PEx. R x. a node. a local routing function

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1 Memory Reurement for Routng n Dstrbuted Networks (Extended Abstract) Cyrl Gavolle y LIP, Ecole Normale Suereure de Lyon (gavolle@ens-lyon.fr) Stehane Perennes z IS, Unverste de Nce-Soha Antols (s@ess.fr) Abstract In ths aer, we deal wth the comact routng roblem on dstrbuted networks, that s mlementng routng schemes that use a mnmum memory sze on each node. We rove that for every shortest ath routng scheme, for any constant ", 0 < " <, and for every nteger d such that d "n, there exsts an n-node network of maxmum degree d that locally reures (n log d) bts of memory on (n) nodes. Ths otmal lower bound means that whatever you choose the routng scheme (nterval routng, boolean routng, rex routng, : : : ), there exsts a network on whch one can not do better than routng tables. Introducton. The Comact Routng Problem The general routng roblem n a network conssts of ndng a routng rotocol, or routng functon, such that, for any ar source-destnaton, any message from the source can be routed to the destnaton. In the comact routng roblem we are nterested to mlement rotocols that reure a low amount of hardware and easly to obtan n a VLSI ont of vew. For nstance, mlementng XY or e-cube routng s smle because routng s locally erformed by resectvely comarson and xor-ng of the local address wth the destnaton address. However, networks do not necessary have the smle structure of meshes or hyercubes. Also, networks may even have no structure at all so that a routng algorthm cannot be clearly dened on them. In these cases, routng tables are generally used: when a node receves a message, t consults ts local table to determne the outut ort of Part of ths work has been done whle the authors were vstng the Wezmann Insttute of Israel. Addtonal suort by the French- Israel assocaton AFIRST. y Suorted by the research rograms PRS and ANM of the CNRS, and by the DRET. z Suorted by the research rogram PRS of the CNRS. the message. Such a rocess s fast, that s the latency of each router s small, and can be aled on all networks. However each router needs to store a large table of sze O(n log d) bts for an n-node network of maxmum degree d. Moreover, snce routers must be relcated n tmes n the network, such a large local memory reurement mles a global nformaton of O(n log d) bts for the whole network, just for routng. Many authors roosed methods to comress the routng table (see for nstance [,, 9]). Among them, nterval routng [, 6, 9, 0] was chosen by Inmos for ts C04 routng ch [5]. Unfortunately, t was roved n [6, 7] that the number of ntervals er lnk necessary to route n any network s not bounded when usng shortest ath routng (actually t was recently roved n [4] that ths number s (n) for an unbounded degree n-node network). Snce the C04 routng ch only allows a few ntervals er lnk, t means that there exst networks such that no shortest ath routng rotocol on them can be mlemented usng C04 chs. In ths sense, ths ch s not \unversal", that s t cannot be used for all ossble networks. To formalze ths concet, let us dene a routng scheme to be a functon that returns a routng rotocol R (or a routng functon) for any network G. For nstance, the shortest ath nterval routng scheme s the selecton, for any network G, of a shortest ath routng functon R whose mlementaton mnmzes the number of ntervals er lnk. For any network, the obtaned routng functon R may reure a large number of ntervals but t exsts, as routng tables do. The shortest ath nterval routng scheme s therefore called unversal because t ales to all networks. On the contrary, the shortest ath -nterval routng scheme, that s the routng scheme that returns a shortest ath routng functon that can be mlemented usng a unue nterval er lnk, s only artal n the sense that there exst networks on whch ths routng scheme s not dened [7]. The unversalty of a routng scheme s counterbalanced by the large memory that may reure routers to mlement the routng functons of ths scheme. Partal routng schemes reure less memory, but they do not aly on all networks. In the last PODC Symosum, n [8], t s roved that any unversal routng scheme on n-node networks, reures a total of (n ) bts of memory to store a shortest ath routng functon on some networks. In fact, the worst-cases networks that reure the total of (n ) bts of memory have a unbounded maxmum degree, namely (n).

2 In ths aer the general roblem of the memory reurement for shortest ath routng scheme on n-node networks of maxmum degree d s addressed. We rove that, for any constant ", 0 < " <, and for every nteger d, d "n, there exsts an n-node network of maxmum degree d on whch any shortest ath routng functon reures (n log d) bts of local memory on (n) nodes. Ths roves that, n general, one can not do better than routng tables for (n) nodes. It means that whatever the way you choose the shortest aths on ths network, the labels of the nodes n the sace f; : : : ; ng and whatever the way you choose the ort labelng, the total routng nformaton s (n log d) bts. In other words, there s no hoe to nd a unversal routng scheme more comact than routng tables for n- node networks of maxmum degree d. Ths general result mroves revous works on the comact routng roblem studed n [8, 6, 8].. A General Model of Routng Functon A ont-to-ont communcaton network s descrbed by a nte connected symmetrc dgrah G = (V; E). In the followng, we denote by n the number of vertces of such a grah. The vertces reresent the nodes or the routers of the network: we assume that one router s assgned to each rocessor (PE). The model may be easly extended to many rocessors connected to a router. The edges reresent bdrectonal communcaton lnks between the routers (to each edge corresonds two ooste arcs). A secal bdrectonal lnk connects the rocessor to the router (see Fgure ). A vertex can communcate drectly only wth ts neghbors, and messages between nonadjacent vertces are sent along a ath connectng them n the network. The router s comosed of an nut and of an outut nterface and of a local routng functon. The nut nterface sends to the local routng functon the nut ort and the header h of an arrvng message M enterng the router. The local routng functon comutes a new header h 0 and a new outut ort 0, and sends these nformatons to the outut nterface. Fnally, the outut nterface forwards M and ts new header h 0 on the ort 0. Ths model can be used to descrbe an unversal routng scheme. PEx R x a node nut nterface (,h) M Px Hx (,h ) a local routng functon Fgure : A model of router outut nterface In a sense, ndng a routng functon n a symmetrc dgrah G = (V; E) conssts on label vertces (= routers) and arcs (= orts of routers) of G such that the local routng functon can comute a ath between any ar sourcedestnaton of vertces (our goal beng to measure the mnmum number of bts to code and to store the local routng functon n a router). More recsely a vertex-labelng of G s an njecton L : V! IN, that assgns for every vertex x V an unue nteger L(x) IN. Let us denote by E x the set of arcs ncdent to x for every x V. A local arclabelng of a vertex x of G s a functon E x : E x! IN, such that any two arcs ncdent to x have the same label f and only f they reresent the same edge (= bdrectonal lnk). It means that we label by the same ort an nut lnk and outut lnk of a same edge ncdent to x. The collecton E = fe x j x V g s called an arc-labelng or a ort labelng of G. Wthout loss of generalty, we assume that headers are coded by ostve ntegers. Thus, the sets of labels (vertces and orts) and headers are subsets of IN. Wthout loss of generalty, we also assume that lnks between the router and the PE are labeled 0. Denton gves a formal denton of a routng functon on a grah G. Note that t s slghtly derent from the model gven by Peleg and Ufal n [8] (the ntalzaton functon s not consdered). Denton (Routng Functon) Let G = (V; E) be a symmetrc dgrah. A routng functon on G s a functon R : VININ! ININ such that there exsts a vertex-labelng L and an arc-labelng E = fe x j x V g satsfyng: for every coule (x; y) of vertces of G, there exst: a seuence fv V j 0 kg of k + vertces, formng a ath from x to y n G a seuence f IN j 0 kg of k + ort labels, labelng the arcs between two consecutve vertces of the ath a seuence fh IN j 0 kg of k + headers, obtaned by successve modcaton of the orgnal header along the ath such that: v 0 = x, 0 = 0, h 0 = L(y) ( ; h ) = R(v? ;? ; h? ), for every < k v k = v k? = y, k = 0, h k = h k? Remark. In ths aer we focus on nte connected symmetrc dgrah wthout mult-arc. In the followng we always refer to such dgrahs by the smle term grah. From Denton, f R s a routng functon on a grah G = (V; E), the ath s bult by R as follows. If the rocessor assgned to the router x wants to send a message M to the rocessor assgned to the router y, t sends the label of the destnaton, L(y), by ts secal lnk labeled 0. The router x reares a header h and an outut ort by comutng the ar (; h) = R(x; 0; L(y)). The header h s attached to the message M, and both are sent on ort. Now, suose that the message M wth a header h 0 arrves at some router z by the ort 0. The router comutes ( 00 ; h 00 ) = R(z; 0 ; h 0 ). If 00 = 0, the routng rocess ends,.e. the message M wth the header h 0 = h 00 s sent on the secal lnk 0 to the rocessor assgned to the router z. Otherwse, z must forward the message M wth the new header h 00 (relacng the old header h 0 ), attached to the message, and sends the message on ort 00. Notatons. The dstrbuted reresentaton of R on G = (V; E) s gven by the collecton fr x j x V g such that

3 R x (; h) = R(x; ; h), for every x V, and every (; h) IN. For every x V, R x s called the local routng functon of R n x. Moreover R can be vewed as a ar of functons (P; H), where P s the ort functon and where H s the header functon. Smlarly P x and H x denote the local ort functon and the local header functon resectvely (see Fgure ). They are dened by P x (; h) = P (x; ; h) and H x (; h) = H(x; ; h), for every x V and every (; h) IN. In ths model, we consder oblvous routng functons,.e. routng functons R such that the ath nduced by R from x to y deends on x and y only. Such functons are also called statc or non-adatve routng functons. Of course the results of the local header functon H x used to comute the next header deend on the current header and the enterng ort of the message. Smlarly, the results of a local ort functon P x also deend on ths header and ths ort. Ths allows to emloy exble tral-and-error routng functons, that s the aths nduced by R can contan loos. Ths roerty may be able to decrease the local memory reurement for the storage of routng nformaton because t allows to vst many nodes to gather nformatons that are stored n the headers. Ths ossblty has been ecently used n the aroach of [8]. Snce our model s more general, our lower bounds on the memory reurement also aly for the usual routng functons where headers always contan the name of the destnaton only. The length of the ath from x to y nduced by R s the number of arcs of G used by the ath. A routng functon R s sad a shortest ath routng functon f and only f the aths nduced by R are always shortest aths n G. In ths aer, we consder shortest ath routng functon only. Tradeos between memory reurement and length of the routng ath have been studed n [,, 8]. However, lower bounds gven n these aers are not tght for near-shortest ath routng schemes, and do not nclude the maxmum degree of the network as arameter.. Memory Reurement Denton (Memory Reurement of a Vertex) Gven a grah G, a routng functon R on G, and a vertex x of G, the memory reurement of x, denoted by MEM(G; R; x), s the mnmum number of bts necessary to store the local routng functon R x. In a sense the memory reurement of a vertex x s the nformaton needed to evaluate R x on some nuts, or the mnmum ROM (Read Only Memory) sze necessary to descrbe R x. It can be formally dened as the Kolmogorov comlexty of the functon R x dened as the sze of the smallest rogram that returns R x and halts [5]. Just as the Kolmogorov comlexty, the memory reurement s dened for a xed rogrammng language. Of course comutaton of the functon R x tself may reure RAM (Random-Access Memory) that s not taken n account here. The memory reurement s only a lower bound on the sace memory reured by a router. Clearly, usng some trcks, every grah G and every vertex x of G suorts a routng functon R x that can be descrbed wth at most O() bts. Indeed labelngs (vertex and ort) are not taken nto account n the memory reurement. Therefore we could use the label of each vertex to encode the full routng nformaton (ncludng for examle the adjacency matrx of G) and each ort label to store the label of a node. Ths aroach reduces memory sze of the router to a constant, but forces the PEs to know what are vertex-labels avalable n the network, that may waste a lot of bts n the nner PE's memory. Ths s why we wll lmt n ths aer the class of routng functons to those that make use of a vertex-labelng from f; : : : ; ng and a local arc-labelng from f; : : : ; deg(x)g, where deg(x) s the degree of x. Note that the lmt to f; : : : ; ng for the vertexlabelng corresonds to the model of many comact routng schemes, ncludng the nterval routng schemes, and s mlctly used n the lterature for lower bounds (e.g. see [8] and [8]). However, certan uer bounds use longer labels (see [,, 4,,, 8]). Under ths assumton, let us consder the comlete grah K n of order n. In general, n a vertex x, a local routng functon R x on K n can be stored wth (n log n) bts for a random ort labelng of x or for a ort labelng chosen by an adversary. Indeed, an adversary can choose some ermutaton between the n? ort labels and the n? neghbors of x n a such way that to reach any neghbor of x mles to know the full descrton of,.e. dlog ((n? )!)e = (n log n) bts. However, some routng functon R 0 x on K n may be stored n x n a more comact way usng O(log n) bts. For examle vertces are labeled by ntegers from f; : : : ; ng and orts by ntegers from f; : : : ; n? g such that the orts connectng vertces x and y s the nteger y f x > y and y? f y > x. Note that we do not make derence between the vertex x and ts label. If headers are always labels of destnaton, the routng functon n x, R 0 x = (Px; 0 Identty), s dened by ts ort functon Px. 0 It runs the followng algorthm: f the destnaton s x then the message s arrved, otherwse forward the message on the ort that has the same name as the destnaton f x > y, else on the ort y?. Clearly R 0 = fr 0 x = (P 0 x; H 0 x) j x V g denes a shortest ath routng functon on K n, and the revous algorthm can be descrbed wth only O(log n) bts for the tem x. In general we are nterestng to nd the \best" routng functon on a gven grah, that s the best vertex and ort labelng, and the best choce of routng aths that mnmze codng of the routng functon. These examles lead to the followng denton of the memory reurement of a grah. Denton (Memory Reurement of a Grah) Gven a grah G = (V; E), the memory reurement of the grah G s the value denoted by MEM local (G) : MEM local (G) = mn R max MEM(G; R; x) xv where R s mnmzed over all shortest ath routng functons on G, such that vertex-labels are from f; : : : ; jv jg and ort labels of every x V are from f; : : : ; deg(x)g. The comact routng roblem conssts of ndng a routng scheme that, for every grah G, returns a routng functon on G that locally needs the mnmum number of bts. For examle the memory reurement for H n, the hyercube of order n, s O(log n), by usng e-cube routng. It s known that all trees [9], outerlanar grahs [0] and all unt crcular-arc grahs [7] of order n and of maxmum degree d,

4 have a memory reurement of O(d log n) bts, because the -nterval routng scheme ales to these grahs ( nterval er arc). Note that our aroach s local. To rove lower bounds for the memory reurement, we made the model as general as ossble. We summarze below some conseuences of Dentons, and : () Headers are of unrestrcted sze. () Routng functons are of shortest aths. () Networks are nte connected symmetrc dgrahs wthout mult-arc and wth unt cost on the arcs. (4) Vertex are labeled by nteger n f; : : : ; ng. (5) For every vertex x, orts of x are labeled by nteger n f; : : : ; deg(x)g..4 Lower ound on the Memory Reurement Ths secton shortly resents the man results of ths aer. Theorem Let ", 0 < " <, be a xed constant. For every nteger d, d "n, there exsts a grah of order n and maxmum degree d that has a memory reurement of (n log d) bts on (log d n) vertces. To rove ths result we construct a grah where a small art of the vertces, namely (log d n), needs to know all together a large amount of routng nformaton, namely a total of (n log n) bts, to route along shortest aths. Ths grah of order n and maxmum degree d s denoted by G n;d. The roof s gven n Secton. In Secton 4 we rove a general result that allows to ncrease the number of routers havng to know (n log d) bts of routng nformaton. Ths transformaton, aled to a G (n);d grah allows to derve a grah of order n and maxmum degree d that s denoted by Slt-G n;d. The followng corollary s roved n Secton 4. Corollary Let ", 0 < " <, be a xed constant. For every nteger d, d "n, there exsts a grah of order n and maxmum degree d that has a memory reurement of (n log d) bts on (n) vertces. Ths corollary roves that there exsts a case where the global routng nformaton s (n log d) bts, mlyng that routng tables are ncomressble n the general case, or that there s no comact dstrbuted routng algorthm for ths network. We resent below a table that summarzes bounds on the memory reurement, and the number of routers usng such a reurement as a functon of the maxmum degree d and the order of the grah. The second table s the sum of the memory reurement on all the routers of these grahs. It s well known that every grah of maxmum degree (aths P n and rngs C n ) has O(log n) bts of memory reurement on all vertces, because a -nterval routng scheme can be aled (see [0]). In ths table, " s an arbtrary ostve constant <. Of course our lower bounds can be aled to the memory reurement of networks wth non unform costs on ts lnks. maxmum degree memory bts # routers grahs O(log n) n P n, C n d "n (n log d) (log d n) G n;d d "n (n log d) (n) Slt-G n;d n? O(log n) n K n grahs total memory bts P n, C n O(n log n) G n;d (n log n) Slt-G n;d (n log d) K n O(n log n) Matrces and Grahs of Constrants Comlexty of roofs s due to the fact that for each grah we allow to choose the set of shortest aths, the vertex-labelng and the ort labelng that mnmze the local routng nformaton. The man dea of our lower bounds s based on the fact that n some grah G some vertces can be connected by a unue shortest ath. Communcatons between these vertces are constraned,.e. every shortest ath routng functon on G has to select the same arc for communcaton between such a ar of vertces. A set of constrants s an nvarant nformaton for routng on G, under any vertex and ort labelngs. Then we try to buld grahs from sets of constrants. More recsely, for each set of constrants, called matrx of constrants, we buld a grah that ossesses a set of unue shortest ath satsfyng these constrants. We argue that certan set of constrants (even after any vertex and ort ermutaton) needs a large memory to be encoded n comarson wth the sze of the grah bult. Roughly seakng, every routng functon on such a grah reures to store ths amount of memory. Recall the noton of matrx and grah of constrants ntroduced n [9]. The basc dea s to aly a countng argument to a set of grahs on whch any shortest ath routng functon \comutes" a nte set of nteger matrces. Denton 4 (Matrx of Constrants) Let G = (V; E) be a grah. A matrx of constrants of G s a nteger matrx M = (m ;j ), I = f; : : : ; g and j J = f; : : : ; g, such that there exst two sets of vertces A = fa j Ig V and = fb j j j Jg V, satsfyng that, for every shortest ath routng functon R = f(p x ; H x ) j x V g on G, there s a way to label the ncdent arcs of vertces of A such that: 8(; j) IJ; m ;j = P a (0; b j ) Informally f M = (m ;j ) s a matrx of constrants of a grah G, Denton 4 means that there exsts two subsets of vertces, A and, such that every shortest ath from a A to b j n G starts wth the arc e ;j whch can be labeled by the ort m ;j. In the followng vertces of A are called constraned vertces and vertces n are called target vertces for M.

5 Denton 5 ( Relaton) Let M = (m ;j ) and M 0 = (m 0 ;j) be two nteger matrces, I = f; : : : ; g and j J = f; : : : ; g. M M 0 f and only f there exst two ermutatons r of I and c of J, and ermutatons of f; : : : ; j [ jj fm ;j gjg, I, such that: 8(; j) IJ; m 0 ;j = (m r();c(j) ) The relaton s an euvalence relaton snce r, c and, for f; : : : ; g, are bjectve functons. M 0 M means that M 0 s obtaned from M by row and column ermutaton, and by a ermutaton of the set of entres for each row of M. Permutatons can be seen as a ort labelng whereas r and c can be seen as a vertex-labelng (recsely, vertex-labelng of constraned and target vertces). Notaton. For every set M of nteger matrces, we denote by M = the set of euvalence classes of M under the euvalence relaton. Denton 6 (Grahs of Constrants) Let M be a set of nteger matrces. A set of grahs G s a set of grahs of constrants of M f t satses that for every M M there exsts a grah G G that ossesses M as matrx of constrants. The followng lemma establshes the fundamental roerty of grahs of constrants relatve to the memory reurement. It roves that grahs of constrants of M, f exst, must reure some nformaton deendng of jm = j and the number of constraned vertces. Theorem s based on ths lemma. Lemma Let M be a set of nteger matrces. For any set G of grahs of constrants of M, there exsts a grah G G such that the sum of the memory reurement of vertces of G s at least where n s the order of G. jm = j log? n? O(log n) Sketch of the roof. Let M M, there exsts G = (V; E) G havng n vertces, and two subsets of vertces A, of sze and resectvely, such that for every routng functon on G, A conssts of the constraned vertces and conssts of the target vertces for M. Let R = fr x = (P x ; H x ) j x V g be any shortest ath routng functon on G, let R A = fr a = (P a ; H a ) j a Ag and let f : f; : : : ; ng! f0; g the characterstc seuence of the set n f; : : : ; ng (f (x) = f and only f there exsts a vertex b that s labeled x). We show that the knowledge of R A, f and O(log n) bts s sucent to comute a matrx M 0 M. Indeed, we have no nformaton about the vertex and the ort labelng of G (n artcular of vertces of A) mnmzng the codng of functons of R A. Let m the mnmum number of bts to necessary store R A. We show that m + log? n + O(log n) log jm = j, and therefore that the sum of the memory reurement on routers s at least m bts where m log (jm = j=? n )?O(log n). Remark. If all grahs of G are of order at most n then one can recse Lemma, based on a smle countng argument, showng that at least jm = j= grahs of G have a sum of the memory reurement on vertces of at least log (jm = j=? n )? O(log n) bts. Grah Constructons In ths secton we wll construct a set of grahs of constrants of a artcular set of matrces wth maxmum cardnalty n order to aly Lemma. Frst to sharen the roof of Theorem, we gve a basc constructon for low degree d n = and whenever n = (d? ), for any constant 0 and for any nteger. Then we generalze the constructon for every n and for every degree d.. Constructon for n ower of = (d? ) For every ar of ntegers (; ), let A ; = (a ;j ) be the nteger matrx such that a ;j = (b(j?)=?+ c mod )+, for every f; : : : ; g and for every j f; : : : ; g. Each column of A ; can be seen as a word of length from the alhabet f; : : : ; g, and the set of columns can be seen as the set of all derent words of length from the alhabet f; : : : ; g. For examle A ; = Lemma For every ar of ntegers (; ), and, there exsts a grah H ; that satses: () H ; has at most 5 vertces () H ; s of maxmum degree + () H ; ossesses A ; as matrx of constrants Sketch of the roof. The grah H ; s dened by nducton on. H ; has two dstngushed subsets of vertces: A, the set of constraned vertces, that are labeled f; : : : ; g, and, the set of target vertces, that are labeled by all the words of length from the alhabet f; : : : ; g. H ; s somorh to K ;. We denote by T ;, the comlete -ary tree of heght and of leaves. H +; s comosed of a coy of H ; and a coy of T ; connected by + + ntermedate vertces (see Fgure ). For the artcular case + =, we do not add the rst = ntermedate vertces. Each vertex of the rst of ntermedate vertces s connected by an edge to a vertex of, formng a new set of vertces denoted 0 labeled lke. The + other vertces form the set + of target vertces of H +; labeled by all the words of length + from the alhabet f; : : : ; g. The leaves of T ; are arttoned n subsets, each one of leaves. A leaf x s labeled, for every f; : : : ; g, f the th chld of the root of T ; s an ancestor of x. Each vertex w = u of +, where s a word of length and x s a letter of the alhabet f; : : : ; g, s of degree two and s connected to u 0 and to one of the leaf x labeled, such that x s connected to at most vertces of the set +. Fgure shows an examle of the H ; grah.

6 H +, A + H, T, + + matchng Fgure : The recursve constructon of the H ; grah. H, constraned vertces A target vertces Lemma For every ar of ntegers (; ), let A ; be the set of ntegers matrces of the form h A; A ;. Then, for every and for every, ( )! (!) ()! ja ; = j = ( log ) where Proof. Let (; ) a ar of ntegers and let A = fm j M X A ; g. Then jaj (!), because any matrx of the form M = h (A ; ) (A ; ), where ; are two ermutatons of columns of each submatrx A ;, s n A. Indeed, aly? on the column of M to get h A ;? ( (A ; )). Therefore ja = j ( )! =(( )!(!) ()!), where ( )!(!) ()! s the maxmum sze of a set of nteger matrces from f; : : : ; g euvalent under (clearly A s a set of nteger matrces from f; : : : ; g). Moreover ja = j = ja ;= j, thus ja ;= j ( )!=((!) ()!). Let L = log ja ;= j. We get L = ( log? log? log ) = (((? ) log? log )). Snce and, L = ( log ) = ( log ). The followng rooston shows that some routers of a grah, called the G n;d grah (see gure 4), can reure (n log d) bts. Prooston Let be a real constant, 0, and be a constant nteger,. For every nteger d, d n =, there exsts a grah of order n = (d? ) and maxmum degree d that has a memory reurement of (n log d) bts on (log d n) vertces. Sketch of the roof. We buld a set of grahs of constrants of order n and maxmum degree d of the set of matrces A ;d? (ntroduced n Lemma ) wth constraned vertces. The grah s comosed of two coes of H ; connected by a erfect matchng between target vertces of the H ; grahs (see Fgure 4). The target vertces Fgure : An examle of the H ; grah for = and =. The shortest aths leadng to the target vertex xyz have to use the ort x from the constraned vertex A, the ort y from the vertex and the ort z from the vertex. of the whole grah are target vertces of only one coy of H ;. Roughly seakng, alyng Lemma and Lemma, the sum of the memory reurement on these vertces s ((d? ) log (d? ) )? O(n). The term O(n) s because log? n x n for every x. Snce n = (d? ) for a constant, t follows that n = ((d? ) ). Therefore the sum of the memory reurement s (n log n) on = (log d? n) = (log d n) vertces, for d. R R - H, R R R R R T-, erfect matchng Fgure 4: The G n;d grah. A grah of constrants of order n and of maxmum degree d = +. Routers R,: : :,R and R 0,: : :,R 0 \comute" the erfect matchng between = (n) vertces.. Constructon for every n and for every d We generalze the A ; matrces by the A ;;k matrces n the followng way: A ;;k s a ( + )(k ) nteger matrx R R R

7 comosed of k A ; matrces, [ A ; A ; : : : A ; ], lus a row [ : : : : : : : : : k k : : : k ]. Each column of the A ;;k matrx can be seen as a word of length +, where the rst symbol s from the alhabet f; : : : ; kg and the last are from the alhabet f; : : : ; g. Note that the A ; matrces are artcular cases of the A ;;k matrces snce A ;; = A +;. For examle " # A ;; = Lemma 4 For every trle of ntegers (; ; k), and k, there exsts a grah H ;;k that satses: () H ;;k has at most (k + 8) vertces () H ;;k s of maxmum degree + () H ;;k ossesses A ;;k as matrx of constrants Sketch of the roof. The H ;;k grah s smlar to a H +; grah where the tree T ;, that was mxed to H +; by a matchng, s relaced by a new tree denoted by Q ;;k where ts root s of degree k. More recsely H ;;k s comosed of H ; wth edges added to the target vertces of H ;, a tree Q ;;k and k addtonal vertces (that are the target vertces of H ;;k ) that are connected by a matchng. The tree Q ;;k s comosed of k -ary subtrees, each havng? leaves at eual dstance from ts root. These k subtrees are connected to a sngle vertex of degree k, that forms the root of Q ;;k. Fgure 5 shows a H ;; grah that ossesses the matrx A ;; (ut on age 7) as matrx of constrants. H,, Fgure 5: The H ;;k grah, for =, = and k =. Lemma 5 For every trle of ntegers (; ; k), let A ;;k be the set of ( + )(k ) ntegers matrces of the form h A;;k A every k, where A A ;;k. Then, for every and for (k )! (k!) (!) ( + )! ja ;;k = j = (k log(k )) Proof. Smlar to the roof of Lemma. Theorem Let ", 0 < " <, be a xed constant. For every nteger d, d "n, there exsts a grah of order n and maxmum degree d that has a memory reurement of (n log d) bts on (log d n) vertces. Proof. Let " ] 0; [ be any xed constant. Let n be a sucently large nteger and let d be any nteger such that d "n. Smlarly to the roof of Prooston, we buld a grah of constrants of order n and maxmum degree d. Ths grah s also denoted by G n;d because, for small values of d, Prooston can be seen as a artcular case of ths theorem. Our grah constructon deends on the degree d. Small degree: d n=0 Let = d? and let the largest nteger such that bn= c 0. Let k = bn= c? 8. An nteger exsts, because n artcular for =, bn= c > n=d? n= n=0? = 9, and bn= c > 9 mles bn= c 0. In seuel, k = d?. The G n;d grah s comosed of two H ;;k grahs that we connected by a erfect matchng. Smlarly to the roof of Prooston, G n;d belongs to a grah of constrants of A ;;k ntroduced n Lemma 5. It has constraned vertces. y denton (k + 8) = bn= c = n? r, where r = (n mod ) f0; : : : ; n? g because n. Thus, (k + 8) n < (k + 8), therefore k = (n). Hence = (log (n=k)) = (log n?log k) = (log d n), because k and log = (log d). As n the roof of rooston, we can add a ath to obtan a grah of order n exactly. Alyng Lemma and Lemma 5, the sum of the memory reurement on vertces s (k log(k ))? O(n) = (n log n) bts. Large degree: n=0 d "n Let n 0 = b(?")nc and = b n 0 =0c?. The constructon of G n;d for large d conssts of a G n 0 ;+ grah, where we add a coy of a K ;d? tree connected by ts root wth an edge to a non constraned vertces of G n 0 ;+. G n 0 ;+ s of degree at most + n 0 =0 d. Clearly the maxmum degree of such a grah s d, due to the root of the K ;d? tree. Shortest aths between the 0 constraned vertces of G n 0 ;+ are unchanged by the K ;d? added. Fnally we can add, for examle to a leaf of K ;d?, a ath of length n? (n 0 + d) = n? (b(? ")nc + "n) 0 to obtan a grah of exactly n vertces. From the revous case, the sum of the memory reurement on 0 = (log n 0 ) vertces s (n 0 log n 0 ) bts. ecause n 0 = (n) and log = (log n) = (log d), then the sum of the memory reurement on (log d n) vertces s (n log n) bts. Therefore, for every degree d such that d "n, t follows that the sum of the memory reurement on (log d n) vertces of the G n;d grah s (n log n) bts. Let 00 = (log d n) ths number of constraned vertces. Thus the memory reurement s (n log n)= 00 = (n log d) bts on at least one constraned vertex among 00. Snce O(n log d) bts are enough to route along shortest aths then ( 00 ) = (log d n) routers need (n log d) bts each. Some authors have suggested to relax the assumton of shortest aths. The stretch factor, ntroduced n [8], denotes the maxmum rato between the length of the rout-

8 ng ath and the dstance, overall ars source-destnaton. In ractce we are nterested by near-shortest ath routng schemes usng a small stretch factor, because clearly routng schemes of stretch factor of O(n) can be descrbe n a very comact way. The (n ) bts lower bound on the global memory reurement was roved for every stretch factor < n [8]. In [9] a tght lower bound on the memory reurement (locally) of (n log n) bts for every stretch factor <, but ths large memory reurement was reured by at most O(n " ) vertces (" constant < ) n a worst-case grah of maxmum degree (n). Usng Theorem, the G n;b n=0c grah constructon allows to easly rove a tght lower bound of (n log n) bts of memory reurement on a constant number of routers, but for every stretch factor < 5= and for a maxmum degree of O( n). 4 Global Memory Reurement vertces (as shown n Fgure 6) to construct a new grah G 0 of order 9n, of maxmum degree d, and wth a global memory reurement of (nf(n)) bts. x x G Fgure 6: A constraned vertex of G slt n 4 constraned vertces n G 0 A x x G In the lterature concernng the comact routng roblem, lower and uer bounds have been gven for the global memory reurement (e.g. see [4, 8, 8]). Even though the local memory reurement seems to be more mortant n ractce for the concrete desgn of routng chs, the global memory reurement can gve a theoretcal nformaton about routng n the network. Denton 7 (Global Memory Reurement of a Grah) Gven a grah G = (V; E), the global memory reurement of the grah G s the value denoted by MEM global (G) : MEM global (G) = mn R X xv MEM(G; R; x) where R s mnmzed over all shortest ath routng functons on G, such that vertex-labels are from f; : : : ; jv jg and ort labels of every x V are from f; : : : ; deg(x)g. In general we are nterested by the local memory reurement of a router rather than the global memory reurement of the whole grah, because a global value does not gves n general the way to dstrbute and balance the routng nformaton. The followng result shows that from a lower bound of the local memory reurement we can derve a lower on the global memory reurement by a grah transformaton. Lemma 6 Let M be a set of nteger matrces. Let G be a set of grahs of constrants of M such that every grah G G s of order at most n and maxmum degree d, and such that every constraned vertces x of G satses deg(x) n=. Then for every nteger n 0 9n, there exsts a grah G 0 of order n 0 and of maxmum degree d that has a global memory reurement of (nm=) bts, where m = log (jm = j=? n 0 )? O(log n 0 ). Ths lemma has an mortant alcaton. Assume that a grah of constrants G from a set of matrces M wth n vertces and maxmum degree d satses that ts constraned vertces have degree n= and a memory reurement of at least f(n) bts each. Then we can slt every constraned An alcaton of Theorem and Lemma 6 s the followng corollary. Corollary Let ", 0 < " <, be a xed constant. For every nteger d, d "n, there exsts a grah of order n and maxmum degree d that has a memory reurement of (n log d) bts on (n) vertces. Ths result roves that the global memory reurement of a grah can be (n log d) bts. Proof. Theorem shows that there exsts a grah G bn=9c;d of order bn=9c and maxmum degree d, for every d "n, " beng any xed ostve constant <, that has a memory reurement of (n log d) bts on (log d n) constraned vertces. Ths reurement s due to a large set of matrx of constrants M such that log jm = j = (n log n). The maxmum degree of constraned vertces s = O( n) (see the constructon n the roof of Theorem ) and moreover n= = (n= log d n) = (n= log n). For a sucent large nteger n, O( n) (n= log n),.e. n=. Therefore we can aly Lemma 6 and buld a grah, denoted Slt-G n;d, of order n and maxmum degree d, that has a memory reurement of (n log d) bts on (n) vertces. As n the end of the revous secton, we can derve a lower bound of the global memory reurement for nearshortest routng schemes: (n log n) bts for every stretch factor < 7=5 and for a worst case n-node network of maxmum degree O( n) (for more nformaton see []). 5 Concluson and Oen Problems We have bult an n-node network of maxmum degree d for whch there s no shortest ath routng scheme more comact than routng tables. Our constructon s based on a random erfect matchng. Moreover our network can be descrbed wth at most O(n log n) bts. In the case of large maxmum degree, we rove therefore by alyng Corollary than the memory reurement of (n) routers of our network can be (n log n) bts each. It means that there s

9 no ecent way to dstrbute (n memory sace) a shortest ath routng functon on ths network, snce (n) routers reure to store the full descrton of the network. However, our network reurng (n log d) bts of routng nformaton s based on connectng two very structured networks. It would be nterestng to know f a d-regular random grah have the same roerty for the routng memory reurement, esecally for maxmum degree o( n). More recsely: What s, on average, the memory reurement for a grah of order n and maxmum degree d? It seems not easy to show that a grah of maxmum degree reures (n ) bts, on average, of global memory reurement, although we know that such a grah exsts. Note that the average case has been studed recently n [4]. In artcular, they show that for \almost all grahs" of order n O(n) bts er router are necessary and sucent for routng of shortest aths. However \almost all grahs" have a maxmum degree n (n) and therefore the roblem stays oen for degree d xed. The best uer bounds roosed on the comact routng roblem are based on routng schemes that generate aths that are not always shortest ath routng schemes, but of length close by a factor s (called the stretch factor) to the mnmal ossble length. Moreover these routng schemes use vertex-labels of O(log n) bts sze (see [,, 8]). In the assumton of vertex-labels of O(log n) bts sze, what s the lower bound of the memory reurement? Acknowledgments: The authors are grateful to Davd Peleg and El Ufal for helful dscussons and remarks about these results. References []. Awerbuch, A. ar-noy, N. Lnal, and D. Peleg, Imroved routng strateges wth succnt tables, Journal of Algorthms, (990),. 07{4. []. Awerbuch and D. Peleg, Routng wth olynomal communcaton-sace trade-o, SIAM Journal on Dscrete Mathematcs, 5 (99),. 5{6. [] E. M. akker, J. van Leeuwen, and R.. Tan, Lnear nterval routng, Algorthms Revew, (99),. 45{6. [4] H. uhrman, J.-H. Hoeman, and P. Vtany, Otmal routng tables, n 5 th Annual ACM Symosum on Prncles of Dstrbuted Comutng (PODC), May 996. To aear. [5] F. Desrez, E. Fleury, and M. Lo, T9000 et C04 : La nouvelle generaton de transuters, Research Reort 9-0, LIP, Ecole Normale Suereure de Lyon, 6964 Lyon Cedex 07, France, Feb. 99. [6] P. Fragnaud and C. Gavolle, Interval routng schemes, Research Reort 94-04, LIP, Ecole Normale Suereure de Lyon, 6964 Lyon Cedex 07, France, Jan [7], Otmal nterval routng, n Parallel Processng: CONPAR '94 - VAPP VI,. uchberger and J. Volkert, eds., vol. 854 of Lecture Notes n Comuter Scence, Srnger-Verlag, Set. 994,. 785{796. [8], Memory reurement for unversal routng schemes, n 4 th Annual ACM Symosum on Prncles of Dstrbuted Comutng (PODC), ACM PRESS, ed., Aug. 995,. {0. [9] P. Fragnaud and C. Gavolle, Unversal routng schemes. Manuscrt submtted to JDC, June 995. [0] G. N. Frederckson and R. Janardan, Desgnng networks wth comact routng tables, Algorthmca, (988),. 7{90. [], Ecent message routng n lanar networks, SIAM Journal on Comutng, 8 (989),. 84{857. [], Sace-ecent message routng n c-decomosable networks, SIAM Journal on Comutng, 9 (990),. 64{8. [] C. Gavolle, Comlexte memore du routage dans les reseaux dstrbues, PhD thess, Ecole Normale Suereure de Lyon, 46, allee d'itale, Jan [4] C. Gavolle and E. Guevremont, Worst case bounds for shortest ath nterval routng, Research Reort 95-0, LIP, Ecole Normale Suereure de Lyon, 6964 Lyon Cedex 07, France, Jan [5] A. N. Kolmogorov, Three aroaches to the uanttatve denton of nformaton, Problems Inform. Transmsson, (965),. {7. [6] E. Kranaks and D. Krzanc, Lower bounds for comact routng, Tech. Re. TR-95-8, Carleton Unversty, July 995. To aear n STACS '96. [7] E. Kranaks, D. Krzanc, and S. S. Rav, On mult-label lnear nterval routng schemes, n 9 th Internatonal Worksho on Grah - Theoretc Concets n Comuter Scence - Dstrbuted Algorthms (WG), vol. 790 of Lecture Notes n Comuter Scence, Utrecht, June 99, Srnger-Verlag,. 8{49. [8] D. Peleg and E. Ufal, A trade-o between sace and ecency for routng tables, n 0 th Annual ACM Symosum on Theory of Comutng (STOC), Chcago, IL, May 988,. 4{5. [9] N. Santoro and R. Khatb, Labellng and mlct routng n networks, The Comuter Journal, 8 (985),. 5{8. [0] J. van Leeuwen and R.. Tan, Interval routng, The Comuter Journal, 0 (987),. 98{07.