On Equivlene Between Network Topologies Tre Ho Deprtment of Eletril Engineering Cliforni Institute of Tehnolog tho@lteh.eu; Mihelle Effros Deprtments of Eletril Engineering Cliforni Institute of Tehnolog effros@lteh.eu; Shirin Jlli Center for Mthemtis of Informtion Cliforni Institute of Tehnolog shirin@lteh.eu; Astrt One mjor open prolem in network oing is to hrterize the pit region of generl multi-soure multi-emn network. There re some existing omputtionl tools for ouning the pit of generl networks, ut their omputtionl omplexit grows ver quikl with the size of the network. This motivtes us to propose new hierrhil pproh whih fins upper n lower ouning networks of smller size for given network. This pproh sequentill reples omponents of the network with simpler strutures, i.e., with fewer links or noes, so tht the resulting network is more menle to omputtionl nlsis n its pit provies n upper or lower oun on the pit of the originl network. The ur of the resulting ouns n e oune s funtion of the link pities. Surprisingl, we re le to simplif some fmilies of network strutures without n loss in ur. I. INTRODUCTION Fining the network oing pit of networks with generl topologies n ommunition emns is hllenging open prolem, even for networks onsisting of noiseless pointto-point links. Informtion theoreti inequlities n e use to oun network pities, ut it is in generl omplex tsk to fin the est omintion of inequlities to ppl. While vrious ouns (e.g. [1], [2], [2], [3]) n e otine lever hoies of inequlities, we woul like to hve sstemti generl tehniques for ouning the pit in ritrr network prolems. We hope to erive these ouns in w tht llows us to oun the ur of the otine ouns n to tre off tightness n omputtion. The LP outer oun [4], whih gives the tightest outer oun implie Shnnon-tpe inequlities n hs een implemente s the softwre progrms Informtion Theoreti Inequlities Prover (ITIP) [5] n XITIP [6], hs omplexit exponentil in the numer of links in the network n n thus onl e use to ompute pit ouns for reltivel smll prolem instnes. Inner ouns n e otine restriting ttention to slr liner, vetor liner or elin oes e.g. [7], [8], ut the omplexit of suh pprohes lso grows quikl in the network size. This motivtes us to seek sstemti tools for ouning the pit of network the pit of nother network with fewer links n hrterizing the ifferene in pit. In this pper we introue novel pproh for nlzing pit regions of li networks onsisting of pitte noiseless links with generl emns. Inspire [9], we emplo hierrhil network nlsis tht reples piees of the network equivlent or ouning moels with fewer links. At eh step of the proess, one omponent of the network is reple simpler struture with the sme inputs n outputs. The moel is n upper ouning moel if ll funtions tht n e implemente ross the originl network n lso e implemente ross the moel. The moel is lower ouning moel if ll funtions tht n e implemente ross tht moel n lso e implemente ross the given omponent. If the sme moel is oth n upper ouning moel n lower ouning moel for given omponent, then the omponent n its moel re equivlent. Where possile, we tr to fin upper n lower ouns tht hve ientil strutures, sine ouning the ur of the resulting pit ouns is esier when the topologies of the upper n lower ouning networks mth. The orgniztion of this pper is s follows. Setion II esries the sstem moel. The prolem of fining equivlent or ouning networks of smller size n the properties of suh networks is isusse in Setion III. Setions IV n V esrie vriet of opertions for fining suh networks. Setion VI trets ur ouns. The networks onsiere in this pper re ssume to e li. The effet of les n el is isusse in Setion VII. Finll, Setion VIII onlues the pper. II. SYSTEM MODEL We minl use the moel n nottion introue in [10]. The network is moele n li irete grph N = (V,E), where V n E V V enote the set of noes n links, respetivel. Eh irete link e = (v 1,v 2 ) E represents noiseless link of pit C e etween the noes v 1 n v 2 in N. For eh noe v, let In(v) n Out(v) enote the set of inoming n outgoing links of noe v respetivel. We ssume tht the soure noes (S) n sink noes (T ) re istint, i.e., S T T = /0, n eh soure (sink) noe hs onl outgoing (inoming) links. There is no loss of generlit in this ssumption sine n network tht violtes these onitions n e moifie to form network tht stisfies these onitions n hs the sme pit. After these moifitions, eh noe v V flls into one of the following tegories: i) soure noes (S), ii) sink noes (T ) n iii) rel noes (I ). Rel noes hve oth inoming n outgoing links, n the o not oserve n externl informtion. Their role is to filitte t trnsfer from the soure noes to the sink noes.
A sunetwork N s = (V s,e s ) of network N = (V,E) is onstrute se on suset of rel noes I s I s follows. The set of soures n sinks of the sunetwork N s re efine s S s = {v V : v / I s, v I s,(v,v ) E}, n T s = {v V : v / I s, v I s,(v,v) E}. Then, V S = S s S Is S Ts, n E s = {e E : e = (v,v ),v,v V s }. A oing sheme of lok length n for this network is esrie s follows. Eh soure noe s S oserves some messge M s X s = {1,2,...,2 nr s }. Eh sink noe t T is intereste in reovering some of the messges tht re oserve soure noes. Let β(t) S enote the set of soure noes tht the noe t is intereste in reovering. The oing opertions performe eh noe n e tegorize s follows 1) Enoing funtions performe the soure noes: For eh s S, n e Out(s), the enoing funtion orresponing to link e is esrie s g e : X s {1,2,...,2 nc e }. (1) 2) Rel funtions performe t rel noes: If v / S S T, then for eh e Out(v), the rel funtion orresponing to the link e is esrie s g e : e In(v) {1,2,...,2 nc e } {1,2,...,2 nc e }. (2) 3) Finll, for eh t T, n eh s β(t), eoing funtion is efine s g s t : e In(t) {1,2,...,2 nc e } X s. (3) A rte vetor R orresponing to the set {R s } s S is si to e hievle on network N, if for n ε > 0, there exists n lrge enough n oing sheme of lok length n suh tht for ll t T n s β(t) P( ˆM (t) s X s ) ε, (4) where ˆM s (t) enotes the reonstrution of messge M s t noe t. Let R (N ) enote the set of hievle rtes on network N. Throughout the pper, vetors re enote ol upperse letters, e.g. A, B, et. Sets re enotes lligrphi upper-se letters, e.g. A, B, et. For vetor A = ( 1, 2,..., n ) of length n n set F {1,2,...,n}, A F enotes vetor of length F forme the elements of the vetor A whose inies re in the set F in the orer the show up in A. III. EQUIVALENT AND BOUNDING NETWORKS The prolem we onsier is efine formll s follows. For given network N, we wish to fin network N with fewer links for whih the set of hievle rtes either ouns R (N ) from elow (R (N ) R (N )), ouns R (N ) from ove (R (N ) R (N )) or esries it perfetl (R (N ) = R (N )). We tke hierrhil pproh, sequentill ppling opertions to simplif the given network. Following [9], eh opertion reples sunetwork of the network with ouning moel. Sunetwork N 2 is n upper ouning moel for sunetwork N 1 with the sme numer of input (soure) n output (sink) noes (written N 1 N 2 ) if ll funtions { f t } t T of soures tht n e reonstrute t the sinks of N 1 n lso e reonstrute t the sinks of N 2 n lso e reonstrute t the sinks of N 2. Here, eh funtion f t, for t T, is funtion of the informtion soures tht re ville t soure noes. Sunetworks N 1 n N 2 re equivlent if N 1 N 2 n N 2 N 1. When eriving upper n lower ouning networks, it is esirle to fin upper n lower ouning networks tht hve the sme topologies. In this se, we n oun the ifferene etween the pit of network N n pities R (N l ) n R (N u ) of lower n upper ouning networks N l n N u using oun from [11]. Note tht hving ientil grphs, we lso require tht ll links hve non-zero pit in oth networks. For ompring two networks N l n N u whih hve ientil topologies, efine the ifferene ftor etween N l n N u s C (N l,n e (N u ) u ) mx e E C e (N l ), (5) where C e (N l ) n C e (N u ) enote the pities of the link e is N l n N u respetivel. Note tht (N l,n u ) 1. Let R l (R u ) enote the pit region of N l (N u ). Then while R l R R u, (6) R u (N l,n u )R l. (7) IV. BASIC SIMPLIFICATION OPERATIONS One of the simplest opertions for eriving n upperouning network for given network is merging noes. Colesing two noes is equivlent to ing two links of infinite pit from eh of them to the other one. This is preisel the pproh emploe in ut-set ouns. Beuse of these infinite-pit links, omining noes, unless one wisel, potentill n result in ver loose ouns. However, we show tht in some ses, noes n e omine without ffeting the network pit. One simple exmple is when the sum of the pities of the inoming links of noe v is less tht the pit of eh of its outgoing links. In this sitution, the noe n e omine with ll noes w suh tht (v,w) E. Another possile opertion for getting upper or lower ouning networks is reuing or inresing the link pities. As speil se of suh opertions, one n reue the pit of link to zero whih is the sme s eleting the link. In some ses reuing/inresing link pities is helpful in simplifing the network. Another tpe of opertion for simplifing networks is se on network ut-sets. A ut P etween two sets of noes W 1 n W 2 is prtition of the network noes V into two sets V 1 n V 2 suh tht W 1 V 1,W 2 V 2. The pit of ut is efine s the sum of pities of the forwr links of
the ut, i.e. links (v,w) suh tht v V 1,w V 2. Links (v,w) suh tht v V 2,w V 1 re lle kwr links of the ut. For exmple, if we fin minimum ut seprting sink from its orresponing soures n ll other sink noes, n onnet the forwr links iretl to the sink noe while preserving ll the kwr links, this results in n upper-ouning network. If inste of keeping the kwr links, we elete them, lower-ouning network is otine. In the se where there re no kwr links, this proeure results in n equivlent network. We n of ourse repet this proeure for ever sink. Another simplifition opertion involves removing set A of links or omponents n possil repling it with itionl pit tht might e spre over the remining network. A simple lower ouning network n e otine removing set A, while n upper ouning network n e otine from repling set A ing suffiient pit to the remining network to e le to rr n informtion tht oul hve een rrie the set A. For exmple, if the remining network ontins pths from the inputs to the outputs of the set A, we n formulte liner progrm, se on generlize flows, to fin the minimum ftor k whih sling up the pities of the remining links uniforml gives n upper ouning network 1. Sine the lower ouning network otine just removing the set A iffers from the upper ouning network the sling ftor k, this gives multiplitive oun of k 1 on the potentil pit ifferene ssoite with the upper n lower ouning opertions. V. Y-NETWORKS AND GENERALIZATIONS Consier the network shown in Fig. 1 onsisting of four noes n three irete links with pities (r 1,r 2,r 3 ). This topolog is prol the simplest network in whih noes re shring resoures to sen their informtion. We will refer to suh network s Y-network Y(r 1,r 2,r 3 ). Now onsier the network shown in Fig. 2 whih onsists of two Y-networks with shre input n output noes. The following lemm shows tht in some speil ses this network is equivlent to nother Y-network. This simplifition reues the numer of links 3 n the numer of noes 1. Lemm 1: Consier the network shown in Fig. 2, N 1, when ã = α, = α n = α. This network is equivlent to N 2, Y-network Y((1+α),(1+α),(1+α)). Proof: Clerl, Y-network Y((1 + α),(1 + α),(1 + α)) is n outer oun for the network of Fig. 2. Hene, we onl nee to show tht it lso serves s n inner oun. For the rest of the proof ssume tht α is rtionl numer (If it is not rtionl, it n e pproximte rtionl numers with ritrr preision). Consier oe of lok length n tht runs on network N 2. The mile noe mps the n(1 + α) its reeive from x 1 1 One w to think of this is to ssoite ommoit with eh link or pth segment in A. Conversions etween ommoities tke ple oring to the link pit rtios in A. The liner progrm minimizes k sujet to flow onservtion of these ommoities. x 1 x 2 Fig. 1. r 1 r 2 r 3 A Y-network n the n(1+α) its reeive from x 2 to n(1+α) its sent to noe. In orer to run the sme oe on N 1, onsier using the network m times, where m is hosen suh tht αm/(1+α)n n m/(1+α)n re oth integers. Note tht sine α is rtionl ssumption, it is lws possile to fin suh m. Let k 1 αm/(1+α)n n k 2 m/(1+α)n. During these m hnnel uses the intersetion noe t the left hn sie reeives m its from x 1 n m its from x 2. This is equl to the its reeive the intersetion noe in N 2 uring m/(1+α)n = k 2 oing sessions. Therefore, using the oe use on N 2, these its n e mppe into k 2 n(1 + α) = m its tht will e sent to. Similrl, the numer of its reeive the intersetion noe on the right hn sie uring m hnnel uses is equl to the its tht woul hve een reeive the interession noe on N 2 uring k 1 oing sessions. Lemm 1 serves s useful tool in our network simplifitions. For n exmple of how to emplo this result, onsier the network shown in Fig. 3. This network n e onsiere s omintion of two overlpping Y-networks. Fig. 2. links x 1 x 2 ã 2 Y-networks with shre soure n sink noes n seprte rel Lemm 2: Let β +. If β + (1 β), then the network shown in Fig. 3 is equivlent to Y-network Y(,+,). Proof: Clerl Y-network Y(, +,) is n upperouning network for the network of Fig. 3. We show tht if the onstrints in the lemm is stisfie, it lso serves s lower-ouning network. To fin lower ouning network, onsier reking the links in Fig. 3 into prllel links s in Fig. 4, where = 1 + 2 + 3, = 1 + 2 + 3, et. The network ontins two
x 1 x 2 PSfrg 1 1 1 2 1 2 1 2 Fig. 3. Two noes ommuniting with one sink noe vi two rel noes Fig. 5. Breking the network in Fig. 3 into two Y-networks pit-isjoint Y-networks s illustrte in Fig. 5. Our gol is to omine these two Y-networks ppling Lemm 1, n in orer to e le to o this, we require 2 = 2 = α 1, 1 = α 1 n 1 = 1 = 2 α. The omintion of these two Y-networks will e Y-network Y((1+α) 1,(1+α) 1,(1+ α) 1 ) whih is lower-ouning network for our originl network. Now hoosing 1 =, 1 = n α = from the link pities onstrints, we shoul hve (1+ ) 1, (1+ ) 1, 1 + 1. (8) From these inequlities, if β+(1 β), we n hoose 1 = (1 β) n 1 = (1 β) n get lower-ouning Y- network Y(,+,). Fig. 4. 1 2 3 1 2 3 1 2 3 1 2 1 Breking the links in Fig. 3 into prllel links In orer to get etter unerstning of the require onstrint stte in Lemm 2, onsier the following speil ses: 2 0 1 2 k. Fig. 6. Generliztion of the network of Fig. 3 i., ii. = + 2, iii.. Agin onsier the network shown in Fig. 3 where ll the links hve pit 1 exept for the link of pit. From Lemm 2, for 0.5 1 + 0.5 1 = 1, this network is equivlent to Y-network T(1,2,1). Our proof pproh in Lemm 2 oes not s nthing out the se where < 1. It might e the se tht even for some vlues < 1, still this equivlene hols. As we show through n exmple this nnot hppen, n for < 1 the Y-network of Y(1,2,1) is strit upper oun for our network. Assume tht 1 n ( 2, 3 ) re ville t x 1 n x 2 respetivel, where i {0,1}. The gol is reonstruting 1 2 + 1 2 + 3 t noe, where ll opertions re in GF(2). This n e one esil in the Y- network Y(1,2,1), ut is impossile in the originl network for < 1. Fig. 6 shows generliztion of the network of Fig. 3, where inste of 2 intermeite noes, there re k + 1 intermeite noes. Let α 0 1, α i i / 0, for i {1,...,k}, 1 2 k 1, 1+ k α i i=1
n 1. 1+ k α i i=1 B extening the proof of Lemm 2 to this more generl se, we get Lemm 3. Lemm 3: If, for i {1,...,k}, i 1 j=0 α j 1 + k j=i α j 1 i, (9) then the network shown in Fig. 6 is equivlent to Y-network Y(, k i,). i=0 Another possile generliztion of Y-network is shown in Fig. 7. Here, while the numer of rel noes is kept s two, the numer of soure noes is inrese. For this network, we n prove Lemm 4 with strightforwr extension of Lemm 2. Lemm 4: For i = {2,...,k}, let Then, if n i=1 β i i. i β i + n i=1 (1 β i ), (10) the two intermeite noes n e omine without hnging the performne. x 1 z 1 2 2 k k Fig. 7. Another generliztion of the network of Fig. 3 Agin onsier the network shown in Fig. 3, n ssume tht < β +(1 β). As shown in the previous setion, Y-network Y(,+,), N u, serves s strit upper oun for the network of Fig. 3. In orer to get lower ouning Y-network, we reue the pities of the links n ftor δ, suh tht = βδ+(1 β)δ. Hene, δ = β+(1 β), n from our ssumption 0 < δ < 1. Using this, we n gin invoke Lemm 2 n get lower-ouning Y-network of... Fig. 8. Fig. 3. min(, ) min(,) A pir of upper n lower ouning networks for the network of (δ,+,δ) enote N l. Compring the link pities of the upper n lower ouning Y-networks, their ifferene ftor is (N l,n u ) = δ. If β + (1 β), then δ 1, mening tht the ouns eome ver loose in suh ses. To solve this prolem, onsier nother possile pir of upper n lower ouning networks shown in Fig. 8, whih hve topolog ifferent thn Y-network. Here the ssumption is tht <. It is es to hek tht these re inee upper n lower ouning networks. For eriving the upper-oun, the inoming informtion from links n re ssume to e trnsmitte iretl to the finl noe, n sine the inoming pit of the finl noe is, the informtion sent to it from link n e pture iret link of pit min(,). For the lower oun, sine <, ll the informtion on link n e irete to the finl noe. B this strteg, the remining unuse pit of link is whih n e eite to link. If min(, ) =, then +, then the upper n lower ouning networks oinie, n we hve n equivlent network. The more interesting se is when > +, n therefore min(, ) =. In this se the ifferene ftor of the upper n lower ouning networks is t lest 1 /. Choosing the est oun epening on the link pities, given,, n the worst ifferene ftor is min mx{ β+(1 β),1 } = +β+(1 β). (11) As n exmple, for the se where = = = = 1, the worst se ifferene ftor is 0.5 whih orrespons to = 0.5. This mens tht hoosing the est pir of ouns for ifferent vlues of, the ifferene ftor of our selete pir is lws lower oune 0.5. We onlue this setion isussing how the pproh in the setion generlizes to lrger lss of network topologies. Consier the network N 1 shown in Fig. 9. A Y-network Y( i, i, i ) is lws n upper oun for the network N 1. On the other hn, if we n show, s in the proof of Lemm 2, tht N 1 ontins two pit-isjoint Y-networks of the form Y(α i,α i,α i ) n Y((1 α) i,(1 α) i,(1 α) i ), then Y( i, i, i ) is lso lower oun for, n thus equivlent to, N 1. Sufigures () n () provie two simple exmples of networks where this is the se. This onstrution n lso e generlize repling the si Y -shpe topolog with str-shpe topologies with ritrr numers of inputs n outputs.
x... n 1 2 1 2... m e v 1 v 2 v 1 v 2 1 2... l z Fig. 10. N 1 N 2 Networks N 1 n N 2 whih re with n without link e respetivel. x r (1-r) -' ' (1-r) z r + (1-r) - ' () Exmple 1 () Network N 1 x r z (1-r) r + (1-r) () Exmple 2 Fig. 9. A fmil of omponents tht n e reple with n equivlent omponent. The lels represent the link pities, whih stisf = 1 + 2 +... n, = 1 + 2 +... m, = 1 + 2 +... l, n 0 < r < 1. VI. EFFECT OF LINK CAPACITIES Consier two networks N 1 n N 2 whih hve ientil topologies n link pities exept for some link e whih hs pit C e,1 in N 1 n pit C e,2 < C e,1 in N 2. Let R 1 n R 2 enote the set of hievle rtes on N 1 n N 2 respetivel. The question is how this ifferene ffets the performnes of these two networks. One w of oing this omprison is se on wht ws mentione erlier, i.e., to ompute the rtio etween C e,1 n C e,2. However, this might not lws give the est possile oun. The reson is tht it might e se tht while C e,1 n C e,2 re oth smll ompre to the pit region of the networks, (N l,n u ) = C e,1 /C e,2 is ver lrge. In this setion, we stu this prolem in more etils. Note tht the link of pit C e,1 in network N 1 into two prllel links of pities C e,1 C e,2 n C e,2. Clerl this proess oes not ffet the pit region of N 1. B this trnsformtion, network N 2 is equivlent to this new network with link of pit ε C e,1 C e,1 eing remove. Therefore, in the rest of this setion, inste of hnging the pit of link, we ssume tht link of pit ε is remove s shown in Fig. 10, n prove tht t lest in some ses hnging the pit of link ε nnot hve n effet lrger thn ε on the set of hievle rtes, i.e., if the rte vetor R is hievle on N 1, rte vetor R ε1, where 1 enotes n ll-one vetor of length S, is hievle on N 2 s well. One suh exmple is the se of multist networks. In tht se, the pit of the network is etermine the vlues of the uts from the soures to the sinks. Therefore, sine removing link of pit ε oes not hnge the vlues of the uts more thn ε, the pit of the network will not e ffete more thn ε. Another exmple, is the se where ll soures re onnete iretl link to super-soure noe whih hs therefore ess to the informtion of ll soures. Without loss of generlit, let S = {1,2,..., S }. As efore, ssume tht eh sink noe t T is intereste in reovering suset of soures enote β(t). Theorem 1: For the esrie network with super sourenoe n ritrr emns, removing link of pit ε n hnge the pit region t most ε. Proof: Assume tht the rte vetor R = (R 1,R 2,...,R S ) is hievle on N 1. Sine in the single soure se, pit regions orresponing to zero n smptotill zero proilit of errors oinie [12], we n ssume there exist oing sheme of lok length n tht hieves rte R on N 1 with zero proilit of error. Bse on this zero-error oe, we onstrut nother oe for network N 2 tht hieves rte R ε1 with smptotill zero proilit of error. B our ssumption out the network struture, the super soure-noe oserves messge vetor M = (M 1,...,M S ), where M s {1,2,...,2 nr s }, for s {1,2,..., S }. Now onsier the link e = (v 1,v 2 ) in network N 1 whih hs een remove in network N 2. During the oing proess on N 1, the its sent ross this link n tke on t most 2 nε ifferent vlues. Consier inning the messge vetors M = (M 1,M 2,...,M S ) into 2 nε ifferent ins se on the it strem sent over this link uring their trnsmission. Sine the oe is eterministi oe, eh messge vetor onl orrespons to one in. Sine there exist 2 n s S R s istint messge vetors, se on the Pigeonhole priniple, there will e t lest one in with more tht 2 n( s S R s ε) messge vetors. Denote the messge vetors ontine in this in the set M 0. In N 2, no messge n e sent from v 1 to v 2 through link e.
Therefore, in orer to run the oe for N 1 on network N 2, we nee to speif the ssumption of noe v 2 out the messge tht it woul hve reeive from noe v 1 in N 1. Let noe v 2 ssume tht it lws reeives the it pttern orresponing to the in ontining M 0. Mking this ssumption n hving the rest of N 2 to perform s in N 1, it is ler tht ll messge vetors in M 0 n e elivere with zero proilit of error on N 2 s well. In other wors, if the input to the super soure noe in N 2 is one of the messge vetors in M 0, then eh sink t T reovers its intene messge M β(t) with proilit one. In the rest of the proof we show how this set M 0 n e use to eliver inepenent informtion to ifferent reeivers over network N 2. Define rnom vetor U = (U 1,U 2,...,U S ) to hve uniform istriution over the elements of M 0. For eh input vetor U, eh sink noe t, reovers U β(t) perfetl. The esrie moel with input U, n ouputs U β(t), for t T is eterministi rost hnnel (DBC) 2 whose pit region is known. Therefore, we n emplo this DBC to eliver informtion on N 2. Before oing this, we slightl hnge the set of sink noes, n reple the set of sinks T T e s esrie in the following. This moifition oes not ffet the funtionlit of the network, ut simplifies the sttement of the proof. Divie eh sink noe t into β(t) sink noes, suh tht eh one hs ess to ll the informtion ville to the noe t, ut is onl intereste in reonstruting one of the soures. Let T e enote this expne set of sinks. Consier suset T s of size S of T e suh tht eh soure s S is reovere one of the sinks in T S. Sine eh sink in T s onl reovers one soure, hene there shoul e one-to-one orresponene etween the elements of S n T s. Now onsier the DBC with input U n outputs {U t } t Ts. Sine the oe is zero-error, n there is one-to-one orresponene etween the elements of S n T s, {U s } s S n e reple {U s } s S. The pit region of this DBC, s expline in Appenix A, n e esrie the set of rtes (r 1,r 2,...,r S ) stisfing for ll A S. From our onstrution, r s H(U A ), (12) s A H(U S ) = H(U 1,U 2,...,U S ) n( R s ε). (13) s S On the other hn, for eh A S H(U S ) = H(U A )+H(U S\A ). (14) 2 DBCs n their pit regions re riefl esrie in Appenix A. Therefore, H(U A ) = H(U S ) H(U S\A ) n( s S n( s S R s ε) H(U s ) s S\A R s ε) nr s s S\A = n( R s ε). (15) s A It is es to hek tht the point (r 1,r 2,...,r S ) = n(r ε1) stisfies ll inequlities require (12). Hene, using the network N 2 s DBC, the rte vetor R ε n e trnsmitte to noes in T s. We now rgue tht the esrie DBC oe, with no extr effort, elivers the rte vetor R ε on network N 2 to ll sinks, not just those in T s. The reson is tht for eh messge vetor M M 0, n onsequentl for eh input U, ll sink noes in T e tht re intereste in reovering soure s, i.e., for ll t T e suh tht β(t) = s, reeive M s with proilit one. Hene, if the esrie oing sheme rte R s is elivere to one of them, then the rest re le to reeive the sme t strem t rte R s s well. For some simple ses of multi-soure multi-emn se networks, for instne when the link of pit ε is iretl onnete to one of the soures, the sme result still hols, i.e. removing tht link nnot hnge the pit region more thn ε. However, for the generl se, it remins open s to how muh removing link of pit ε n ffet the pit region of the network. VII. CYCLES AND DELAYS In [13], the pit of hnnel is efine in terms of its per hnnel use. In the se where there re no les, this efinition is equivlent to efining pit in terms of its trnsmitte per unit time. However, we o not know whether in generl network with les n els, these two efinitions, i.e., its per hnnel use versus its per unit time, oul result in ifferent pit regions. For exmple, onsier two networks N 1 n N 2 tht re ientil exept for single omponent le, illustrte in Fig. 11. In N 2 the trnsmission el from v 1 to v 2 is lrger thn in N 1 ue to the e noe v 0, ssuming tht eh link introues unit el. Now, we o not know whether hieving pit on N 1 oul require intertive ommunition over feek loop, where eh suessive smol trnsmitte v 1 epens on previous smols reeive from v 2 n vie vers. If we tr to run the sme oe on N 2, the numer of smols per unit time tht n e exhnge noes v 1 n v 2 is erese euse of the itionl el introue noe v 0. Hene, if the rte is efine in terms of its ommunite per unit time, this itionl el will reue the pit region. However, if the pit is efine in terms of its ommunite per hnnel use, letting eh noe wit until it hs reeive the informtion it requires to generte its next smol, the
replements v 1 v 2 v 0 v 1 v 2 Y 1 = f 1 (X) Y 2 = f 2 (X) () Component of N 1 () Component of N 2 X DBC Fig. 11. noe Chnging the el etween two noes introuing n extr Y K 1 = f K 1 (X) Y K = f K (X) elivere rtes n onsequentl the pit region will not e ffete. 3 In most of the opertions introue in this pper, the numer of hops etween noes is ifferent in the originl network ompre to its simplifie version. Therefore, suh opertions in li grph with el n potentill hnge the pit region in ws tht re not preite our nlsis. Hene, in this pper we hve restrite ourselves to the se of li grphs. However, stuing the effet of these opertions in generl network with les remins s question for further stu. VIII. CONCLUSIONS In this pper, we propose new tehniques for effiientl pproximting or ouning the pit of networks with omplex topologies n ritrr ommunition emns, suh s non-multist n funtionl emns. We propose to tke new pproh se on sstemti grph opertions tht n e pplie reursivel to simplif lrge networks, so s to filitte the pplition of omputtionl ouning tools. Besies its generlit, nother motivtion for suh reursive grph-se pproh is tht it lens itself to trtle ivien-onquer lgorithms for nlzing ver lrge networks n llows omputtion to e tre off ginst tightness of otine ouns oring to the extent of simplifition. Tehniques propose for networks of irete noiseless point-to-point links (it pipes) n e reil pplie to networks of nois multi-terminl hnnels using the results of Koetter et l. [9] to reple eh nois link with ouning moel onsisting of suh it pipes. However, it m e possile to otin etter ouns eveloping simplifition tools tht re iretl pplile to nois networks. Thus, we will seek to exten the ove work to networks ontining nois rost, multiple ess n interferene links. APPENDIX A DETERMINISTIC BROADCAST CHANNEL Deterministi rost hnnels (DBC) re spil ses of generl rost hnnels. In K-user DBC, P((Y 1,...,Y K ) = ( 1,..., K ) X = x) {0,1}. In other wors, sine in BC the pit region onl epens on the mrginl istriutions [13], K-user DBC n e 3 Note tht in the se of multist, sine it is known tht pit-hieving oes o not require suh ommunition over feek loops, the pit region remins unhnge uner oth efinitions, even in the se of li grphs. Fig. 12. Deterministi rost hnnel esrie k funtions ( f 1,..., f k ) where f i : X Y i, (A-1) n Y i = f i (X) for i {1,2,...,k}. In (A-1), X n Y i refer to the hnnel input lphet n the output lphet of the i th hnnel respetivel. The pit region of k-user DBC n e esrie the union of the set of rtes (R 1,R 2,...,R k ) stisfing for some P(X) [14], [15]. R i H(Y A ), A {1,...,k}, i A ACKNOWLEDGMENTS (A-2) This work hs een supporte in prt the Air Fore Offie of Sientifi Reserh uner grnt FA9550-10-1-0166 n Clteh s Lee Center for Avne Networking. REFERENCES [1] Krmer. G. n S. Svri. Cpit ouns for rel networks. In Pro. 2006 Workshop Inf. Theor Appl., L Joll, CA, fe 2006. [2] M. Aler, N. J. Hrver, K. Jin, R. Kleinerg, n A. R. Lehmn. On the pit of informtion networks. In SODA, 2006. [3] N. J. A. Hrve, R. Kleinerg, C. Nir, n Y. Wu. A hiken n egg network oing prolem. In Pro. IEEE Int. Smp. Inform. Theor, Nie, Frne, June 2007. [4] R.W. Yeung. A frmework for liner informtion inequlities. Informtion Theor, IEEE Trnstions on, 43(6):1924 1934, nov. 1997. [5] A.T. Surmnin n A. Thngrj. A simple lgeri formultion for the slr liner network oing prolem. ArXiv e-prints, Jul 2008. [6] Xitip - informtion theoreti inequlities prover. http://xitip.epfl.h/. [7] M. Mér n R. Koetter. An lgeri pproh to network oing. IEEE/ACM Trnstions on Networking, pges 782 795, 2003. [8] T. H. Chn. Cpit regions for liner n elin network oe. In NETCOD, Sn Diego, CA, Jnur 2007. [9] R. Koetter, M. Effros, n M. Mér. A theor of network equivlene, prts i n ii. ArXiv e-prints, Jul 2010. [10] R. W. Yeung. Informtion Theor n Network Coing. Springer, 2008. [11] M. Effros. On pit outer ouns for simple fmil of wireless networks. In Informtion Theor n Applitions Workshop, Sn Diego, CA, 2010. [12] T. Chn n A. Grnt. On pit regions of non-multist networks. In Pro. IEEE Int. Smp. Inform. Theor, Austin, TX, June 2010. [13] T. Cover n J. Thoms. Elements of Informtion Theor. Wile, New York, 2n eition, 2006. [14] K. Mrton. The pit region of eterministi rost hnnels. In Pro. IEEE Int. Smp. Inform. Theor, Pris-Chn, Frne, 1977. [15] M. S. Pinsker. Cpit of noiseless rost hnnels. Prol. Inform. Trnsm., pges 92 102, 1978.