discuss v arious cost allo cation
axioms in Section W e turn to issues of implemen tation in Section where w e discuss the abilit y of our
simple onepass approac h to satisfy the axioms presen ted in Section W e conclude in Section with a
discussion of v arious directions for future w ork
Before turning to the basic mo del w e rst discuss some related w ork There is a rapidly gro wing literature
on pricing in computer net w orks See References for a few represen tativ e examples Ho w ev er
this literature do es not t ypically relate the prices c harged to an y underlying cost of net w ork usage The
emphasis is on the strategic utilit y maximizing b eha vior of users and the prot maximizing b eha vior
of rms rather than on the equitable distribution of costs whic his what w e consider here This is an
imp ortan t dierence In theory prop er pricing requires kno wledge of the users utilit y functions at least
in aggregate suc h as in a demand function in practice pricing also in v olv es man y other nontec hnical
mark eting factors W e consider none of those issues here W e instead fo cus exclusiv ely on the allo cation
Calculating the costs for a unicast o w is a dicult problem Ho w do es one quan tify congestion costs Ho w do es one
relate capital and main tenance costs to the cost assigned to users W e do not address these questions Instead no matter
ho w they are calculated w e address the issue of ho w to split them among the users
Our treatmen t here only considers a o w with a single source the generalization to m ultiple senders is discussed in Section

of costs
There is a large economics literature on cost allo cation see Reference and references therein for a
brief o v erview Cost allo cation is usually treated as a sp ecial case of a c o op er ative game A co op erativ e
game is one in whic h there are N pla y ers and there is some v alue function V that assigns a v alue to eac h
coalition or subset of pla y ers The question addressed b y this literature is how toassigntoeac h pla y er
a share suc h that the sum of the shares equals the v alue of the complete set see References Axiomatic approac hes are often in fact almost exclusiv ely used to analyze suc h cooperativ e games The
axioms presen ted here are v ery similar to those used in the theory of co op erativ e games Megiddo considered a family of co op erativ e games whichha v e the same structure as the problem w e consider here
There is also a v ery extensiv e literature on minimal cost spanning tree games whic h dier from what w e
consider here in that the routing is alw a ys c hosen to minimize the total cost See for a lucid o v erview
of these results and for a m uc h broader discussion of net w ork mo dels in economics W e are not a w are
ho w ev er of an y treatmen t of the mec hanisms needed to implemen t suc h cost allo cation sc hemes see our
discussion in Section While the presen tation in this pap er is geared to w ards the net w orking researc h
comm unit yw eha v e included sev eral fo otnotes whic h relate our w ork to the economics literature Also to
aid the presen tation w e defer all pro ofs to the app endix

The Cost Allo cation Mo del
In this section w e rst presen t our abstract mo del and then illustrate it b y discussing sev eral simple
examples of cost allo cation p olicies
Basic Denitions
In this section w e presen t an abstract mo del of the costsharing problem This mo del will b e the basis for
our axiomatic analysis of cost allo cation The abstract mo del denes a formal description of the net w ork
tec hnology for m ulticast transmission o v er a net w ork The abstract mo del also describ es the structure of
incurred costs of sucha m ulticast transmission The mo del w e describ e b elo w is illustrated in Figure for a simple net w ork scenario
The net w ork tec hnology mo del has three comp onen ts net w ork top ology group mem b ership and
routing function Net w ork top ology describ es the ph ysical connectivit y of the net w ork in terms of
v ertices no des and links N V L T describ es a net w ork N with no des v
i
V dir e cte d

links
v
i
v
j
L and a routing function proto col T Let M describ e the set of m ulticast group mem bers w e denote individual mem bers b y m
M Mem b ers of a m ulticast group can join and lea v e dynamically
and can attac h themselv es to anynode Eachpac k et sen t to this m ulticast group byan y source not
necessarily a mem b er of the group will go to all mem bers M of the group W e will use the terms memb er
and r e c eiver somewhat in terc hangeably The mapping of individual mem b ers to sp ecic net w ork no des
is describ ed b y the lo cation function loc M V The routing function abstraction describ es the
path tak en b y pac k ets when a sender transmits to the group In unic ast r outing the routing function
L
T N R v
i
tak es a net w ork N along with a source ro ot R and a receiv er v
i
and denes the set of
This is not to sa y that costs and prices are completely unrelated F or instance in a comp etitiv e mark et prices are often
set at the true marginal costs of usage In a regulated mark et often prices are legislated to b e link ed to true costs
In the conference v ersion of the pap er w e will refer readers to an online v ersion of the app endix
Notice that links are directed and therefore v i v j v j v i for all non trivial links ie v i v j
directed links that establish the path b et w een R and v
i
While w emak e no particular assumptions ab out
the optimalit y of this unicast route w e do assume that the set of unicast routes from a single ro ot R to all
other no des forms a tree
In this pap er w e assume that m ulticast routing is based on lo opfree unicast
routing that is m ulticast routing creates a source ro oted distribution tree created from the union of the
unicast paths b et w een the ro ot and the set of receiv ers T N R V
v
j
V
T N R v
j
for all subsets
V
V Eac h link in a distribution tree creates a distinction relativ e to itself b et w een upstream and do wnstream
p ortions of the tree the do wnstream p ortion con tains all the no des and mem b ers that incorp orate this
link in their unicast path from the ro ot The upstream p ortion is merely dened as the p ortion of the tree
that is not do wnstream
Weno w turn to the structure of incurred costs Our abstract mo del considers the allo cation of costs for
eac h data o w individuall y ie w e do not consider the join t allo cation of the costs of m ultiple o ws
where data o w refers to a particular source sending to a particular m ulticast group destination Suc ha
o w can b e explicit created b y resource reserv ation or implicit
F or reasons of simplicit yw e asso ciate all costs with the links that the pac k ets tra v erse

These costs
could arise from man y dieren t asp ects of net w ork usage for instance there could b e costs asso ciated with
eac h pac k et andor costs asso ciated with resource reserv ations bandwidth etc W e lump all of these
dieren t costs in to a single quan tit y Because w e assume that the same qualit y of service is deliv ered to
all receiv ers the link costs are indep enden t of whic h receiv ers or ho wman y are do wnstream Later in
section w e discuss generalizations of the mo del to incorp orate m ultiple qualities of service Moreo v er
w e should note that our abstract mo del is in tended to address the problem of cost allo cation among a
giv en set of group mem b ers with a giv en net w ork top ology this pap er do es not address dynamics either
in mem b ership or in top ology
The set of link costs for a giv en net w ork N V L T is dened as c L
The cost function
of a distribution tree comprised of links L
is merely the sum of all the directed links in L
cf L
c
P
v
i
v
j
L
c v
i
v
j
This function expresses the total cost incurred b y a distribution tree In this pap er
w e explore ho w to allo cate this cost among the individual mem b ers
W e denote a cost allo cati on function b y af N R M l oc c this represen ts a sp ecic cost allo cation
strategy Giv en a net w ork tec hnology dened b y N V L T a group mem b ership dened b y M and
loc and a cost structure dened b y the link costs c the allo cation function af denes the cost that will
b e individual ly attributed to mem b ers of the m ulticast group af
N R M loc c denotes the allo cation
to mem ber This assumption follo ws if whenev er the route from R to no de v i passes through no de v j the route from R to v j is a
subset of the route from R to v iW e also are assuming that routing is only a function of the lo cation of the receiv er Some
routing proto cols allo w receiv ers to request dieren t qualit y of routes QoR whic hw ould violate our assumption
Data pac k ets can b e classied implicitl y in to o ws b y examining their sourcedestination addresses and other signican t
elds in their headers
Although most reserv ation proto cols reserv e link resources and not in trano de resources w eac kno wledge that some in tra
no de resources ma y b e costly lik ein ternal buers CPU cycles etc In trano de costs can b e attributed to links b y attributing
the in trano de cost asso ciated with pro cessing incoming data to the incoming link and the o v erhead for outgoing data to the
appropriate outgoing links
These costs are in no w a y related to the linkmetrics used b y routing algorithms to calculate routes in the net w ork
Of course our approac h in not limited to the completely static case One can apply our approac h to allo cating cost for
time slices and assuming static b eha vior within those time slices The costs asso ciated with transitioning either mem b ership
or top ology is not addressed here

v1
v2
v5
v3 v4
m1
m2,m3
m4,m5,m6 m7,m8
m9
Figure A simple example of a net w ork
F or an y distribution there maybe n umerous w a ys of allo cating costs to receiv ers and our fo cus in this
pap er is on pro viding a rationale for discriminating among these allo cation p olicies W e restrict ourselv es
to p olicies that fully allo cate the costs incurred W e require that
P
m M
af
N R M l oc c cf T N R l oc M c N R M lo c and c
W e call this equalit y the b alanc e d budget condition Moreo v er w e restrict ourselv es to allo cating costs whic h means that all allo cations are nonnegativ e ie no receiv er is p aid for using the net w ork
af
N R M l oc c N R Mloc and c
Examples
The previous section describ ed our abstract mo del of cost allo cation and here w e presen t a few examples
of allo cation strategies W e use the sample problem depicted in Figure to illustrate these example cost
allo cation strategies
The simplest approac h to allo cating costs is to merely divide the total cost equally among all receiv ers
w e call this the Equal T ree Split ETS sc heme In the example depicted in Figure all mem b ers are
allo cated the cost c v
v
c v
v
c v
v
c v
v

The ETS p olicy do es not discriminate b et w een those receiv ers far from the source and those close to the
source and th us do es not attempt to hold receiv ers accoun table for the costs their individual mem b ership
incurs Ho w ev er the cost of a particular link is incurred b ecause there is at least one do wnstream receiv er
and so while all do wnstream receiv ers can b e considered equally resp onsible for the cost all other receiv ers
are not resp onsible at all This leads to a dieren t approac h to allo cating costs one where the cost of eac h
link is split equally among only the do wnstream receiv ers W e call this the Equal Link Split among
Do wnstream mem b ers ELSD sc heme
F or instance in the example ab o v e the cost allo cated to
m is zero since she is not do wnstream of an y links and the cost allo cated to misgiv en b y c v
v

c v
v
An approac h that is midw aybet w een the egalitarian approac h of ETS and the emphasis on individual
accoun tabilit y of ELSD is to assign the cost of a link equally to all the next hop links that are part of
the distribution tree with all receiv ers at that no de b eing treated as part of one other link
This is
motiv ated b y the idea that w e pass costs on to eachdo wnstream nexthop rather than allo cate them to
F or those familiar with the concept this is the Shapley V alue of this problem
W e giv e a more precise denition of this sc heme in Section
the do wnstream receiv ers themselv es and then allo cate the costs of those do wnstream links in a recursiv e
fashion These costs are recursiv ely forw arded un til there are no more do wnstream no des and costs are
fully allo cated W e call this the Equal NextHop Split ENHS sc heme F or the example in Figure the cost allo cated to m is giv en b y c v
v
c v
v
and the cost allo cated to mis c v
v
Of course all of these cost allo cation p olicies should b e compared to the unicast costs The total cost of
separate unicast transmissions is higher than the total cost of the m ulticast transmission and dieren t
allo cation p olicies distribute this surplus dieren tly Unicast is not a real cost allo cation strategy in the
sense of our mo del since it allo cates to mem b ers more than the net w ork costs ho w ev er it do es pro vide
a useful p oin t of comparison

The unicast cost of mem ber is giv en b y
P
v
i
v
j
T N Rloc m
c v
i
v
j

F or the example in Figure the unicast cost of mis c v
v
c v
v

Axiomati c Analysis
W e analyze this abstract mo del of costsharing through the use of axioms W e use as axioms some p ossible
prop erties that desirable costsharing form ula mightha v e W e then in v estigate the implications of assuming
these prop erties What additional prop erties are implied What forms of costsharing rules are allo w ed
W e rst in Section iden tify three b asic axioms that describ e the pro cess of allo cating cost These
axioms describ e whic h asp ects of the problem are relev an t to the costsharing form ula W e assume that
the basic axioms apply throughout the rest of Section W e discuss the implications of these basic axioms
in Section and then presen t a few examples cost allo cation p olicies whic h satisfy them in Section In Section w e iden tify a few additional axioms that express dieren t p olicy ob jectiv es of costsharing
and w e explore the implications of eac h of these axioms when com bined with the three basic axioms
Basic Axioms
The three basic axioms describ e asp ects of the cost sharing problem that should b e irrelev an t to the actual
cost sharing form ula W e start b y observing that a costsharing form ula should b e in v arian t under arbitrary
relab elling of equiv alentmem b ers it should not ha vean in trinsic and arbitrary asymmetry built in to it
Axiom A nonymity
Name lab els by which memb ers ar e identie d ar eirr elevant to c ost al lo c ation
F orm al denition
Giv en an y net w ork N V L T an y link cost function can ymem ber set Man yt w o
placemen t functions loc and loc
and anyt w o mem b ers m
and m
M Note that if the allo cated m ulticast cost is greater or equal to the unicast cost there maybeno incen tiv e to use m ulticast
This do es not mean that there cannot b e systematic asymmetries eg c harging professors more than studen ts on a
campus net w ork but these asymmetries should b e describ ed as a nonequiv alency of mem b ers eg in tro ducing t w o classes
of mem b ers in to the formalism
Notice that this axiom implies that t womem b ers lo cated at the same no de m ust b e allo cated the same cost loc m
loc m af af

if loc m
loc
m
and
loc m
loc
m
and
loc m
loc
m
m
M m
m
then af
N R M loc c af
N R M loc
c af
N R M l oc c af
N R M l oc
c af
N R M loc c af
N R M loc
c m
M m
m
Link costs can come from man y sources suc h as p erpac k et costs p erhour costs and p erconnection costs
They can also come from link transmission costs buering costs etc It is imp ortan t in order to a v oid the
additional complexit y and o v erhead of indep enden tly allo cating all suc h costs that the cost allo cation not
dep end on whether the accoun ting metho d allo cates these costs separately or join tly That is it should
not matter if the v arious comp onen ts of the link costs are com bined in to one single cost to b e allo cated
or if they are separately allo cated This is expressed in the follo wing axiom
Axiom A dditivity
Given any two sets of link c osts the sum of their r esp e ctive c ost al lo c ation functions is e qual to the c ost
al lo c ation function of the sum of the two sets
F orm al denition
Giv en an y net w ork N V L T an ymem b er set Man y placemen t function loc and an y
t w o link cost functions cand c
af N R M loc c c af N R M l oc c af N R M loc c
The cost allo cation form ula m ust dep end on the underlying net w ork top ologyHo w ev er it is preferable
for this top ological dep endency to b e restricted to factors related to cost F or instance if w e to ok a single
link and articially brok eit in to t w o links spreading the cost of the link b et w een the t w o links the cost
allo cation should not c hange A more general form of this in ten tion is that the cost allo cation should
only dep end on the set of costs incurred byeac h subgroup If t wonet w orks incur the same c harges for
ev ery subgroup of mem b ers then they should allo cate the cost in the same w a y This condition that
the net w orks c harge the same for eac h subgroup of mem b ers is actually quite strong in particular ev ery
unicast cost m ust b e the same ev ery cost for a pair of receiv ers m ust b e the same etc An yc hanges in
the net w ork that preserv e this prop ert y mightw ell b e considered irrelev an t to cost allo cation
Axiom Equivalency
Consider two networks with the same single groupofmemb ers if the c ost of serving any sub gr oup of
memb ers is identic al in b oth networks then the al lo c atedc osts must b e the same
F orm al denition
Giv en a single set of mem bers M and t w o dieren tnet w ork scenarios N R loc c and
N R loc c if cf T N R loc M
c cf T N R loc M
c M
M
then af N R M
loc c af N R M
loc c M
M
This axiom implies that the only relev an t asp ect of the problem is the induced co op erativ e game all other asp ects of the
top ology are irrelev an t

The Canonical F orm of Cost Allo cation F orm ulae
The three basic axioms presen ted in the previous section greatly reduce the scop e of allo w able cost allo cation
p olicies In this section w e sho w that all cost allo cation p olicies satisfying these three basic axioms can
b e expressed in a v ery simple canonical form W e consider functions F fZ
g
where w e
use the sym bol Z
to denote the nonnegativein tegers and to denote the nonnegativ e reals
W e
dene the family of functions F to b e those functions F fZ
g
that satisfy the prop erties
z
F
z
z
z
F
z
z
and F
z F
z Note that the set F is con v ex
F G F F G F for all Also w ekno w that F
z F
z z
for
z Weno w use these functions to dene cost allo cation strategies Consider a link v
i
v
j
with a cost c v
i
v
j
where there are n
d
receiv ers do wnstream
and n
u
receiv ers upstream W e use a function
F F
u
F
d
F to determine the fraction of the link cost that is allo cated to eac h upstream or do wnstream
receiv er Mem b ers are allo cated the follo wing costs
F
u
n
u
n
d
c v
i
v
j
for eac h upstream mem ber F
d
n
u
n
d
c v
i
v
j
for eachdo wnstream mem ber The total cost allo cated to a mem b er is the sum o v er all links of its share of eac h link cost
Cost allo cation strategies that can b e expressed in terms of a function F F are called c anonic al strate
gies Note that the rst condition ab o v e in the denition of F ensures that costs are fully allo cated
n
u
F
u
n
u
n
d
n
d
F
d
n
u
n
d

It is fairly clear that all canonical strategies satisfy the three basic
axioms More in terestingly all cost allo cation strategies satisfying the basic axioms are canonical
Theorem Ac ost al lo c ation formula satises the b asic axioms if and only if it is a c anonic al str ate gy
Examples of Canonical Cost Allo cation Strategies
The set F is innite and so there are an innite n um b er of canonical cost allo cation strategies In this
section w e presentsev eral examples of functions F F Some strategies ma y seem in tuitiv e and others
ma y lo ok less than that lik ec harging upstream receiv ers but they all ob ey the three basic axioms Tw o
of the functions w ere already in tro duced in section Since w ekno w that F
d
n
d

n
d
for all F F w e merely describ e the functions on the set Z

b elowwhere Z

denotes the p ositivein tegers Since
the set F is con v ex all linear com binations of the examples b elo w are also in F Equal Link Split among Do wnstream mem b ers ELSD
F
u
n
u
n
d
No cost is allo cated upstream
F
d
n
u
n
d
n
d
Cost is split equally among do wnstream receiv ers
In case our notation is confusing the domain of this function is all pairs of nonnegativ ein tegers except the pair F or our use in this section weneedonly ha vedened F on Z Z where Z denotes the p ositivein tegers but w e
will mak e use of the family of functions again in Section where w e need the larger domain of fZ
g If there are no receiv ers do wnstream then this link is not part of the distribution tree and w e can ignore its costs
The second condition in the denition of F is is unrelated to allo cation s but is merely used to simplify the expression of
Theorem
Equal T ree Split ETS
F
u
n
u
n
d
F
d
n
u
n
d

n u n
d
T otal tree cost is split equally among all receiv ers
Cost is c harged to upstream receiv ers
F
u
n u
F
d

Cost is c harged relativ e to the n um b er of receiv ers upstreamdo wns tream
F
u
n u
n u
n
d
F
d
n
d
n u
n
d
Additional Axioms
The basic axioms address the issue of what factors can b e considered relev an t to cost allo cation These
axioms ha v e substan tial reductivepo w er in that they dene a class of canonical cost allo cation strategies
Ho w ev er as the examples ab o vesho w one can allo cate all costs to upstream no des or to do wnstream
no des or an ywhere in b et w een Th us this family of canonical cost allo cation strategies incorp orates a
wide v ariet y of distributiv e notions W e use the phrase distributive notion to mean standards of equit y
or justice that allo ws one to discriminate b et w een allo cation p olicies Our next step is to examine some
additional axioms that express particular distributiv e notions These axioms can b e used to select a subset
of canonical allo cation strategies
StandAlone and Related Axioms
One suc h distributiv e notion is that a mem b ers cost should reect the b enets of m ulticast Just as the
total net w ork cost of a m ulticast o w is less than the sum of the costs of unicast o ws to eachmem b er
one migh t require that eac h individual allo cated cost in a m ulticast o wnev er b e greater than the cost
incurred b y the appropriate unicast o w This yields the follo wing axiom
Axiom StandA lone
The unic ast c ost of a memb er is an upp er limit on her c ost al lo c ation
F orm al denition
af
N R M loc c cf T N R loc m
c N R M l oc c m
M
The StandAlone axiom protects the individual ev ery individual receiv er is guaran teed that joining a group
can nev er can nev er cost more than her unicast cost Assuming users ha vethe po w er of c hoice in their
net w ork activities and assuming some ev en minimal amoun tofself in terest guides them it is hard to
imagine whyanyuserw ould w an t to join a shared group of receiv ers if she risks an increase in her allo cated
costs Of course in a co op erativ een vironmen t receiv ers mayc ho ose to risk ha ving increased costs if the
total cost distributed to the group decreases

Insisting up on the StandAlone axiom when com bined with the basic axioms means that there is one and
only one applicable cost allo cation strategy

Theorem Ac ost al lo c ation function satises the b asic and StandA lone axioms if and only if it is the
Equal Link Split Downstr e am ELSD function
A stronger form of the StandAlone axiom is the SharingisGo o d axiom This axiom em b o dies the
distributiv e notion that sharing a m ulticast tree with more mem b ers alw a ys b enets ev erybody Axiom SharingisGo o d
The c ost al lo c atedto a memb er never incr e ases when another memb er joins
F orm al denition
af
N R M m
locc af
N R M l oc c N R M l oc c m
and m
M
The ELSD sc heme satises the SharingisGo o d axiom since the share of costs from eac h link strictly
decreases with the n um ber of do wnstream mem b ers and is indep enden tof the n um b er of upstream mem
b ers Clearly an y cost allo cation sc heme ob eying the SharingisGo o d axiom also ob eys the standalone
axiom These t w o axioms b oth describ e an upp er b ound on the cost that can b e allo cated to a particular
mem b er Ho w ev er w e migh t also b e concerned ab out the problem of fr e e riders who are mem b ers who
do not pa y their fair share According to the standalone axiom the most a mem b er should pa y is her
unicast cost and the SharingisGo o d axiom requires that the allo cations decrease as mem b ers join Ho w
m uch canamem b er b enet without b eing a free rider If all mem b ers are lo cated at the same no de then
they all pay jM jth of their unicast cost W e suggest that anymem ber pa ying less than this should b e
considered a free rider
Axiom NoF r e eR ider
The c ost al lo c atedto a memb er is never less than jM jthofher unic ast c ost
F orm al denition
jM j af
N R M l oc c P
c
i
c
j
T N Rloc m c v
i
v
j
N R M loc c and m
M
Eliminating free riders do es not pic k out a sp ecic allo cation sc heme but do es narro w the range of p ossi
bilities
Theorem Ac ost al lo c ation function satisfying the b asic axioms and the NoF r e eR ider axiom must
satisfy F
u
n
u
n
d
F
d
n
u
n
d
n
u
n
d
Z
This result is closely related to the standard axiomatization of the Shapley V alue see in economics Mem bers
attac hed at the ro ot can b e considered dummy memb ers b ecause adding them to a group do es not increase the total cost
incurred W e can dene a dummy memb er axiom that sa ys that no mem b er lo cated at the ro ot can b e c harged
loc m R af N R M loc c N R M loc c and m M
Theorem con tin ues to hold if w e replace the StandAlond axiom with the m uchw eak er dummymem b er axiom The
Equiv alency axiom means that only the co op erativ e game matters ie top ology is irrelev an t aside from the co op erativegame
it induces The basic result due to Shapley is that there is one and only one budget balanced cost allo cation form ula satisfying
the Additivit y Anon ymit y and Dumm y axioms and this form ula is no w kno wn as the Shapley V alue
This is m uchlik e the Unanimit y b ound in economics see
Subset Monotonicit y
The essen tial guiding principle b ehind the Equiv alency axiom is that the cost allo cations should dep end
only on the costs incurred b y the v arious subsets or coalitions of mem b ers Another distributiv e notion
that arises from this principle is that the cost allo cated to a particular mem b er should b e monotonic with
resp ect to these subset costs More preciselyif w e consider t w o cost structures c and c that is w e
consider the net w ork N and the mem b ers M xed and w e merely consider t w o sets of link costs then if
for ev ery subset M
M the total cost of serving M
is no greater under cost c compared to c then
one migh t require that the allo cated costs under cw ould not b e greater than those under c This yields
the follo wing axiom
Axiom Subset Monotonicity
No c ost al lo c ation c an incr e ase when subset c osts al l de cr e ase or stay the same
F orm al denition
Consider an arbitrary tree L
T N R loc M and t w o link costs c and c if cf T N R loc M
c cf T N R l oc M
c M
M
then af
N R M loc c af
N R M loc c m
M
It turns out that this axiom when com bined with the basic axioms determines a unique cost allo cation
strategy in fact only t w o of the three basic axioms are needed for this result
Theorem Ac ost al lo c ation formula satises Equivalency A nonymity and Subset Monotonicity if and
only if it is the Equal T r e e Split ETS formula
Collusion Prev en tion
Another asp ect of allo cation that is imp ortan t to consider is collusion Whenev er a cost is shared among
clien ts it ma y b e p ossible for sev eral clien ts to unite and b e represen ted b y a single clien t who then
forw ards the data on to them This is analogous to the classic cop y and distribute securit y problem
Collusion among some receiv ers ma y increase the cost allo cated to the other receiv ers and ma y decrease
the eciency of sharing the transmission W ew ould prefer that a cost allo cation sc heme not encourage
collusion among the mem b ers W e therefore prop ose the follo wing axiom
Axiom Col lusion Pr evention
The c ost al lo c ation scheme do es not yield b enets for c ol luding memb ers
F orm al denition
Consider an arbitrary net w ork N V L T a set of link costs casetof mem b ers M and
their lo cation function locF or eac h subset M
M and m
M
P
m M
af
N R M l oc c af
N R M M
m
locc
P
m
M
m af
N loc m
loc M
m
c
Ob viously collusion prev en tion is a desirable prop ert y for cost allo cation form ulae Unfortunately w ecan
pro v e that no canonical allo cation strategy satises this axiom
Theorem No c ost al lo c ation formula satisfying the b asic axioms c an satisfy the no c ol lusion axiom
Discussion
W e started this section with three basic axioms whic h narro w ed the space of cost allo cation strategies to the
canonical ones Within this class w e discussed howv arious distributiv e notions p oin ted to w ards dieren t
c hoices Eliminating free riders restricts us to sc hemes that allo cate more to do wnstream mem b ers than
to upstream mem b ers Subset monotonicit y leads us to the ETS sc heme while the standalone axiom
suggests the ELSD sc heme Cho osing b et w een axioms is purely sub jectiv e but c harging a nonzero amoun t
for a mem b er lo cated at the ro ot seems rather outlandish The only canonical allo cation sc heme whic h
alw a ys allo cates zero to mem b ers lo cated at the ro ot is the ELSD sc heme and so p erhaps it is the most
natural c hoice
Our treatmen t here is completely static Consider for a momen t the dynamic p olicy of allo cating to eac h
mem b er the incremen tal cost of adding them to the distribution tree The resulting allo cations dep end on
the order in whichmem b ers joined whic h seems rather unfair It migh t seem appropriate to then a v erage
these incremen tal costs o v er all arriv al orders Indeed this a v eraging pro duces the ELSD allo cations

In this section w e discussed v arious cost allo cations sc hemes from an axiomatic p ersp ectiv e The next
section discusses mec hanisms for implemen ting these cost allo cation sc hemes
Accoun ting Mec hanisms
In this section w e use the term ac c ounting schemes to denote mec hanisms for implemen ting cost allo cation
sc hemes First w e discuss the general structure of one class of accoun ting sc hemes w e call onep ass sc hemes
W e then describ e t w o dieren t mo dels of implemen tation of suchsc hemes mo dels and whic h dier in
the information pro vided ab out do wnstream mem b ers Eac h of these mo dels is examined according to the
cen tral question raised throughout this section what forms of cost allo cation sc hemes can this family of
accoun ting mec hanisms supp ort
OneP ass Accoun ting Sc hemes
Weha v e made the fundamen tal assumption that costs are allo cated among receiv ers and not just assigned to
the sender Because the n umberofreceiv ers can b ecome quite large and widely disp ersed geographically
the k ey concern in designing accoun ting mec hanisms is scalabilit y It is imp ortan t that the trac load
imp osed b y the accoun ting mec hanism on an y particular link should not increase without b ound with the
size in terms of n um b ers or in geographical disp ersion of the m ulticast group Th us scalabilit y concerns
rule out an y kind of cen tralized accoun ting and so w em ust turn to a more decen tralized approac h In
This is the standard motiv ation for the Shapley V alue
The geographic disp ersion is imp ortan t b ecause it means the amoun t of information needed to describ e the tree top ology
is also gro wing

this pap er w e consider only the family of onep ass mec hanisms whereb y the accoun ting con trol messages
mak e a single pass from the source do wn the m ulticast tree to all receiv ers While this is not the only
scalable accoun ting mec hanism one migh t imagine
it certainly seems among the most natural In this
onepass metho d of accoun ting no des allo cate costs to mem b ers as the ac c ounting message passes through
them The information used to mak e the allo cation decisions comes from t w o sources The rst source of
information is m ulticast routing and p erhaps the reserv ation establishmen t proto col if the costs are related
to reserv ations whic h pro vides information ab out the do wnstream links T raditional m ulticast routing
only pro vides information ab out whether or not there exist mem bers do wnstream of a particular link W e
call this mo del It is p ossible p erhaps only b ecause it enables b etter allo cation of costs that m ulticast
routing could pro vide the exact n um b er of mem b ers do wnstream of eac h link W e call this mo del If the link costs are tied to reserv ations then this information ab out the n um ber of do wnstream mem bers
making reserv ations could b e pro vided b y the reserv ation establishmen t proto col W e will consider b oth
mo dels in what follo ws As w e shall see there is an imp ortan t dierence in the functionalit y that can b e
ac hiev ed in the t w o mo dels

The second source of information is the accoun ting message itself The design
of a onepass accoun ting mec hanism essen tially reduces to the question of what information is carried in
the accoun ting message sentdo wnstream T o ensure scalabilit y this information cannot gro w with the size
of the m ulticast group nor with the n um b er of links tra v ersed This prev en ts us from carrying detailed
information ab out eac h upstream link cost and eachmem b ers allo cated cost Instead w ec ho ose to carry
only a single piece of information in the accoun ting message the unallo cated or residual cost passed do wn
from the upstream no de While this is not the most general form of accoun ting message
it seems a
natural and simple c hoice
Th us with this form of accoun ting message the costs are allo cated with the follo wing pro cess The
accoun ting message arriv es at a no de on the incoming link from an upstream neigh b or carrying the
upstream residual cost w e will call this the input cost to the no de and let in v
j
denote the input cost
arriving at the do wnstream no de v
j
The cost allo cation function determines howm uc h of this cost is
allo cated to eac h of the lo cal mem bers and howm uchispasseddo wn to eac h of the do wnstream nexthops
W e will call the costs that are passed on the residual output from a no de and let out v
i
v
j
denotethe
costs that are passed on from no de v
i
to do wnstream no de v
j
The sum of all residual outputs plus the
sum of all lo cally allo cated costs m ust b e equal to input cost as a result of the balanced budget rule

When an accoun ting message is forw arded to the do wnstream neigh b or the cost of the link connecting the
t w o no des is added to the residual costs and this sum is carried in the accoun ting message as input costs
to the nexthop no de th us in v
j
c v
i
v
j
out v
i
v
j
when v
j
is a nexthop do wnstream of v
i
A tleaf
no des all costs are allo cated to the lo cal mem bers A t no des with no lo cal mem b ers all costs are passed
do wn to the do wnstream nexthops in the accoun ting message
Onepass accoun ting is a distributed accoun ting sc heme Indep enden t cost allo cation decisions are made
b y eac h individual no de based on the information pro vided to it bym ulticast routing either mo del or
F or instance one could design an accoun ting metho d that in v olv ed t w o passes of con trol messages one do wnstream and
the other bac k upstream Also one could use an iterativ e accoun ting metho d where the con trol messages con tin ued to circulate
un til an equilibri um had b een reac hed
Todo an y form of cost allo cation wem ust b e able to iden tify all mem b ers at a no de if for no other reason than to giv e
them the feedbac k ab out their allo cated costs The question is then whic h proto col will carry these n um b ers upstream It
app ears to b e simple to mo dify most m ulticast routing proto cols to carry the cum ulativemem b ership n um b ers upstream once
the n um ber of local n um b ers is a v ailable
F rom the p oin t of view of implemen tation of the accoun ting sc hemes themselv es mo del and mo del are essen tially
equiv alen t
One could for instance include information of the previous v e upstream links
It is straigh tforw ard to sho w that if there is one no de with a lo cally un balanced budget then one cannot guaran tee that
the o v erall cost budget is balanced

mo del and the accoun ting message W e assume that no other information ab out top ology or group
mem b ers can b e factored in to the allo cation decision F or mo del since a no de can mak e no meaningful
distinctions b et w een do wnstream links w e require that the residual costs passed on to eac h nexthop are
the same out v
i
v
j
out v
i
v
k
for all links v
i
v
j
and v
i
v
k
in the distribution tree W e further
assume in b oth mo del and mo del that all no des m ust implemen t the same allo cation rules In order
to ac hiev e a consisten t allo cation sc heme all no des if giv en the same information m ust pro duce the same
allo cation
W e will refer to cost allo cation sc hemes that can b e implemen ted with a mo del onepass accoun ting
mec hanism as mo del allo cation sc hemes and similarly for mo del The basic onepass structure of cost
accoun ting imp oses some signican t restrictions on what cost allo cation form ulae can b e supp orted In
particular suc h onepass accoun ting sc hemes can only supp ort cost allo cation sc hemes that satisfy the
StandAlone axiom
Theorem Given a tr e e T N R loc M a setof linkc osts c and a onep ass ac c ounting me chanism
no memb er c an beallo c ateda c ost gr e ater than her unic ast c ost
W e are in terested in howman y of our original axioms are consisten t with our onepass family of accoun ting
mec hanisms Before w e consider mo dels and separately w e can rule out one axiom that do es not apply
to either of them
Theorem No c ost al lo c ation scheme implemente d with a onep ass ac c ounting me chanism c an satisfy the
subset monotonicity axiom
W e rst discuss mo del b ecause our treatmen t of it is closer to our previously dev elop ed results W e then
will return to mo del where w e will need to mo dify our basic set of axioms
Mo del As w e stated ab o v e w e are in terested in the exten t to whic h these accoun ting mec hanisms can supp ort
cost allo cation form ulae that ob ey our previous axioms As w eshowbelo w mo del can supp ort the basic
axioms presen ted in Section In fact there is only one mo del cost allo cation form ula that ob eys the
basic axioms
Theorem ELSD is the only c ost al lo c ation formula that ob eys the b asic axioms and that c an b e imple
mente d with a mo del onep ass ac c ounting me chanism
This follo ws trivially from Theorems and Ho w is the ELSD form ula implemen ted Consider some
no de v
i
in the distribution tree w elet tmem v
i
denote the n um b er of mem b ers lo cated in the subtree
ro oted at v
i
and recall that in v
i
denotes the input costs in the accoun ting message arriving at v
i
The
A stronger result holds No subset M
M can b e allo cated a cost that is greater than their subtree cost
P
m M
af N R M l oc c cf T N R l oc M
c N R M loc M
M and c This means that all onepass ac
coun ting sc hemes pro duce results that are in the c or e see for a denition
Anode v i kno ws the v alue of tmem v j for all immediate do wnstream mem b ers b ecause this is a basic prop ertyof model
The no de can calculate tmem v ib y adding up all the tmem v j for all immediate do wnstream mem b ers and then adding
the n um ber of local mem b ers

cost allo cated to eac h lo cal mem ber is in v
i
tmem v
i
F or eac h link v
i
v
j
in the distribution tree
out v
i
v
j
tmem v
j
tmem v
i
in v
i
and so in v
j
tmem v
j
tmem v
i
in v
i
c v
i
v
j
Notice
that the residual costs passed on to nexthops is prop ortional to the n um b er of receiv ers do wnstream The
cost of the connecting link is passed on fully to the do wnstream nexthop
Mo del Weno w consider accoun ting mec hanisms in the con text of mo del where m ulticast routing only indicates
whic h links ha vedo wnstream mem b ers but not howman y are there
Reduced Basic Axioms
Wew ould liketo in v ok e the basic axioms presen ted in Section Additiv e and Anon ymous cost allo cation
sc hemes can b e supp orted as w e shall see in examples b ymodel accoun ting sc hemes Ho w ev er w e nd
that the Equiv alency Axiom is not consisten t with mo del Theorem No c ost al lo c ation formula that is implementedbyamo del onep ass ac c ounting scheme c an
satisfy the Equivalency Axiom
This theorem can b e in tuitiv ely explained b y observing that the mo del accoun ting sc hemes are physic al
top olo gy dep endent in con trast with the physic altop olo gy indep endenc e of the Equiv alency axiom Th us for
mo del w e will require that the r e duc e d basic axioms of Anon ymit y and Additivit y still hold In the next
section w e dev elop a canonical form for mo del allo cation sc hemes that ob ey these axioms In passing w e
should men tion that these reduced basic axioms lead to the same imp ossibilit y result ab out collusion
Theorem No mo del c ost al lo c ation formula satisfying the r e duc edb asic axioms c an satisfy the no
c ol lusion axiom
The OneP ass Canonical F orm
Consider an y mo del allo cation sc heme that ob eys the reduced basic axioms A t leaf no des all costs
m ust b e allo cated equally to lo cal mem bers A t no des with no lo cal mem b ers all costs m uc h b e passed
on equally to all do wnstream links Th us the only design freedom left is ho wm uc h of the residual cost to
allo cate to the lo cal mem b ers and howm uch to passontodo wnstream mem b ers when b oth are presen t
W e can express the family of p ossible design c hoices with the onepass canonical form of cost allo cation
form ulae supp orting the mo del onepass accoun ting sc heme
Eac h onepass canonical form is asso ciated with a function F F
l
F
r
F in the follo wing w a y Consider
anode v
i
with an input cost of in v
i
If there are n
l
lo cal receiv ers and n
r
next hop no des The allo cated
costs are
F or a lo cal mem ber on v
i
F
l
n
l
n
r
in v
i
F or a next hop no de v
j
out v
i
v
j
F
r
n
l
n
r
in v
i
All mo del allo cation sc hemes ob eying the reduced basic axioms can b e expressed in this form

Theorem Ac ost al lo c ation formula satises the r e duc edb asic axioms if and only if it c an beexpr esse d
in the onep ass c anonic al form
While the onepass canonical form app ears v ery similar to the canonical form discussed in Section there
are imp ortan t dierences The previously discussed canonical form expressed ho w the cost of a particular
link w as allo cated to all upstream and do wnstream mem b ers Here the canonical form only describ es the
allo cation to lo cal mem b ers and to do wnstream links T o nd the resulting allo cation to all mem bers w e
m ust recursiv ely iterate this form ula do wn the tree Th us it is m uc h harder to understand what allo cations
will result from a particular one pass canonical form
Examples
The onepass canonical form allo ws for an innite set of p ossible allo cation strategies Belo ww e list a few
examples As b efore some strategies ma yseem in tuitiv e and others ma y lo ok less than that lik ec harging
upstream receiv ers Since w e kno w that F
l
n
l
n
l
and F
r
n
r

n r
for all n
l
n
r
w e merely
describ e the allo cations on the set Z

Lo cal mem b ers pa y nothing
F
l
n
l
n
r

F
r
n
l
n
r
n
r
Lo cal mem b ers payev erything
F
l
n
l
n
r
n
l
F
r
n
l
n
r

All lo cals are considered as one next hop ENHS
F
l
n
l
n
r

n
l
n r
F
r
n
l
n
r

n r Lo cal mem b ers and Next Hops are allo cated iden tical costs
F
l
n
l
n
r
F
r
n
l
n
r
n
l
n
r
Equal Split b et w een all the lo cal mem b ers and all the nexthops
F
l
n
l
n
r
n
l
F
r
n
l
n
r
n
r
Ma jorit y Loses
if n
l
n
r
F
l
n
l
n
r
n
l
and F
r
n
l
n
r

if n
l
n
r
F
r
n
l
n
r
n
r
and F
l
n
l
n
r

if n
l
n
r
F
l
n
l
n
r
F
r
n
l
n
r
n
l
Additional Axioms
W e already kno w from Theorem that all the onepass allo cation sc hemes m ust satisfy the StandAlone
axiom Ho w ev er not all suc h cost allo cation sc hemes ob ey the SharingisGo o d axiom

Theorem Amodelc ost al lo c ation formula satises the r e duc edb asic axioms and the SharingisGo o d
axiom if and only if the functions F
l
n
l
n
r
and F
l
n
l
n
r
ar e nonincr e asing on fZ
g Note that of our examples only the last one fails this test Th us all mo del allo cation sc hemes ob eying
this mild restriction ensure that the b enets of m ulticast are shared among all receiv ers
Recall that in Theorem a wide v ariet y of canonical cost allo cation p olicies satisfy the NoF reeRider
axiom In particular the ELSD sc heme whic h is the only mo del allo cation p olicy whic h ob eys the basic
axioms satises the NoF reeRider axiom Ho w ev er no mo del allo cation p olicies satisfy the NoF ree
Rider axiom
Theorem Thereis no mo del c ost al lo c ation formula that satises the r e duc edb asic axioms and the
NoF r e eR ider axiom
As long as our functions F
l
F
r
are nonincreasing so that the SharingisGo o d axiom is ob ey ed w eha v e
little to guide us in our c hoice of canonical mo del allo cation sc hemes Allo cating ev erything to the lo cal
mem b ers is unfair as is passing all costs on to the do wnstream links In a v ery rough in tuitiv e sense
fairness seems to dictate that lo cal mem b ers individually are allo cated no more than is passed on to the
individual links F
l
n
l
n
r
F
r
n
l
n
r
and the set of lo cal mem b ers as a whole are allo cated at least as
m uc h as is passed on to the individual links n
l
F
l
n
l
n
r
F
r
n
l
n
r
There is wide latitude b et w een
these t w o extremes of treating lo cal mem bers eachasa do wnstream link and treating the collection of
them as a do wnstream link Ho w can wec ho ose among the v arious p ossibilities that lie within the sp ectrum
F r n
l
n r n
l
F
l
n
l
n
r
F
r
n
l
n
r
One p ossibilit y is to require that the cost allo cated to mem b ers not dep end not on the exact n um ber
of mem b ers at all other no des but only on the set of no des where there are mem bers This condition
mak es little sense for mo del b ecause there w e had the information ab out individual no des do wnstream
Ho w ev er mo del do es not pro vide this information Moreo v er the con texts in whic h mo del migh tbe
deplo y ed ma y not ev en ha v e this information at the lo cal lev el the costs rather than b eing assigned to
individuals migh t really b e assigned to the no des subnets at whic h mem b ers reside F or instance all
costs allo cated to mem b ers on an ethernet at USC w ould b e assigned to USC rather than to the individual
mem b ers This w ould allo w currentm ulticast proto cols whic hdo not k eep trac k of individual mem b ers to
b e consisten t with cost allo cation Giv en that the n umberofmem b ers at a particular no de is not kno wn the
allo cation to other mem b ers should not dep end on it This requiremen t means that F
r
n
l
n
r
F
r
n
r
for all n
l
and th us narro ws the sp ectrum
F r n
l
n r n
l
F
l
n
l
n
r
F
r
n
l
n
r
do wn to the single p oin t
F r n
l
n r n
l
F
l
n
l
n
r
whic h is merely the ENHS sc heme
W e did not em b o dy the considerations in the previous t w o paragraphs in axioms b ecause the distributiv e
notions w ere signican tly less general and comp elling than our previous axioms W e fully admit that this
line of reasoning whic h leads to ENHS is m uchw eak er than our previous results but w e do not see an y
other general principles at our disp osal This is largely b ecause the allo cations that result from mo del ha v e una v oidably p o or prop erties free rider no equiv alency etc and so it is hard to form ulate desirable
distributiv e notions that are ac hiev able in this con text

Discussion
In Section w e discussed a general axiomatization of allo cation p olicies The ELSD sc heme emerged as
the most attractivesc heme W e then turned in this section to issues of implemen tation W e discussed
t w o dieren t mo dels in this section Mo del can implemen t the ELSD sc heme and in fact this is the only
mo del sc heme consisten t with the basic axioms So if w e adopt mo del in our implemen tations there
seems little question that ELSD w ould b e the most appropriate allo cation p olicy Ho w ev er when w e use mo del w eare facedwitham uc h more confusing situation W e cannot ac hiev e
the desired degree of top ological indep endence nor can w eprev en t free riders There are few distributiv e
notions b esides sharingisgo o d and standalone that w e can ac hiev e W e do presentanin tuitiv e line of
reasoning that suggests that ENHS is the b est of this rather sorry lot of allo cation p olicies
Howtoc ho ose b et w een these t w o mo dels F rom the p ersp ectiv e of the accoun ting proto col needed to realize
them the dierence in implemen tations b et weenmodelsand isv ery small The k ey dierence b et w een
the t w o mo dels is in the a v ailabilit y of the exact n um b er of lo cal mem b ers once that n um ber is a v ailable
it seems fairly straigh tforw ard to pro vide it upstream either through m ulticast routing directly or through
the reserv ation establishmen t proto col or ev en through a separate set of accoun ting con trol messages
If
costs are tied to reserv ations then mo del is quite practical since the n um b er of lo cal mem b ers is already
kno wn to receiv erinitiated reserv ation establishmen t proto cols Ho w ev er most m ulticast routing proto cols
do not determine the n um ber of local mem b ers and so if costs are applied more generally w e are faced with
the tradeo b et w een the increased implemen tation dicult y of mo del with the corresp ondingly b etter
allo cation p olicy and the signican tly easier implemen tation of mo del whic h comes with a seriously
a w ed allo cation p olicyAn in termediate p ointis to k eep trac k of the cum ulativen um b er of domains with
in ternal mem b ers instead of the n um b er of individual host mem b ers Only b order routers w ould propagate
the mem b erdomain coun ts upstream eliminating the need for in teriorrouter mo dications and thereb y
sidestepping m uc h of the implemen tation diculties
The discussion in this section fo cused exclusiv ely on the onepass accoun ting mec hanism There are a
wide v ariet y of other approac hes a v ailable wh y narro w consideration to this particular family F rom
a purely mec hanistic p ersp ectiv e the onepass mec hanism has sev eral desirable prop erties simple easy
to implemen t and scalable in that the state carried in the accoun ting message do es not gro w with the
n um b er of mem b ers or with the size of the distribution tree In addition in the con text of mo del the
onepass accoun ting mec hanism can implemen t the ELSD p olicy whic h on purely axiomatic grounds w as
iden tied as the most natural allo cation p olicy The most ob vious dra wbac k with the onepass accoun ting
mec hanism is that when com bined with mo del it cannot implemen t allo cation p olicies that ob ey the
Equiv alency axiom While w e do think it imp ortan t to explore other accoun ting mec hanisms and w e
plan to do so w e b eliev e that it is essen tially imp ossible to ac hiev e Equiv alency if allo cations m ust b e
done without kno wledge of the n um ber of do wnstream mem b ers That is w e b eliev e that the k ey factor
prev en ting Equiv alency is the dierence b et w een mo del and mo del not the features of the onepass
mec hanism itself
If the n um b er of lo cal mem b ers is a v ailable but m ulticast routing do es not propagate these n um b ers upstream then one
could add suc h a function directly to the accoun ting mec hanism and this w ould necessitate a second set of accoun ting con trol
messages tra v ersing up the distribution tree This can b e view ed as an alternativ e implemen tation of mo del since as w e
observ e b elo w the basic allo cation pro cess can b e adequately handled with a onepass mec hanism

F uture W ork
This pap er used a v ery simple mo del to illustrate some of the basic issues in v olv ed in cost allo cation In
this section w e briey discuss sev eral directions in whic h our mo del could b e made more general
Multicast Distribution Mo del
This pap er mo deled m ulticast distributions as source ro oted trees computed from unicast routing
Although this is the most common form of m ulticast distribution new er proto cols use dieren t mo dels
for their distribution CBT adv o cates using a shared tree for all sources core ro oted rather than
source ro oted trees PIM Proto col Indep enden t Multicast in its sparse mo de enables mixing
source ro oted shortest path SP routes for some groups and shared trees for others Our theory
only assumes that the route tak en from a particular source to a particular receiv er is indep endentof
the group mem b ership ie the route will b e the same to a sp ecic group mem b er regardless of who
else has joined the group This remains true for b oth CBT and PIM and th us our results apply
Another asp ect of routing whic h can in v alidate our theory is the abilit y to request alternate routes
T ypically these alternate routes are not automatically lo opfree and so when an a new alternate
route is b eing established the route is follo w ed un til it hits the tree This means that the resulting
routes will dep end on the currentmem b ership
Multiple Sources
Our mo del considered only a single source If the costs b eing allo cated are tied to a p erpac k et
metric then this is sucien t since eac h pac k et comes from a single source This is not sucien t
ho w ev er if the costs are tied to resource reserv ations eg bandwidth RSVP uses the idea of
reserv ation st yles and t w o of these st yles dynamic lter and wildc ardlter allo w a single reserv ation
to b e shared bysev eral sources In terms of our theory this w ould require the ro ot R to b e a set
of no des rather than a single no de This b y itself in tro duces no non trivial c hanges to our theory Ho w ev er if the costs on a link dep ended on the particular set of upstream sources sa y for instance
that the bandwidth reserv ed w as the maximal upstream source bandwidth then our theory needs to
b e extended
Multiple Qualities of Service
In this pap er all mem b ers are equiv alen t except for lo cation Ho w ev er if the net w ork supp orts
m ultiple qualities of services sa y for instance sev eral prioritylev els or sev eral lev els of realtime
service and these dieren t QoS lev els incur dieren tnet w ork costs then the mem b ers are no longer
necessarily equiv alen t This raises an issue that is in some sense orthogonal to the one considered
here In this pap er w e considered the issue of ho w to share costs b et w een equiv alen t users at dieren t
lo cations Ha ving m ultiple QoSs raises the issue of ho w to share the cost b et w een sev eral mem bers
who are in the same place but request dieren t QoS lev els
Clearly for these dieren t QoS requests
to b e merged on to a single m ulticast distribution tree there m ust b e at least a partial set of ordering
relations b et w een the QoSs and the QoS installed on the link m ust b e greater than or equal to all
QoS requests If the set of qualities of service is completely ordered then the problem reduces to
what has b een called in the literature the airp ort game see where the cost of the link is the cost
asso ciated with the highest QoS requested Here the Shapley v alue of this game is easy to compute
Although this assumes that the route used when computing the standalone cost is c hosen using m ulticast rather than
rev erting to a unicast route
Com bining the t w o issues to ha v e dieren t QoSs at dieren t lo cations can only b e done after one has solv ed the t w o
problems indep enden tly
it is v ery m uc h lik e the ELSD sc heme in that ev ery user shares equally the incremen tal cost of all
lev els less than or equal to their requested lev el This cost allo cation form ula for this sp ecial form
of the problem can b e axiomatized in a v arietyof w a ys see and app ears to b e a rather natural
c hoice If the set of QoSs is not completely ordered but only partially ordered then the problem
b ecomes m uc h more complicated
Multiple qualities of service also presen t some problems for implemen tation If there are a discrete set
of QoS lev els then mo dels and can easily b e c hanged to include information ab out the presence
or exact n um b er of mem b ers requesting eac h lev el Ho w ever ifthereisa con tin uum of QoS lev els
eg bandwidth then w em ust rethink the form of the accoun ting message
Other Issues
There are sev eral other areas that weha venot y et explored W e did not discuss the issue of dynamics
Clearly in a large m ulticast group the mem b ership can c hange rather rapidly and these c hanges aect
the costs allo cated to individual mem b ers Is there some principled w a y to describ e and ameliorate
these c hanges Also w eha v e not at all considered the issue of incen tiv es Giv en some mo del of
strategic b eha vior what equilibria result from the v arious forms of cost allo cation metho ds prop osed
here
References
AJ Ballardie P F F rancis and J Cro w croft Core Based T rees CBT Pr o c e e dings of A CM SIG
COMM SanF r ancisc o HW Braun and K Clay A F ramew ork for Flo wBased Accoun ting on the In ternet Pr o c e e dings of
SICON IEEE Singap or e DD Clark S Shenk er and L Zhang Supp orting RealTime Applications in an In tegrated Services
P ac k et Net w ork Arc hitecture and Mec hanism Pr o c e e dings of A CM SIGCOMM August R Co cc hi S Shenk er D Estrin and L Zhang Pricing in computer net w orks Motiv ation F orm ula
tion and Example IEEEA CM T r ansactions on NetworkingDecem ber S Deering and D Cheriton Multicast Routing in Datagram In ternet w orks and Extended LANs A CM
T r ansactions on Computer Systems pages Ma y S Deering D Estrin D F arinacci V Jacobson CG Liu and L W ei An Arc hitecture for WideArea
Multicast Routing A CM SIGCOMM L ondon Septem ber SC Littlec hild and G Ow en A Simple Expression for the Shapley V alue in a Sp ecial Case Manage
ment Scienc e JK MacKieMason and HR V arian Pricing the In ternet In B Kahin and J Kel ler
Eds Public Access to the In ternetPr entic eHal l Englewo o d Clis New Jersey A vailable fr om
ftpgophere c onlsaumiche dupubPap ersMa y JK MacKieMason and HR V arian Pricing Congestable Net w ork Resources Pr eprint Octob er
N Megiddo Computational Complexit y of the Game Theory Approac h to Cost Allo cation for a T ree
Mathematics of Op er ations R ese ar ch
H Moulin Uniform Externalities Tw o Axioms for F air Allo cation Journal of Public Ec onomy p
H Moulin W elfare Bound in the Co op erativ e Pro duction Problem Games and Ec onomic Behavior
V olume p H Moulin and S Shenk er Av erage Cost Pricing v ersus Serial Cost Sharing An Axiomatic Comparison
Journal of Ec onomic The ory p RB My erson Game Theory Harvar d University Pr ess Cambridge Massachusetts H P eyton and I Y oung Eds Cost Allo cation Metho ds Principles Applications Elseviers Scienc e
Publishers BV A mster dam The Netherlands and New Y ork NY USA AE Roth Ed The Shapley V alue an Essa y in Honor of Llo yd S Shapley Cambridge University
Pr ess Cambridge
J Sairamesh DF F erguson and Y Y emini An Approac h to Pricing Optimal Allo cation and Qualit y
of Service Pro visioning in HighSp eed P ac k et Net w orks Toapp e ar in Info c om LS Shapley A V alue for NP erson Games In HW Kuhn and AW T ucker Eds Con tributions
to the Theory of GamesV ol II A nnals of Mathematics Studies No Princ eton NJ Princ eton
University Pr ess WW Shark ey Net w ork Mo dels in Economics Handb o ok of Op er ations R ese ar ch and Management
Scienc e Networks to app ear
S Shenk er Service Mo dels and Pricing P olicies for an In tegrated Services In ternet In B Kahin and
J Kel ler Eds Public Access to the In ternetPr entic eHal l Englewo o d Clis New Jersey
L Zhang S Deering D Estrin S Shenk er and D Zappala RSVP A New Resource Reserv ation
Proto col IEEE Networks MagazineSeptem ber
v1
v2
v3 v4
v2
v4
0
0 c(v2,v4)
c(v2,v4)
m1,m3,m4
m2
m2 m1,m3,m4
Figure Merging no des with no cost link the Equiv alency axiom
App endix
Theorem Ac ost al lo c ation formula satises the b asic axioms if and only if it is a c anonic al str ate gy
It is straigh tforw ard to v erify that an y canonical form satises the basic axioms W eno w sho w that an y
cost allo cation form ula that ob eys the basic axioms can b e expressed in the canonical form Consider an y
cost allo cation form ula that ob eys the basic axioms W e b egin with the most general case of a net w ork
N V L T and a tree L
T N R loc M Since the allo cation function is additiv e w e can restrict
our atten tion to cost functions c
ij
whichha v e nonzero cost only on the link v
i
v
j
and ha v e unit cost on
that link More sp ecically w ekno w that
P
ij v
i
v
j
L
c
ij
c v
i
v
j
c
and so
P
ij v
i
v
j
L
af
N R M loc c
i j
c v
i
v
j
af
N R M l oc c N R l oc M and M
Wem ust nowsho w that the cost allo cations that result from cost functions c
ij
can b e expressed in terms
of the canonical form
Consider the r e duc e d net w ork with the single link v
i
v
j
and the follo wing loc function
loc
m

v
i
If m
is upstr eam to v
i
v
j
in the or ig inal L
v
j
Otherwise
The subset costs of the reduced net w ork problem are the same as the original problem and the Equiv alency
axiom requires that the cost allo cations b e the same F or example Figure sho ws no des v
v
v
merge
to a single no de v
in the reduced net w ork The anon ymit y condition requires that equiv alen t mem bers
be c harged the same in particular all mem b ers at the same no de b e allo cated the same cost Th us the
allo cations in this reduced problem are c haracterized byt w o quan tities the allo cation to the upstream
mem b ers and the allo cation to the do wnstream mem b ers These allo cations can dep end on the n um ber of
upstream and do wnstream mem b ers so they are expressed as functions F
u
n
u
n
d
and F
d
n
u
n
d
These
costs m ust b e nonnegativ e and the budget balance requiremen t means that
n
u
F
u
n
u
n
d
n
d
F
d
n
u
n
d
This is precisely the canonical form

QED
Theorem Ac ost al lo c ation function satises the b asic and StandA lone axioms if and only if it is the
Equal Link Split Downstr e am ELSD function
Consider an y canonical form F that ob eys the StandAlone axiom The StandAlone axiom implies that
The conditions F u n d and F d n u in the denition of F are irrelev an t to the actual allo cations

R
v2
v3 v4
C1
C3 C2
m2 m1,m2,m3
v5
R
v2
v3 v4
C1
C3 C2
m2 m1,m2,m3
v5
R’
0
C4 C4
R
m1,m2,m3,m4
R’
C1+C2+C3+C4
R
v2
v3 v4
0
00
m2 m1,m2,m3
v5
R’
C1+C2+C3+C4
0
(a) (b) (c) (d)
Figure Equal T ree Split transformations cases abcd
F
u
n
u
n
d
n
u
Com bining this with the budget balance condition n
u
F
u
n
u
n
d
n
d
F
d
n
u
n
d

yields F
d
n
u
n
d
n
d
n
u
whic h is the ELSD form ula It is straigh tforw ard to v erify that the
ELSD form ula satises the StandAlone axiom so the con v erse holds as w ell QED
Theorem Ac ost al lo c ation function satisfying the b asic axioms and the NoF r e eR ider axiom must
satisfy F
u
n
u
n
d
F
d
n
u
n
d
n
u
n
d
Z
Consider an y canonical form F The F ree Rider axiom is ob ey ed if and only if
n u n
d
F
d
n
u
n
d
whenev er
n
d
Ho w ev er com bining this with the budget balance condition n
u
F
u
n
u
n
d
n
d
F
d
n
u
n
d
yields
n
u
F
d
n
u
n
d
F
u
n
u
n
d
Th us whenev er n
u
n
d
Z
w em ust ha v e F
d
n
u
n
d
F
u
n
u
n
d

QED
Theorem Ac ost al lo c ation formula satises Equivalency A nonymity and Subset Monotonicity if and
only if it is the Equal T r e e Split ETS formula
Clearly the ETS sc heme satises the Anon ymit y Equiv alency and Subset Monotonicit y axioms W em ust
nowsho w that an y allo cation function whic h ob eys the Anon ymit yEquiv alency and Subset Monotonicit y
axioms m ust b e the ETS p olicy Figure pro vides an example that follo ws throughout the pro of Consider
an y net w ork N V L T with a ro ot R set of mem bers M lo cation function loc and cost function c
case a F rom it w e buld a second net w ork illustrated in case b b y adding a new link l
R
R with
no cost
N
V R
L l
T
L
T
N
R
loc M L
l
c
c fc
R
R g Clearly all of the subset costs are the same under case a and case b so allo cations m ust not c hange due
to the Equiv alency axiom W enowk eep the net w ork xed and consider a new cost function illustrated
as case c
c
R
R cf L
c
The cost of the whole tree is on RR
c
v
i
v
j
v
i
v
j
L
Note that no subset cost ha v e decreased therefore the subset monotonicit y condition implies that no
individual allo cated costs can decrease Since the total costs ha v e remained the same the balanced budget
implies that total allo cated costs m ust remain unc hanged Under these t w o constrain ts it m ust b e that
the allo cations for this new cost function m ust b e the same as for the old cost function Case d is created

v1
v2
v3 v4
m1 m2
c1
0
10 10
v1
v2
v3 v4
m1 m2
c2
10
10 0
Figure Subset monotonicit y vs OneP ass mo del
b y merging all the no des do wnstream in to R whic h according to the equiv alency axiom do es not c hange
the allo cated costs With all the mem bers in one node R the anon ymit y axiom implies that they all
share the cost of the link equally so eac hmem ber is allocated c

R
R jM j cf L
c jM j This is the
ETS p olicy QED
Theorem No c ost al lo c ation formula satisfying the b asic axioms c an satisfy the no c ol lusion axiom
Consider an y canonical form F and assume that is satises the no collusion axiom Consider the net w ork
with a single link with n
u
mem b ers at the ro ot and n
d
mem bers do wnstream Because F
u
F
d
w e either ha v e F
u
or F
d
Since our pro of w orks with either case
let us assume without loss of generalit y that F
u
The no collusion condition requires that
F
u
n
u
n
d
F
u
n
u
n
d
for an y n
u
b ecause otherwise one elemen tw ould collude with another
Recursing this inequalit yw e nd F
u
n
u
k n
d
k F
u
n
u
n
d
Cho ose n
u
n
d
k and then w e nd that F
u
F
u
This is a violation of the canonical form th us a
con tradiction QED
Theorem Given a tr e e T N R l oc M a set of link c osts candaonep ass me chanism no memb er
c an beallo c ateda c ost gr e ater than her unic ast c ost
F rom the denition of mo del w ekno w that no des are budget balanced so their output and allo cations
are b ounded b y their input and in v
j
c v
i
v
j
out v
i
v
j
for eac h v
j
do wnstream to v
i
Th us
in v
j
c v
i
v
j
in v
i
If w e iterate this inequalit y for eac h no de along the path from R to anynode
v
k
and collapse the inequalities w e get the follo wing result in v
k
cf T N R v
k
c QED
Theorem No c ost al lo c ation scheme implemente d with a onep ass ac c ounting me chanism c an satisfy the
subset monotonicity axiom
Consider an accoun ting sc heme that satises the subset monotonicit y axiom Consider the example in
Figure of a single tree T N R loc M with t w o dieren t link costs c c The cost allo cations for c are straigh tforw ard af
af
Because the total cost of the tree remains the same while no
subset cost decreases subset monotonicit y implies that the cost allo cations m ust b e the same for c and c
Toac hiev e this allo cation for cit m ust b e that out v
v
But w em ust ha v e out v
v
out v
v
whic h violates the lo cal budget balance Con tradiction QED
Theorem No c ost al lo c ation formulae that is implementedbya mo del onep ass ac c ounting scheme c an
satisfy the Equivalency Axiom
Consider a mo del onepass accoun ting sc heme that satises the Equiv alency axiom Consider the follo wing

v1
v2
v3 v4
m1,m2 m3
loc1
0 0
c(v1,v2)
Figure OneP ass mo del vs Equiv alency
example of a net w ork illustrated in gure where c v
v
and c v
v
c v
v
The
Equiv alency axiom implies that the cost allo cations to mem b ers on v
and v
should b e the same af
af
af
Ho w ev er the denition of mo del implies that in v
in v

c v
v
Therefore the
cost allo cations cannot b e the same Con tradiction QED
Theorem No mo del c ost al lo c ation formula satisfying the r e duc edb asic axioms c an satisfy the no
c ol lusion axiom
This pro of is iden tical to the one in theorem replacing F
u
F
d
b y F
l
F
r
n
u
b y n
l
and n
d
b y n
r
QED
Theorem Ac ost al lo c ation formula satises the r e duc edb asic axioms if and only if it c an beexpr esse d
in the onep ass c anonic al form
It is straigh tforw ard to v erify that an y onepass canonical form satises the reduced basic axioms W e
no w sho w that an y mo del cost allo cation form ula that ob eys the basic axioms can b e expressed in the
onepass canonical form Similar to the pro of for theorem w e b egin with the most general case of a
net w ork N V L T and a tree L
T N R l oc M Since the allo cation function is additiv e w ecan
restrict our atten tion to cost functions c
ij
whichha v e nonzero cost only on the link v
i
v
j
and ha v e unit
cost on that link W em ust no w sho w that the cost allo cations that result from cost functions c
ij
can b e
expressed in terms of the canonical form
The anon ymit y axiom implies that all lo cal mem b er are allo cated iden tical costs The denition of mo del
requires that the residual costs passed on to all next hops are the same These allo cations can dep end on
the n um b er of lo cal mem b ers and do wnstream nexthops so they are expressed as functions F
l
n
l
n
r
and
F
r
n
l
n
r
These costs m ust b e nonnegativ e and the budget balance requiremen t means that n
l
F
l
n
l
n
r

n
r
F
r
n
l
n
r
Using Additivit yw e can sho w that the allo cations m ust b e prop ortional to the incoming
costs so
Lo cal mem b ers in v
i
F
l
n
l
n
r

Residual costs out v
j
v
k
in v
i
F
r
n
l
n
r
for eac h immediate do wnstream no de v
k
In mo del the input costs to the next hop v
k
include b oth residual cost and the cost of the link connecting
them in v
k
in v
i
F
r
n
l
n
r
c v
i
v
k
This is precisely the OneP ass canonical form QED
Theorem Amo del c ost al lo c ation formula satises the r e duc edb asic axioms and the SharingisGo o d
axiom if and only if the functions F
l
n
l
n
r
and F
l
n
l
n
r
ar e nonincr e asing on fZ
g
v1
v2
v3 v4
10
11
m1 m2,m3
Figure NoF reeRider axiom in mo del It is straigh tforw ard to sho w that the SharingisGo o d axiom conditions are exactly iden tical to the no
increasing form ulae conditions When adding a lo cal mem b er to a no de already on the distribution tree
wem ust ha v e
F
l
n
l
n
r
F
l
n
l
n
r
n
l
n
r
Z
Z
F
r
n
l
n
r
F
r
n
l
n
r
n
l
n
r
Z
Z
When adding a mem b er to a no de whichw as not on the distribution tree at the nearest no de on the tree
wem ust ha v e
F
l
n
l
n
r
F
l
n
l
n
r
n
l
n
r
Z
Z
F
r
n
l
n
r
F
r
n
l
n
r
n
l
n
r
Z
Z
When w e com bine these equations with the second condition in the denition of F w e get the nonincreasing
condition on all of fZ
g QED
Theorem Thereisnomodelc ost al lo c ation formula that satises the r e duc edb asic axioms and the
NoF r e eR ider axiom
Assume there is a mo del cost allo cation form ula that satises the reduced basic axioms and the NoF ree
Rider axiom In Theorem w eha vesho wn that all mo del cost allo cation form ulae ob eying the reduced
basic axioms can b e expressed in the onepass canonical form Consider the tree detailed in gure with
no mem b ers in v
F
r
for all form ulae resulting in in v
With no nexthops do wnstream
of v F
l
for all form ulae making the allo cation to eac hmem ber on v
exactly whic his less
than unicast cost jM j QED

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Description

Shai Herzog, Scott Shenker, and Deborah Estrin. "Sharing the "cost" of multicast trees: An axiomatic analysis." Computer Science Technical Reports (Los Angeles, California, USA: University of Southern California. Department of Computer Science) no. 614 (1995).

Asset Metadata

Creator Estrin, Deborah (author), Herzog, Shai (author), Shenker, Scott (author)

Core Title USC Computer Science Technical Reports, no. 614 (1995)

Alternative Title Sharing the "cost" of multicast trees: An axiomatic analysis (title)

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

Tag OAI-PMH Harvest

Format 26 pages (extent), technical reports (aat)

Language English

Unique identifier UC16269206

Identifier 95-614 Sharing the cost of Multicast Trees An Axiomatic Analysis (filename)

Legacy Identifier usc-cstr-95-614

Format 26 pages (extent),technical reports (aat)

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

Internet Media Type application/pdf

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

Access Conditions The author(s) retain rights to their work according to U.S. copyright law. Electronic access is being provided by the USC Libraries, but does not grant the reader permission to use the work if the desired use is covered by copyright. It is the author, as rights holder, who must provide use permission if such use is covered by copyright.

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

Repository Email csdept@usc.edu

Inherited Values

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)