Topology Information Condensation in Hierarchical Networks.

Transcription

1 Topology Information Condnsation in Hirarchical Ntworks. Pit Van Mighm Dlft Univrsity of Tchnology a ABSTRACT Inspird by th PNNI protocol of th ATM Forum, this work focuss on th problm of nod aggrgation within pr groups and link aggrgation btwn pr groups. It is assumd that th (larg) ntwork is alrady dividd into pr groups. Th objctiv is to maximally condns topology information subjct to a givn accuracy constraint. Th QoS masurs can b rducd into thr distinct classs: an additiv QoS class, a min-max on and th last, a combination of additiv and min-max QoS masurs. A nw mthod for nod aggrgation of th additiv QoS class (vn with multipl QoS masurs) is prsntd. Th min-max class is discussd and L s optimal solution (L, 95) for a singl min(max) QoS masur is rviwd. Finally, w discuss th xtnsion of our nw mthod to th combination of additiv and min-max QoS masurs. A dtaild xampl illustrats th prsntd algorithm for a singl additiv QoS masur. Subjct to a givn rlativ accuracy ε, it shows how to prform nod and link aggrgation on diffrnt hirarchical lvls and how to stablish th whol hirarchical structur of th original ntwork. KEYWORDS: hirarchical ntworks, information condnsation, nod and link aggrgation SUBMITTED TO : Computr Ntworks and ISDN Systms Corrsponding author: Pit Van Mighm Tlcommunication and Traffic Control Systms Group Faculty of Information Tchnology and Systms Dlft Univrsity of Tchnology Postbus GA Dlft Th Nthrlands Tl.: Fax: mail: P.VanMighm@its.tudlft.nl a Part of this work was don whil at Alcatl Corporat Rsarch Cntr in Antwrp.

2 1. HIERARCHICAL STRUCTURING In today s ntworks, th numbr of nods (switchs, routrs, trminals,...) is growing fast. Just as in dictionaris, tlphon books and larg data systms, also in larg ntworks hirarchical structuring provs fficint and highly dsirabl. Th ATM-forum has incorporatd this principl of hirarchical structuring in its PNNI (privat ntwork-ntwork intrfac) spcification. In graph thory, thr xists svral standard rprsntations of a ntwork topology (Cormn t al., 90), such as a topology matrix, an adjacncy list, a link stat tabl, tc... Roughly, w may considr a hirarchical structur of a ntwork as a pr-procssing of a standard topology rprsntation to nhanc routing. Th hirarchical structuring of vry larg ntworks consists of (a) a partitioning of th ntwork in smallr substs of nods (Van Mighm, 98a) and (b) a rprsntation of this partitioning in an fficint, layrd form, calld a hirarchy. Th partitioning of th ntwork is, in fact, a rcursiv procss bcaus substs of nods may in turn again b groupd into an vn smallr numbr of substs. Th rcursiv procss nds aftr N rcursions whn on st contains all undrlying substs. This is th highst lvl N subnt (or pr group). Hnc, th rcursion crats a hirarchical tr in which child nods rprsnt th ntwork in mor dtail than thir parnt nods. Each rcursion rflcts a diffrnt hirarchical lvl. An intrsting proprty of this rcursiv partitioning procss is, that th union of all substs of hirarchical lvl k again rprsnts th original ntwork, howvr, with diffrnt granularity. Th lowst lvl - also calld th physical lvl and furthr rfrrd to as th k = 0 hirarchical lvl -posssss th finst granularity sinc that lvl consists of substs with th original nods as lmnts. Nods ar assumd to b indivisibl into smallr parts. Th subsqunt lvls hav substs as lmnts. Th whol tr comprising all hirarchis is calld th hirarchical structur of th ntwork as shown in Figur 1. Logical Link A B C Highst-lvl Pr Group A.1 PG(A) A.2 A.3 B.1 PG(B) B.2 B.3 B.4 C.1 PG(C) C.2 A.1.2 A.1.1 A.1.3 A.1.4 Figur 1 A hirarchical structur basd on th spcifications of PNNI whr a subnt is coind a pr group, dnotd as PG(.). An attractiv fatur of hirarchical structurs is th fficint rprsntation. Sinc ach hirarchical lvl actually rprsnts th ntwork, a full dtaild dscription of ach lvl lads to a N-fold rdundancy. Thus, th hirarchical structur naturally calls for (topology) information condnsation. It maks sns to rprsnt a subst on th nxt highr lvl in som condnsd form, which is calld a complx nod or an aggrgatd nod. Actually, hirarchical structuring is basd on a gnral principl stating that th immdiat nighborhood is dsird to b known in gratr dtail than th farthr surroundings. For instanc, it sms rasonabl that a citizn of Antwrp roughly knows th strt map of Antwrp and vn that of Brussls, but it sounds odd to assum that h also knows th strt map of Tokyo qually dtaild.

3 Prhaps th most important bnfit of a hirarchical structuring of a ntwork is scalability lading to a dcrasd routing complxity (Van Mighm, 97), smallr routing tabls (Klinrock and Kamoun, 77; Plg and Upfal, 88) and an nhancd human insight in th ntwork. Howvr, th information condnsation can also rsult in routing solutions that ar lss optimal. In th litratur, w found som articls on hirarchical aggrgation and dcomposition (s.g. rfrncs in Huang and Zhu, 96), but ths wr limitd to a routing algorithm that optimizs on QoS masur. Th majority of articls daling with hirarchical structurs confin thmslvs to th proposal and valuation of hirarchical routing algorithms ddicatd to spcial ntwork rquirmnts. Montgomry and d Vciana (98) prsnt a hirarchical sourc routing algorithm basd on th implid cost of a connction which masurs th xpctd incras in futur blocking that would occur from accpting this connction. Van Mighm (98) shows that a top-down stratgy for routing in a hirarchy as shown in Figur 1 offrs clar advantags ovr bottom-up stratgis. Dirctly rlatd to Intrnt scalability problms but ignoring QoS issus, Bhrns and Garcia-Luna-Acvs (98) xplain a hirarchical routing algorithm, calld ara-basd link-vctor algorithm (ALVA), which uss link-stat information to comput optimal paths but without rplicating th complt topology information at vry nod. Hr, w concntrat on topology information condnsation, in particular, on th qustion how w can rprsnt a subst (dtrmind abov) as aggrgatd nod on a nxt highr lvl (as shown in Figur 2). For an altrnativ point of viw, w rfr to W. L (95a). W considr th information condnsation procss as consisting of two stps: first nod aggrgation, followd by link aggrgation. Th final rsult aftr link aggrgation is advrtisd. Th discussion, although inspird by PNNI, is vry gnral including th full impact of quality of srvic (QoS), and hnc it is valid for gnral hirarchical ntworks (.g. a futur Intrnt Hirarchical protocol, say H-OSPF). 2. NODE AGGREGATION. B A A B C C Figur 2 Th original ntwork (lft) and th rduction of th ntwork to a complx nod (right). Th nods with connctivity outsid th ntwork (.g. thos with links lading to A, B, C) ar calld ingrss or grss nods. Nod aggrgation, as illustratd in Figur 2, is concrnd with th complx nod rprsntation of th original ntwork. Idally, whn making abstraction of th dtails of th original ntwork (nclosd by th dottd lin in Figur 2), its rplacmnt by a complx nod should b transparnt for th narst nighbor nods (A, B, C in Figur 2). Just as in systms thory, th ntwork to b condnsd is rgardd as a black box and only via input/output rlations (transfr functions) charactristics of this ntwork ar known. In information ntworks, th input/output rlations ar spcifid via QoS masurs (such as dlay, availabl bandwidth, link usag cost, packt loss, ) btwn ingrss and grss. W can catgoriz th QoS masurs into thr important QoS classs: a class consisting of only additiv QoS masurs, of only min(max) QoS constraints or of a combination of both additiv and min(max) QoS masurs. W assum that th all ingrss/grss QoS masur(s) can b computd via a routing algorithm. This sttls th givn data. Th scond stag handls th rprsntation or nodal structur of th complx nod. Thr xists two xtrm situations: th full msh rprsntation containing all pairwis QoS masurs btwn ingrss/grsss and th symmtric-point cas whr th ntir ntwork is collapsd into on point and whr mrly an all-in-on charactrizing paramtr (known as th diamtr) is advrtisd (L, 95). Th lattr cas also rflcts th ultimat information condnsation possibl. In btwn ths xtrms lis th nuclus-spok rprsntation as

4 adoptd in PNNI and illustratd in Figur 3. Othr rprsntations (such as.g. mor than on nuclus b ) sm to complicat mor than to nhanc th fficincy of nod aggrgation. A B PNNI A x y z B C C Figur 3. Th complx nod structur in PNNI. Th vrtics x, y, z conncting a fictitious nuclus and th ingrss nods ar calld spoks (dottd lins) whil an xcptional link (full lin) is a dirct link btwn an ingrss/grss pair. PNNI assums that th pr groups can b rprsntd by a whl with a nuclus and svral spoks. Th concpt of a nuclus is attractiv, bcaus it rprsnts in som way th point of gravity of th undrlying pr group. Hnc, th concpt assums a rasonabl amount of symmtry in th sns that ach nod of th undrlying pr group lis at approximatly th sam QoS-masur-distanc from that nuclus. In cas th path nds in a complx nod, th nuclus is th bst rprsntativ for ach nod in th undrlying pr group. This proprty maks th concpt attractiv. Ntwork asymmtry is dalt with via xcptional links (s Figur 3 and th discussion blow). In spit of th attraction of th nuclus-spok concpt, thr ar also limitations. Considr as an xampl th routing from a physical nod B.4.x to anothr on B.1.y in Figur 1. Using th top-down lvl routing principl (Van Mighm, 98), a path is computd in pr group PG(B) from th nuclus of B.4 to th nuclus of B.1. Lt us assum that th path runs ovr th complx nod B.2. Aftr routing at this lvl, th grss port at B.4 and th ingrss port at B.1 ar known. From ths ports, a path is computd on th nxt lowr lvl (in this cas th physical lvl) to B.4.x and B.1.y rspctivly. In this way, th complt, hirarchical path from B.4.x to B.1.y is known. Sinc th nuclus is th bst rprsntativ for th ntir pr group, it possibl that th nuclus of B.4, for instanc, is not th bst rprsntativ for th particular nod B.4.x such that a bttr path to B.1.y may xist (which travrss.g. B.3 instad of B.2). This phnomnon is, in gnral, charactristic for any information condnsation procss. If small losss or inaccuracis ar not tolrabl, condnsation may not b possibl. Th ovrall objctiv is to achiv maximal information condnsation, dfind as minimizing th numbr of xcptional links in th nuclus-spok rprsntation whil simultanously approximating ach ingrss/grss QoS masur to within a givn accuracy. Th optimum is rachd whn only spoks ar ndd in th complx nod Figur 3 and th ultimat limit, th symmtric-point condnsation, can b achivd for th particular ntwork whos ingrss/grss quations ar th sam for ach pair. 2.1 ADDITIVE QOS MEASURES. An additiv QoS masur has th proprty that th valu along a path is th sum of th valus of th dgs along that path. Typical xampls of additiv QoS masurs ar th dlay, th hop count, cost, Singl QoS masur. W assum that w dispos of th bst QoS valu for paths AB, AC, BC. Ths ingrss/grss valus QoS AB, QoS AC, QoS BC ar computd from th original ntwork (Figur 2) via som shortst path algorithm and ar all non-ngativ. According to th nuclus-spok structur (Figur 3), ths ingrss/grss valus QoS AB, QoS AC, QoS BC must b mappd into th spok valus QoS x, QoS y, QoS z. W hav 3 unknown and prcisly 3 quations, QoS x + QoS y = QoS AB b Placing additional nucli is analogous to using Stinr points in a multicast tr to obtain a bttr ovrall tr

5 QoS x + QoS z = QoS AC QoS y + QoS z = QoS BC Th solution of this linar st is straightforward and th xact solution is found in th appndix. Howvr, in gnral w hav m vrtics conncting th nighbors (via symmtrical links). Ths m vrtics corrspond to m(m-1)/2 ingrss-grss pairs (all combinations of 2 out of m). Only in th cas m = 3, it happns that m(m-1)/2 = 3. Thus, in gnral, w hav an ovr-dtrmind st of m(m-1)/2 quations in only m unknowns. Although an xact solution clarly dos not xist, w can always find th bst c possibl st of m unknowns subjct to th m(m-1)/2 linar quations. Th linar st has an intrsting structur, QoS 1 QoS QoS 2 QoS QoS 3 QoS x QoS m = QoS 1m (1) QoS QoS QoS m-1,m W furthr dnot this st by M m(m-1)/2 x m Q m x 1 = F m(m-1)/2 x 1 whr th componnt F ij rfrs to th bst ingrss/grss valu corrsponding to th bst QoS path from ingrss i to grss j. On ach row thr ar prcisly two non-zro lmnts and ths non-zro lmnts ar qual to on. Gomtrically, ach quation rprsnts a hyprplan in th m-dimnsional spac orthogonal to th bisctric-vctor d of two axs i and j and intrscting thos axs at a distanc QoS ij from th origin. Th wll spcifid form of th matrix M ncourags a furthr analysis. Th last-squar solution of M m(m-1)/2 x m Q m x 1 = F m(m-1)/2 x 1 follows (Lanczos, 88) by multiplying both sids with th transpos of M, rsulting in M T M Q = M T F, such that Q = (M T M) -1 M T F (2) whr w comput th invrs of a symmtrical matrix. Sinc th matrix M is known, th right hand sid can b pr-calculatd as dmonstratd in th appndix. So far, w ar in th position to dtrmin a bst possibl solution (QoS 1, QoS 2,..., QoS m ) that charactrizs th spok valus for a crtain QoS. Howvr, a valid solution QoS i must b non-ngativ to mak any sns. Gomtrically, this mans that th bst solution must hav all its coordinats non-ngativ. Whn rasoning in thr dimnsions (s Figur 4), w obsrv that such a point can b found in cas th distancs of th plans from th origin ar not too far apart (i.. whn th QoS ij ar comparabl in magnitud). If th distancs broadly vary, th point of intrsction is likly not to li in th positiv octant. In highr dimnsions m > 3, th probability that th point of intrsction lis in th only positiv subspac dcrass sharply, bcaus thr is only 1 such a positiv subspac out of 2 m similar subspacs! Hnc, w s that th non-ngativ condition of QoS i puts additional constraints to th problm. Som paths may b significantly bttr in QoS prformanc than othrs. Th m unknowns ar sriously affctd by th xcptional QoS valus of a particular path, vn to th xtnd that a valid bst fitting solution dos not xist bcaus th non-ngativity condition is violatd. In ordr to circumvnt this problm, an xcptional link btwn a particular ingrss-grss is introducd. This link valu is immdiatly advrtisd without affcting th dtrmination of th othr spok valus ((QoS x, QoS y, QoS z ) in Figur 3 and (QoS 1, QoS 2,..., QoS m ) in gnral). Thus, w hav mans to always find a good rprsntation by adding nough xcptional links. Clarly, th maximum numbr of xcptional links for on QoS masur quals m(m-1)/2, a situation oftn c Bst possibl is to b undrstood in th last squars sns. This stms from th minimization of th squar of th rror r. Spcifically, r 2 = M.Q - F whr. dnots th Frobnius norm, which can also b writtn as r 2 = (M.Q - F) T (M.Q - F) and furthr as r 2 = Q T M T MQ - 2 F T M.Q + F T F. Hnc, minimizing this quadratic form via formal drivation dr 2 /dq = 0 yilds M T MQ = M T F that is quivalnt to (2). Th gnral thory rlis on singular valu dcomposition (SVD) for which w rfr to th book of Golub and Van Loan (89). d This is th vctor at an angl of 45 o lying in th plan formd by two axs. Mathmatically, of cours, w can dal with ngativ valus. Howvr, if w rprsnt th undrlying ntwork by a particular hirarchical lvl, th complx nods on this lvl should rflct physical QoS masurs.

6 coind as a full msh solution. In this xtrm situation, thr is no fictitious nuclus and information condnsation has faild bcaus th complt st of QoS valus btwn ingrss/grss pairs is unaltrd advrtisd. z QoS AC QoS BC QoS BC QoS AB y QoS AC 1 QoS AB 2 x Figur 4 Gomtry in thr dimnsions of intrscting bisctric-plans. In thr dimnsions, w hav a uniqu solution which is sn as th intrsction point of th thr dottd straight lins. Two situations ar drawn for two valus of QoS BC lading to th solutions 1 and 2. W obsrv that solution 1 has all thr componnts positiv whil solution 2 has a ngativ z-componnt which is attributd to th fact that, in cas 2, QoS BC is much smallr than th QoS AB and QoS AC. A last point concrns th accuracy of th rsult. Th non-ngativity condition is ncssary for th physical intrprtation, but not sufficint to guarant a crtain lvl of accuracy. Th accuracy is takn into account by th maximum rlativ rror, dnotd by ε Stratgy : Optimizing condnsation and accuracy simultanously. Th analysis abov naturally lads to a stratgy that dtrmins whn to advrtis xcptional links: 1. Dtrmin Q m x 1 from quation (2). 2. if ( Σ 1 j m M i j.qos j - F i ε F i ) for ach i, go to 3 ls go to 4 3. if Q m x 1 = (QoS 1, QoS 2,..., QoS m ) obys QoS i 0 for ach i, w ar don ls go to 4 4. Sarch for th minimum componnt in F m(m-1)/2 x 1. Advrtis th path and QoS valu that minimizs this componnt in F as an xcptional link. Omit th corrsponding linar quation and go to 1. In ach condnsation cycl (stp 2 to stp 4), th accuracy of th solution Q m x 1 is chckd (in stp 2): th squar rror pr componnt (i.. ingrss/grss pair) must li within a givn rlativ rror ε. Stp 3 vrifis th

7 non-ngativity of th rsult and stp 4 slcts an xcptional link basd on gomtrical considrations as outlind abov. If ε = 0, thr will b no information condnsation whil for ε, thr is maximal condnsation, but no accuracy control. Obviously, th valu of ε is critical. Th worst cas complxity of th proposd stratgy occurs whn ε = 0, and it is dtrmind according to th notion xplaind in th appndix, C worst = Σ 0 j < m(m-1)/2 [C 1 (j) + C 2 (j) + C 3 + C 4 (j)] whr th worst cas complxitis of ach stp in th stratgy ar C 1 (j) = O((m(m-1)/2-j) 3 ) which ignors th spcial structur of M, C 2 (j) = O(m 2 (m-1)/2-mj)), C 3 = O(m) and C 4 (j) = O(m(m-1)/2 j). Hnc, C worst = O(Σ 0 j < m(m-1)/2 ((m(m-1)/2-j) 3 ] = O(m 8 /64) This worst cas complxity should b confrontd with th bst cas (only stp 1, 2 and 3 onc) complxity C bst = O(m 3 /2) as drivd in th appndix. It is instructiv to rlat our stratgy to th concpt of a t-spannr of a graph G which has bn broadly studid in graph thory (Plg and Shäffr, 89; Althöfr t al., 93). A t-spannr of a graph G(V,E) is a subgraph G (V,E ) of G whr for all nods u,v {V} holds that th distanc from u to v in G is at most t tims longr than th distanc in G. Althöfr t al. (93) hav proposd a simpl algorithm to comput a t-spannr, strongly basd on Kruskal s minimum spanning tr algorithm. In addition, thy dmonstrat that for an undirctd graph, thr xists a polynomially constructabl (2t+1)-spannr such that th numbr of links E < V.[V 1/t ] (whr [x] dnots th intgr just xcding or qual to x) and th sum of all th dg wights of G is smallr than th (1+V/(2t)) tims th wight of th minimum spanning tr of G. Roughly, th maximum rlativ rror ε plays a rol analogous to t. Th diffrnc clarly is that, in our proposal, th rsulting complx nod (nuclus-spok structur with xcptional links) is not a subgraph of th original graph so that thr is no confinmnt to minimum spanning tr tchniqus. Thrfor, it is xpctd that, for a sam accuracy, a closr agrmnt to th original graph can b obtaind with a smallr numbr of links. Finally, from xprincs via simulation, w found that this stratgy constituts a vry robust tchniqu that prfctly dals with possibly rdundant information (.g. triangl (in)qualitis such as QoS AB +QoS Bc = (<) QoS Ac ) Multipl QoS masurs. Suppos now ach link is charactrizd by n additiv masurs and assum that for all ingrss i -grss j pairs, a bst rprsntativ QoS vctor, QoS ij = [QoS ij1, QoS ij2,..., QoS ijn ] is availabl. How this bst QoS vctor is computd is hr rgardd as byond th scop, but w rfr to our QoS routing algorithm, TAMCRA, a Tunabl Accuracy Multipl Constraints Routing Algorithm (D Nv and Van Mighm, 98, 98a). W furthr considr vry componnt as qually important. Thn, all prvious unknowns bcom unknown n x 1 vctors. Rasoning on th xampl drawn in Figur 3, this mans that QoS x transforms to QoS x = [QoS x1, QoS x2,..., QoS xn ] and likwis all othr paramtrs. Thus, w hav, instad of 3 unknown now 3 n unknowns. Th linar st (1) thn gnralizs to I nxn I nxn O nxn O nxn O nxn... O nxn QoS 1 QoS 12 I nxn O nxn I nxn O nxn O nxn... O nxn QoS 2 QoS 13 I nxn O nxn O nxn I nxn O nxn... O nxn QoS 3 QoS I nxn O nxn O nxn O nxn O nxn... I nxn x QoS m = QoS 1m (3) O nxn I nxn I nxn O nxn O nxn... O nxn QoS 23 O nxn I nxn O nxn I nxn O nxn... O nxn QoS O nxn.. O nxn I nxn I nxn QoS m-1,m whr I nxn and O nxn ar th n x n idntity and zro matrix, rspctivly. Formally, th structur of (1) is maintaind if th lmnts ar substitutd by th corrsponding block matrics. In fact, using th Kronckr product, th block matrix in (3) can b writtn as M I nxn. This formal rsmblanc is blivd to b advantagous bcaus fficint matrix manipulations analogous to thos prsntd in th appndix sm possibl.

8 Th choic of xcptional links is inspird by th gomtrical considrations of th on-dimnsional cas abov. Spcifically, whn th QoS i vctor has ngativ componnts, it triggrs th advrtismnt of an xcptional link. Now, w slct that xcptional link that gomtrically lis farst from (or narst to) th origin bcaus, by rmoving th corrsponding block row, th rmaining st of block rows will only b bttr approximatd (in last squar sns) if th most xtrm-valud block row disappars. To comput th distanc, various distanc mtrics ar possibl, for instanc, th Holdr norm QoS i p = Σ 1 j n QoS ij p (whr p is ral and p 1) which rducs in th cas p = 2 to th simpl, wll-known Euclidian distanc masur, whil for p, th maximum componnt quals th norm. Th chosn norm should agr with th on usd in th QoS routing algorithm,.g. TAMCRA is basd on th p norm. In th squl, w confin ourslvs to th simplr on-dimnsional cas, bcaus w hav shown hr how to xtnd to multipl, indpndnt, additiv QoS masurs. 2.2 MIN (MAX) FUNCTION OF QOS MEASURES ALONG A PATH. A min(max) QoS masur has th proprty that th valu along a crtain path in a topology consists of th minimum (maximum) of th valus of th dgs that constitut that path. Typical xampls of min (max) QoS masurs ar th bandwidth, (policy rlatd) transit flags,... Th boolan ANDing of QoS masurs that ar boolan numbrs (ithr 0 or 1) is a subclass of min(max) QoS masurs. Again starting from th simpl xampl in Figur 3, w now hav th following non-linar quations in th 3 unknown, min(qos x,qos y ) = QoS AB min(qos x,qos z ) = QoS AC min(qos y,qos z ) = QoS BC or, quivalntly, QoS x QoS AB and QoS y = QoS AB or QoS y QoS AB and QoS x = QoS AB QoS x QoS AC and QoS z = QoS AC or QoS z QoS AC and QoS x = QoS AC QoS y QoS BC and QoS z = QoS BC or QoS z QoS BC and QoS y = QoS BC Without loss of gnrality, w assum QoS AB QoS AC QoS BC for othrwis, w r-labl x, y, z in Figur 3. In cas QoS AB < QoS AC < QoS BC, w nd up with conflicting rquirmnts. To s this, considr first th inqualitis. Th rsult obying QoS x QoS AB and QoS x QoS AC is QoS x QoS AC and similarly, QoS y QoS BC and QoS z QoS BC. But, th quations on th first lin of th quivalnt st rquir that at last on of th valus (QoS x,qos y ) achivs th minimum valu QoS AB. Thr ar solutions in dgnrat cass whr two of th right hand sid valus ar idntical. Lt QoS AB = QoS AC < QoS BC. Thn, th solutions ar QoS x = QoS AB and ithr QoS y = QoS BC and QoS z QoS BC or QoS z = QoS BC and QoS y QoS BC. In cas QoS AB = QoS AC = QoS BC, w radily vrify that QoS x = QoS y = QoS z. This situation corrsponds with a prfct symmtrical cas. Howvr, in cas QoS AB < QoS AC = QoS BC, thr is again no solution. Curiously, all anomalis will not occur in our problm, bcaus w hav mor information in that th st QoS AB, QoS AC and QoS BC ar not arbitrarily non-ngativ numbrs, but, in fact, corrlatd. Indd,.g. whn th QoS masur is maximum bandwidth, w hav in addition that min(qos Ac, QoS BC ) QoS AB min(qos AB, QoS BC ) QoS AC min(qos AB, QoS AC ) QoS BC which implis that at last two ingrss-grss Qos valus must b qual. Just ths cass ar shown to hav a solution without xcptional links. Bfor procding with th gnral cas whr m > 3, w will invok rsults obtaind from L s (95) spanning tr mthod. If w rlax th complx nod rprsntation and only ask for a rduction in th m(m-1)/2 ingrss-grss rlations without dmanding a nuclus-spok structur, an optimal condnsation is possibl using proprtis of a minimum spanning tr (Cormn t al., 90). L (95) has dmonstratd that min(max) QoS masurs - h concntrats on maximum bandwidth - can b xactly condnsd into m-1 rlation, just th numbr of links in a spanning tr. Th complxity to comput an optimal spanning tr is O(m log m).

9 A maximum wight spanning tr f has th valuabl proprty that th wight of th link conncting a pair of nods must b boundd from abov by th minimum wight among th links along th uniqu path conncting th nods on th spanning tr. Indd, othrwis, on can incras th total wight of th spanning tr by substituting this link for th minimum wight along th uniqu path. Furthr, L (95) obsrvs that th bandwidth of th uniqu path btwn any pair of nods on a spanning tr cannot b largr than th bandwidth of th original ingrss-grss valu of that sam pair of nods. Combining both bounds, it follows that th fullmsh basd on bandwidth btwn ingrss-grss pairs can b xactly ncodd by a maximum wight spanning tr. In othr words, th original ingrss-grss bandwidth quals th minimum of th bandwidths of th uniqu path along th spanning tr conncting ths particular ingrss and grss. A A y x x C y B C y z B x > y minimum spanning tr structur z x > y nuclus-spok structur Figur 5. Transformation btwn minimum spanning tr structur and a nuclus-spok structur. Concntrating again on th gnral cas analogous to (1) whr m > 3, L s rsults implis that at most m-1 of th m(m-1)/2 right hand sids min(max) QoS valus ar diffrnt. It is doubtful whthr thr actually xists a pur nuclus-spok structur without xcptional links. For, th minimum oprator rquirs that, for all 1 j m QoS j = max({qos kl }) whr ithr k = j or l = j Hnc, all quations with a right hand sid valu smallr than min j (QoS j ) (with 1 j m) cannot b obyd. W may wish thn to advrtis all ths quations as xcptional links. Th difficulty lis in th dtrmination of that numbr of xcptional links. Rcall that th ovrall stratgy is to condns information as much as possibl. Apart from this, th rmaining quations ar not guarantd to b consistnt. In conclusion, th attmpt to forc an information condnsation b conform to a complx nod rprsntation with d facto spoks, dos not sm to b a satisfying stratgy. Hnc, th nuclus-spok structur is th prfrabl candidat for additiv QoS masurs, whil th spanning tr structur is mor suitd for a singl min(max) QoS masur. Howvr, th xtnsion to multipl min(max)-qos masurs rquirs a spanning tr algorithm for multipl min(max) QoS masurs which is vry likly to b NP-complt. Morovr, th basic proprty of a maximum wight spanning tr, drivd in th singl min(max)-qos cas, dos not xtnd to multipl min(max)-qos bcaus it is unclar whthr thr always xist a spanning tr that is a good rflction of th full msh btwn ingrss-grss pairs. 2.3 COMBINATION OF ADDITIVE AND MIN (MAX) QOS MEASURES ALONG A PATH. Th minimum spanning tr mthod of L (95) only yilds approximat rsults for th additiv QoS masur, though an uppr bound on th rror can b drivd. Howvr, L did not suggst ways to adjust or control this uppr bound on th rror. Morovr, in th computation, th additiv QoS masur (dlay) is dominatd by th min(max) QoS masur (bandwidth). In cas bandwidth and dlay ar tratd qually fair, th attractiv, xact proprty of a maximum wight spanning tr dos not hold any longr. Indd, sinc dlay is f A maximum wight spanning tr of a topology is a spanning tr that maximizs th total link wight.

10 snsitiv to th lngth of a path (bcaus it is an additiv masur) whras maximum bandwidth is not, thr may xist paths with largr bandwidth btwn ingrss and grss if th dlay componnt is ignord as illustratd in Figur A 10 7 B D 12 6 C 7 8 F G E 6 H 10 Figur 6. Th link mtrics is maximum bandwidth. Th bst maximum bandwidth path from nod A to nod D, dnotd as P = A-B-F-G-H-D, is shown in bold and can carry 8 bandwidth units. If, in addition, ach link has a unit dlay componnt, th dlay along that path P quals 5 dlay unit. In cas bandwidth and dlay ar tratd qually fair, th path A-D with (7,1) can b prfrabl ovr path P with (8,5) dpnding on th comparison critrion. Th combination of additiv and min(max) QoS masurs assums a critrion that allows to compar two (maximum bandwidth, dlay) vctors,.g. a convx functional F ij = w B B ij -1 + w d d ij whr w B and w d ar som positiv (givn) wights rflcting a cost and B ij = min p (B p ) whil d ij = Σ p B p and p dnots a path btwn i and j. This critrion should b th sam as th on usd to comput th bst vctor valus btwn all in- and grss pairs. An xtnsion of our stratgy amounts in solving a nonlinar st of quations in unknowns for th spok vctors subjct to non-ngativity rquirmnt for componnts. Again, applid to th simpl xampl in Figur 3 and using th convx functional suggstd abov, th non linar st bcoms w B min -1 (B x,b y ) + w d (d x + d y ) = F AB w B min -1 (B x,b z ) + w d (d x + d z ) = F AC w B min -1 (B y,b z ) + w d (d y + d z ) = F BC Ths sttings ar asily gnralizd to th cas m>3. Th complication clarly lis in finding th solution of such a non-linar st of quations with boundary conditions. In addition, th choic of th convx functional (a cost function) will always b dbatabl. In conclusion, optimal information condnsation of a combind st of min(max) and additiv QoS masurs is much mor complicatd than trating homognous sts and, in fact, still an opn problm.

11 A B b d a c D C B A X x link y Y D C Figur 7 Link aggrgation: th dashd links, a,b,c,d,, btwn two pr groups ar rprsntd on th nxt hirarchical lvl by th bold aggrgatd link. 3. LINK AGGREGATION In Figur 2, th rlvant bordr nods (ingrsss or grsss) ar visualizd by links lading towards A, B and C. For simplicity, ths links wr thought of as singl, not-aggrgatd links. Th gnral problm of link aggrgation is shown in Figur 7. W confin ourslvs to additiv QoS masurs. W propos a nod aggrgation first on th two subnts (pr groups) via our stratgy (sc. 2.12). Clarly (s Figur 8), th ingrss-grss points of th pr groups ar charactrizd via th link-sts {A,B,a,b,c,d,} and {C,D,a,b,c,d,}, rspctivly. B A X x 1 x 2 x 5 x 4 x 3 a b c d y 3 y 4 y2 y 1 y 5 x Y C D Figur 8 Link aggrgation aftr nod aggrgation Figur 8 assums that th complt nod aggrgation procdur has bn prformd and that th spok valus x i and y i (and possibly th xcptional links that ar also drawn in Figur 8) for ach QoS masur ar computd. Th QoS valus of th links a, b, c, d, ar known. From all possibl paths btwn th nucli of pr group X and Y, w choos th bst QoS masur valu and advrtis this valu in th aggrgat pictur (undrnath in Figur 7). Thus, in cas th QoS masur is additiv, w advrtis on th aggrgatd link that st {QoS x, (xcptional link) x, QoS link, (xcptional link) y, QoS y } that minimizs th sum of th QoS valus (possibly not all 5 path sgmnts ar ndd). Th xcptional links ar important bcaus thy possss th bst QoS masur valu. Hnc, although th path from nuclus X to nuclus Y using xcptional links can pass along mor path sgmnts, th QoS masur along that path may b th bst on.

12 3.1 DISCUSSION AND ALTERNATIVES. Bcaus first nod aggrgation is prformd, this stratgy for link aggrgation is fairly computationally intnsiv. In th nd, possibl xcptional links may hav disappard from th pictur. Nvrthlss, th bst (final) valu for a crtain QoS, {QoS x, (xcptional link) x, QoS link, (xcptional link) y, QoS y } may crucially dpnd on thos xcptional link valus. From th point of viw of information condnsation, link aggrgation as prsntd hr, contributs most significantly. On may wondr if th final rsults obtaind by first aggrgating th links (or what is mor important th numbr of ingrss-grss combinations) ar still comparabl, although obtaind with considrably lss ffort. Judgd a priori, w ar in doubt about th quality of th rsults bcaus it is far from obvious how to aggrgat th links first. For instanc, it is not difficult to condns th links a, b, c, d, in Figur 7 to on with th bst QoS valu of th fiv original ons, but how do w connct th rsulting condnsd link to pr group nods in X and Y? Th procdur of link aggrgation conncts complx nods only by on (th bst ) logical link. Apart from simplicity, th advantag is a highr dgr of uniformity ovr th hirarchical lvls (at th lowst lvl vry nod is but connctd by on link) rsulting in th us of a sam routing algorithm for th whol hirarchical structur. On th othr hand, th drawback lis in th fact that this on link (which also is a physical link) is th only on usd, vn if thr ar mor links possibl to pass from on pr group to anothr. This implis that ths bst links may rapidly b outdatd and no longr th bst QoS masur links. Hnc, rgular updats of th hirarchy (at last th link aggrgation) sm rquird. An altrnativ consists of advrtising a numbr of k bst QoS masur paths. This will rsult in lss condnsation, mor complicatd routing, but, a lss rapid nd to updating th hirarchy. Still anothr ida is to just omit th port numbrs and to connct th nucli of both pr groups by advrtising th bst valu; on a lowr layr, th routing may find out th bst port and th corrsponding spcific link. Th rol of th hirarchy is thn to announc ovr which pr groups a path must b followd (not which grss or port to us). 3.2 ADDITIONAL REMARKS. Although th prsntd procdur is static, a dynamic xtnsion may consist of a rcomputation of (parts of) th hirarchy whr significant changs hav causd nw flooding of topology stat lmnts. W dfin an ssntial in- or grss nod on th physical lvl k = 0 as a nod that has connctivity to othr ntworks not blonging to th larg original ntwork. An important proprty of ssntial in()grsss is that thy do not disappar on som lvl of th hirarchy du to condnsation (s Figur 9, whr nod a and o ar ssntial grsss). Hnc, on ach lvl th ssntial in()grss appars as a portnod in som complx nod. Ths ssntial in()grss may b usd by ntwork managmnt to achiv a mor optimal hirarchical structur, as ntry points for masurmnts and tsts in th hirarchy or as attachmnt points for mobil ntworks (Dykman t al., 97).

13 4. EXAMPLES FOR ADDITIVE QOS MEASURES. 4.1 INFORMATION CONDENSATION OVER ONE HIERARCHICAL LEVEL. To illustrat th proposd stratgy for nod and link aggrgation, w hav prformd th computations for th additiv QoS masur maxctd or simply maximum dlay. Th ntwork, randomly gnratd, is drawn in Figur 9. a 0 8 b 8 c d g 77 f p k h j i l m n o Figur 9 All links ar bi-dirctional. Th arrows indicat th shortst path from nod a to any othr nod in th ntwork. Th QoS-distanc (hr th maximum dlay) from any nod to nod a is th numbr insid th circl. Th bold, dashd lin shows two partitions of th ntwork. Only nod a and o ar in-grsss of th orginal ntwork Th shortst dlay path from nod a to nod o is {a,c,d,,p,l,m,n,o} and quals 175 units. W now prsnt a nod aggrgation of th two parts of this ntwork sparatd by th bold dashd lin in Figur 9, and thraftr, a link aggrgation. Th rsult of this information condnsation is compard with th valu of th bst dlay path, namly 175 units Nod Aggrgation without accuracy (ε ). Th nod aggrgation of th whol ntwork is shown in Figur 10. In both th subntwork at lft hand sid, dnotd by L, and th subntwork at th right hand sid, dnotd by R, w hav to dtrmin th unknowns {x 1, x 2, x 3, x 4 } and {y 1, y 2, y 3, y 4, y 5 }, possibly augmntd with xcptional links (not yt drawn in Figur 10).

14 a L x 1 h 72 i x 2 y 2 j y x3 g 3 29 x k 4 y 5 p x y 4 y 1 R o Figur 10 Th aggrgatd nod rprsntation of th ntwork drawn in Figur 9. For th L subntwork, w hav to solv rror x x x x 4 = W invok quation (2) to find {x 1, x 2, x 3, x 4 } = {54.66, 11.16, 26.16,.66}. Sinc all componnts ar positiv, thr is no nd to introduc an xcptional link. Notic that th sum of th rrors (~man rror) is almost zro (as xpctd sinc th ovrall squar rror was minimizd) and that th distribution of th rrors around th man is symmtric. Similarly, for th R subntwork, th st linar quations ar y y y y 4 = y and th corrsponding bst fitting solution {y 1, y 2, y 3, y 4, y 5 } = {70.83,.5, 10.5, 12.83, 16.83}. Again, all componnts ar positiv and xcptional links ar not ndd. Th rason is that th dlay valus in th ntwork ar mor or lss comparabl. In ordr to giv a fling for th quality of th nod aggrgation, w may simply fill in th rsults in th ovrdtrmind linar st and compar th rsulting valus with th right hand sid. Ths valus ar givn in th last column, sparatd by a short lin. Clarly, information condnsation (without accuracy chck as in stp 2) givs ris to inaccuracis. LINK Aggrgation. Sinc thr ar no xcptional links, th link aggrgation is a quit asy procss. W mrly choos and advrtis thos valus that minimiz th dlay path btwn th two nucli. From Figur 10, w hav that x y 2 = = x y 3 = = x y 4 = = x y 4 = = x y 5 = = 55.49

15 Th bst path is clarly that corrsponding to th fourth quation. Whn prforming th link aggrgation, th rsulting rprsntation bcoms a k x o L R Figur 11 Th rsult of information condnsation (nod and link aggrgation) of th original ntwork drawn in Figur 9 Comparing th bst dlay, computd from th original ntwork in Figur 9, and that in th rsulting aggrgatd rprsntation, w obtain against 175, which is not too bad without accuracy chck, aftr all Information condnsation with nhancd accuracy. Using our stratgy with an accuracy ε = 10%, w obtain for th sam ntwork (Figur 9), th rsult aftr aggrgation plottd in Figur 12. On may vrify that th original masur,.g. from a to h, ar approximatd to within 10% using th bst path from a to h (taking into account also xcptional links). a h g j i x 52.5 o L 28.5 k 31.5 p R Figur 12 Th aggrgatd nod rprsntation of th ntwork drawn in Figur 9 with a 10% accuracy constraint. Th valus of th xcptional links in L ar hg = 35 and h = 22. Thos in R hav th valus ij = 21, ik = 41, jk =, jp = 32 and kp = 12. Th final pictur aftr link aggrgation is shown in Figur 13 with as bst dlay from a to o quals to xactly 175 (wll within th rquird 10%) whr as th prvious not accuracy-constraind approach finishd at a p x o L R Figur 13 Th rsult of information condnsation (nod and link aggrgation) of th original ntwork drawn in Figur 9 with a 10% accuracy constraint.

16 4.2 INFORMATION CONDENSATION OVER TWO HIERARCHICAL LEVELS. Lt us now illustrat, basd on th sam original ntwork, how nod aggrgation (with ε ) on two hirarchical lvls work. W divid th original ntwork of Figur 9 in four diffrnt pr groups (as shown in Figur 14) instad of just two. In addition, only th nods a,f,l,o ar assumd to hav links to othr ntworks. a c d g f A.2 h 72 j 21 i n A b 68 A k A.4 27 l m o p Figur 14 Th original ntwork of Figur 9 is organizd in 4 diffrnt pr groups, A.1, A.2, A.3 and A First Lvl Nod Aggrgation. c a x 1 x x 1 x 2 x 3 = A.1 x 3 Figur Nod aggrgation of pr group A.1 Whn solving th linar st g for A.1 (s Figur ), w obtain {x 1, x 2, x 3 } = {-3,23,75}. Hr, an xcptional link is ndd. W choos as xcptional link th on with lowst valu, namly ac. Th rmaining quation x 1 + x 3 = 72 and x 2 + x 3 = 98 constitut an undrdtrmind linar st. As additional quation, w choos x 1 = x 3 in ordr to stimulat symmtry. Th solution thn is {x 1, x 2, x 3 } = {36,43,36} complmntd with xcptional link x ac =. g It is intrsting to obsrv what th impact is of a small rrors. Th corrct linar st for A.1 has as right hand sid componnts {,76,96} lading to {x 1, x 2, x 3} = {0,,76}. Hnc, in th corrct cas, w do nithr hav an xcptional link, nor a sparat nuclus (bcaus nod a is nuclus).

17 f x 1 d x 4 A.2 g x 3 x 2 h x 1 x 2 x 3 x 4 = Figur 16 Nod aggrgation of pr group A.2 Th solution of th linar st for A.2 (s Figur 16) using quation (2) givs us th xact solution, {x 1, x 2, x 3, x 4 } = {0,,,}. This is not at all surprising, bcaus w hav trid to forc a solution in th form as shown in Figur 16 bcaus on th nxt hirarchical lvl, w will hav to connct nod f with anothr outsid ntwork. 0 f A.2 a A.1 36 c d 6 h g 29 k j 27 i 26.5 m o A.3 p 6 12 l A.4 Figur 17. Th rsult aftr nod aggrgation on th first hirarchical lvl Th computations for A.3 and A.4 ar similar and w mrly giv th rsults rfrring to Figur 17 for th structur of ach aggrgatd nod. For A.3, w obtain as bst fitting solution {x 1, x 2, x 3, x 4 } = {52,,26.5,22.5} and th accuracy of th solution (by substituting th rsult in th linar quation) is {72, 5/2,149/2,91/2,85/2,48} as compard to th right hand sid {74,100,50,21,65,50}. For A.4, similar to A.1, w must introduc an xcptional link and hr w hav chosn th worst valu (to illustrat th diffrnc with th proposd stratgy). Th rsult is {x 1, x 2, x 3 } = {6,6,12} complmntd with xcptional link x ki = First Lvl Link Aggrgation. Basd on Figur 17, w hav to comput th shortst dlay path from a nuclus to his nighbor nucli. This opration is quit basic. Th rsult is drawn in Figur 18.

18 0 f A.2 c 22 d h a g 29 j o A.1 k A.3 6 p 6 12 l A.4 Figur 18 Th final rsult (aftr nod & link aggrgation) on th first hirarchical lvl Scond Lvl Nod Aggrgation. Th last stp is to aggrgat th subnt consisting of th aggrgatd pr groups A.1, A.2, A.3 and A.4. Sinc w nd connction with othr ntworks via th nods a,f,i,o, th rprsntation on this scond hirarchical lvl is as drawn in Figur. f x 2 A a x 1 i x 4 x x 1 x 2 x 3 x 4 = Figur Nod aggrgation of pr group A Bfor concntrating on th solution, w would lik to point out that th goodnss of th first lvl information condnsation can b xtractd from Figur. Indd, w s that th shortst path from nod a to nod o (th rfrnc takn bfor), now quals compard to th xact valu 175. Hnc, th quality of this information condnsation is slightly lss than in prvious cas (whr ε ) with just two pr groups. Now, solving th st in Figur with (2) rsults in {x 1, x 2, x 3, x 4 } = {62.5,13,94,41} and an indication of th rror follows via substitution of this rsult in th original quation, yilding {75.5, 6.5, 103.5, 107, 54, 135}. Ths valus should b compard with th right hand sid of th quation givn in Figur. This finally nds our scond lvl nod aggrgation. Now, w obsrv that th shortst path from nod a to nod o is 6.5 to b compard to th xact valu of 175. Hnc, as xpctd, th mor information condnsation, th lowr th quality of th rsults.

19 5. CONCLUSIONS For (multipl) additiv QoS masurs, w hav prsntd how to ffctivly construct a hirarchical structur. Th procdur consists of two basic stps: nod aggrgation followd by link aggrgation. Our stratgy (sc ) is th bst way to aggrgat nodal topology information for (multipl) additiv QoS masurs subjct to a givn accuracy (th rlativ rror ε). This mans that, for additiv QoS masurs, always an optimal (in th sns that information is maximally condnsd) nod aggrgation with a guarantd accuracy can b constructd. Th procdur for link aggrgation advrtiss th on, bst link btwn two complx nods as th aggrgatd logical link, which again offrs maximal condnsation. In addition, it nhancs uniformity in th data structur of th hirarchy, which in turn simplifis routing, at th compnsation of a mor rapid updating of th hirarchy. Th accuracy paramtr ε allows fast vrification whthr a connction dmanding crtain QoS guarant, QoS*, should b rfusd (if QoS > QoS* ) or not. Indd, th routing procss can focus on that lvl of hirarchy, say k, that ncloss both sourc and dstination. If th computd path on that lvl k lads to a QoS valu xcding (1-kε) QoS*, th connction is blockd, ls, signaling is activatd with th propr pathlist h. For on min(max) QoS masur (lik bandwidth), w hav rviwd L s optimal minimum spanning tr approach. Although th mthod is uniquly suprm for on min(max) masur, it is doubtful whthr it can b gnralizd to multipl min(max) QoS masurs faturing th sam, lgant proprtis. In addition, th xistnc of good stratgis for th link aggrgation of min(max) QoS masurs is qustionabl. Finally, optimal nod and link aggrgation for combind additiv and min(max) QoS masurs, is still blivd to b an opn problm. 6. ACKNOWLEDGMENTS Discussions with various collagus, in particular Hans D Nv, ar much apprciatd. Valuabl commnts of th rviwrs ar acknowldgd. 7. REFERENCES Althöfr, I., G. Das, D. Dobkin, D. Josph and J. Soars, 93, On Spars Spannrs of Wightd Graphs, Discrt Comput. Gom., vol. 9, pp ATMF, 96, Privat Ntwork Ntwork Intrfac, spcification vrsion 1.0, March Bhrns, J. and J. J. Garcia-Luna-Acvs, 98, Hirarchical Routing Using Link Vctors, IEEE INFOCOM 98. Cormn, T. H., C. E. Lisrson and R. L. Rivst, 90, Introduction to Algorithms, Th MIT Prss, Cambridg, Massachustts. D Nv, H. and P. Van Mighm, 98, A Multipl Quality of Srvic Routing Algorithm for PNNI, Procdings of th IEEE ATM Workshop, May 26-29, Fairfax, USA, pp D Nv, H. and P. Van Mighm, 98a, TAMCRA: A Tunabl Accuracy Multipl Constraints Routing Algorithm, submittd to Journal of ACM. Dykman, D., I. Iliadis, P. Scotton, L. Frlchoux and S. Ray, 97, PNNI routing support for mobil ntworks, ATMF Golub, G. H. and C. F. Van Loan,89, Matrix Computations, North Oxford Acadmic, Oxford. Huang, G. M. and S. Zhu, 96, A Nw HAD Algorithm for Optimal Routing of Hirarchically Structurd Data Ntworks, IEEE Trans. on Paralll and Distributd Systms, vol. 7, No. 9. Klinrock, L. and F. Kamoun, 77, Hirarchical routing for larg ntworks. Prformanc valuation and optimization, Computr Ntworks 1, pp h In PNNI, this path-list is calld th dsignatd transfr list (DTL).

20 Lanczos, C.,88, Applid Analysis, Dovr Publications, Inc., London. L, W. C., 95, Spanning Tr Mthod for Link Stat Aggrgation in Larg Communication Ntworks, IEEE INFOCOM 95, pp L, W. C., 95a, Topology Aggrgation for Hirarchical Routing in ATM Ntworks, ACM Sigcomm., vol. 25, No. 2, April, pp Montgomry, M. and G. d Vciana, 98, Hirarchical Sourc Routing Through Clouds, IEEE INFOCOM 98. Plg D. and A. A. Shchäffr, 89, Graph Spannrs, Journal of Graph Thory, vol. 13, No. 1, pp Plg D. and E. Upfal, 88, A trafoff btwn spac and fficincy for routing tabls, ACM Procdings, pp Van Mighm, P., 97, Estimation of an Optimal PNNI Topology, Procdings of th IEEE ATM 97 Workshop, May 26-28, Lisboa, Portugal, pp Van Mighm, P., 98, Routing in a Hirarchical Structur, Procdings of th first IEEE Intrnational Confrnc on ATM, ICATM 98, Jun 22-24, Colmar, Franc. Van Mighm, P., 98a, Dividing a Ntwork into Pr Groups to Build a Hirarchical Structur, Procdings of th first Intrnational Workshop on th Dsign of Rliabl Communication Ntworks, May 17-, Brugg, Blgium, O APPENDIX Lmma. Lt P = (M T M) -1 M T and M m(m-1)/2 x m as dfind in (1). Thn P quals M T whr th ons in M T ar changd for 1/(m-1) and th zro s for -1/(m-1)(m-2). This gnral solution of (2), Q = P. F, is computd vry rapidly. In matrix algbra, th amount of work involvd or th complxity is usually xprssd in th numbr of multiplicativ oprations (Golub and Van Loan, 89). Hnc, th complxity to comput Q = P. F quals C(Q) = m 2 (m-1)/2 flops (floating point oprations). In this papr, w hav usd th lss prcis notion of ordr to xprss th complxity, thus, C(Q) = O(m 3 /2) for larg m. Th vrification of th lmma is as follows. Lt us first concntrat on th simplst cas m = 3. Th st (1) can b solvd xactly sinc th invrs of M xists, namly M -1 = 1/ Hnc, th solution (QoS 1, QoS 2, QoS 3 ) = 1/2 (f 1 + f 2 - f 3, f 1 - f 2 + f 3, -f 1 + f 2 + f 3 ) will only hav a ngativ componnt if on of th componnts of F is largr than th sum of th two othr componnts, thus, if f i + f j < f k for any combination of (i,j,k). Th matrix M in cas m = 5 quals and th corrsponding (M T M) -1 M T is /4 1/4 1/4 1/4-1/12-1/12-1/12-1/12-1/12-1/12 1/4-1/12-1/12-1/12 1/4 1/4 1/4-1/12-1/12-1/12-1/12 1/4-1/12-1/12 1/4-1/12-1/12 1/4 1/4-1/12-1/12-1/12 1/4-1/12-1/12 1/4-1/12 1/4-1/12 1/4-1/12-1/12-1/12 1/4-1/12-1/12 1/4-1/12 1/4 1/4 Whn m = 6, w hav

21 with corrsponding (M T M) -1 M T, 1/5 1/5 1/5 1/5 1/5-1/ -1/ -1/ -1/ -1/ -1/ -1/ -1/ -1/ -1/ 1/5-1/ -1/ -1/ -1/ 1/5 1/5 1/5 1/5-1/ -1/ -1/ -1/ -1/ -1/ -1/ 1/5-1/ -1/ -1/ 1/5-1/ -1/ -1/ 1/5 1/5 1/5-1/ -1/ -1/ -1/ -1/ 1/5-1/ -1/ -1/ 1/5-1/ -1/ 1/5-1/ -1/ 1/5 1/5-1/ -1/ -1/ -1/ 1/5-1/ -1/ -1/ 1/5-1/ -1/ 1/5-1/ 1/5-1/ 1/5-1/ -1/ -1/ -1/ 1/5-1/ -1/ -1/ 1/5-1/ -1/ 1/5-1/ 1/5 1/5 Th lattr matrix has th following structur (mor asily sn if w rplac 1/5 by 1 and -1/ by 0): W obsrv that th first row consists of (m-1) ons. Th rmaining (m-1) rows xhibit in th first (m-1) columns th I (m-1)x(m-1) idntity matrix, in th (m-2) rmaining row, in th nxt (m-2) columns, th I (m-2)x(m-2) idntity matrix and so on. Th othr lmnts on th rows ar (m-i) conscutiv on s followd by zro s. Th sum of th ons on ach row quals (m-1). This obsrvation is gnral and illustrats th lmma. Now, th solution will hav a ngativ componnt if for crtain prmutations of th st (j 1, j 2,...,j m ) holds that m 1 f i= 1 < 1 m(m 1)/ 2 f j i m 2 j i i = m (A.1) Obsrv that if all f i = f, th rlation (A.1) is nvr satisfid bcaus th lft hand sid quals (m-1)f and th right hand sid quals (m-1)f/2. Indd, in this prfct symmtrical situation, th gnral solution is prcisly (QoS 1, QoS 2,..., QoS 3 ) = f/2 (1,1,...,1). Furthrmor, if m = 3, (A.1) rducs to th wll-known triangular inquality. No simpl mthod to chck (A.1) a priori sms to xist. Thrfor, th proposd algorithm (with possibly many itration) is hard to simplify. Morovr, in cas of itrations, i.. whn xcptional links ar introducd, th rgular structur of M is dstroyd by dlting rows corrsponding to an xcptional path. Thrfor, no such simpl solution as Q = P.F can b hopd for and numrical computation will incras significantly.