Constructions of Given-Depth and OptimalMultirate Rearrangeably Nonblocking Distributors

Yang Wang, Hung Q. Ngo, and Thanh-Nhan Nguyen

Abstract The theory of multirate switching networks, startedin the late 80s, has been very practically useful. In particular, ithas served as the theoretical foundation for the development ofmost ATM switching systems.

Rearrangeable multirate multicast switching networks are cus-tomarily called distribution networks, or distributors for short.It has been known for more than 15 years that distributors withcross-point complexity O(n log2 n) can be constructed, where nis the number of inputs. The problem of constructing optimaldistributors remains open thus far.

In this paper, we give a general method for constructing given-depth rearrangeable multirate distributors. One of the rewardsof our construction method is a distributor with cross-pointcomplexity O(nlogn), which we then show to be optimal. Wethus settle the aforementioned open problem.

I. INTRODUCTION

Multi-rate switching networks are switching networks thatsupport varying bandwidth connections. The theory of mul-tirate switching networks, perhaps started with the papers byNiestegge [1] and Melen and Turner [2], has proved to bevery useful in practice. For example, this theory has served asthe theoretical foundation for the development of most Asyn-chronous Transfer Mode (ATM) switching systems from majorATM equipment manufacturer [3]-[5]. Roughly speaking, asopposed to space switching where each connection request canonly be carried on an internal or external link of a switch, themultirate switches allow for connections with varying "rates"or bandwidths to be carried on a single link, as long as thetotal connection rates does not exceed the link's capacity.

In the unicast case, one particularly fruitful line of researchon multirate switching networks has been on the multirate re-arrangeability of the Clos network [6], represented by the (stillopen) conjecture by Chung and Ross in 1991 [7] which statesthat the Clos network C(n, m, r) is multirate rearrangeablynonblocking when the number m of middle-stage switches isat least 2n -1. This conjecture is interesting because it pointstowards a possible generalization of the Konig's theorem foredge coloring bipartite graphs. Later developments on thisconjecture and related problems were reported in [8]-[13]. Seealso [14], [15] for several related lines of research.

In the multicast and broadcast cases, there have been notablyfew known results, though. The works presented in [14], [16]-[18] concern conditions for the Clos network to be multicastcapable. The study presented in [19] (the journal version is

The work of Hung Q. Ngo was supported in part by NSF CAREER AwardCCF-0347565.

The authors are with the Computer Science and Engineering depart-ment, State University of New York at Buffalo, U.S.A. Emails: (yw43,hungngo, nguyen9) @cse . buffalo. edu

[20]) was the only one that deals directly with more generalconstructions and complexities of multicast multirate switch-ing networks. In their paper, using Pippenger's network [21],the authors constructed a rearrangeable multirate distributorwith cross-point complexity O(n log2 n). (Distributor, alsocalled generalized connector, is a standard name referring tomulticast switching networks.)

The problem of constructing optimal multirate multicastswitching networks remains open thus far. In this paper, wegive a general method for constructing rearrangeable multiratedistributors. One of the rewards of the method is a multicastdistributor with cross-point complexity 0 (n log n). We thenshow that this is optimal, thus settling the aforementioned openproblem.

The rest of the paper is organized as follows. Section IIpresents basic definitions and several fundamental composi-tions of networks. Section III gives the definition of a versionof multirate concentrators, which is crucial in constructingmultirate distributors. Section IV contains the main results,including a general given-depth construction. The constructiongives rise to a multirate n-distributor of size O(n Ig n) whichis then shown to be optimal. Lastly, Section V concludes thepaper with a few remarks and discussions on future works.

II. PRELIMINARIES

A. Multirate networks

In the rest of the paper, let [m] {1, ... , m} and Zm={0, ... . m -1} for any positive integer m. For any finite setX, let 2x denote the power set of X. For any positive integerk, we use (xk) to denote the set of all k-subsets of X. Graphtheoretic terminologies we use here are fairly standard. See[22], for instance.An (nl, n2)-network is a directed acyclic graph (DAG) JV

(V, E; X, Y), where V is the set of vertices, E is the set ofedges, X is a set of n1 nodes called inputs, and Y - disjointfrom X - is a set of n2 nodes called outputs. The verticesin V -X U Y are internal vertices. The in-degrees of theinputs and the out-degrees of the outputs are zero. The sizeof a network is its number of edges. The size of a networkis the equivalence of the cross-point complexity of a switch.The DAG model is standard for studying the complexity ofswitching networks [23], [24]. The depth of a network is themaximum length of a path from an input to an output. Forshort, we call an (n, n)-network an n-network.

In the multirate environment, a constant Q < 1 is oftenused to represent the capacity of the inputs and outputs ofAV. Input nodes have capacity (normalized to) 1. The factor

1-4244-1 206-4/07/$25.00 ©2007 IEEE

1/' is often referred to as the speed advantage of the system.This internal speedup is a common technique for designingbroadband switches [2], [25], [26].

Given an n-network AV = (V, E; X, Y), a distributionrequest (or multicast request) is a triple

D = (x,S,w) C X x 2y x [b,B].As we are only concerned with distribution networks in thispaper, the term "request" should be implicitly understoodas "distribution request" henceforth, unless it is explicitlyspecified otherwise. The weight or rate w of the requestsatisfies b < w < B for some given lower- and upper-bounds0K b < B < Q < 1.A distribution assignment is a set D of requests satisfying

the following conditions: (a) total weight of requests comingfrom any particular input does not exceed Q, and (b) totalrequest weight to any output does not exceed Q, in other words

E w</3, VyeY.(x,S,w)EDyES

A request D is compatible with a distribution assignment Diff D U {D} is also a distribution assignment.A distribution route R for a request D = (x , w) is a

(directed) tree rooted at x whose leaves are precisely the nodesin S. We also say that R realizes D, and call w the weight ofR. A state of AV is a set R of distribution routes, where thetotal weight of routes containing any node does not exceedthe capacity of that node. Each state of AV realizes a uniquedistribution assignment, one route per request. A distributionassignment D is realizable iff there is a network state realizingit. A request is compatible with a state if it is compatible withthe distribution assignment realized by the state.We are now ready to define the central notions of nonblock-

ingness in the multirate environment. In defining differentnotions of distributors, we drop the "multirate" qualificationto avoid being wordy. Distribution networks in this papers areimplicitly understood as multirate distribution networks.A rearrangeable (RNB) n-distributor (or just n-distributor

for short) is an n-network in which any distribution assignmentis realizable.A strictly nonblocking (SNB) n-distributor is an n-network

AV in which, given any network state R realizing a distributionassignment D and a new request D compatible with D, thereexists a route R such that 'R U {R} is a network state realizingD U {D}.As requests come and go, a strategy to pick new routes

for new requests is called a routing algorithm. An n-networkis called a widesense nonblocking (WSNB) n-distributor withrespect to a routing algorithm A if A can always pick a newroute for a new request compatible with the current networkstate. We can also replace A by a class of algorithms A. Ingeneral, an n-network A/ is WSNB iff it is WSNB with respectto some algorithm.We will consider two classes of functions on each network

type: (a) the minimum size of a network, and (b) the minimumsize of a network with a given depth. One of the key problemsaddressed in this paper is the tradeoff between networks'depths and their sizes.

TABLE I

KNOWN RESULTS ON s(n, k)

Depth k ISize s(n, k)

2 e) ( n_log2n A [29]Vlog log n3 e) (n log log n) [30]2d, 2d + 1, d > 2 e) (nA(d, n)) [31], [32]In particular, for k = 4, 5 e (2n log* n) [31], [32]8) (a (n) ) 83e(n) [32]

Given the parameters j3, b, and B as described above,let mrdo[b,B] (n), mwdo[b,B] (n), and msdo[b,B] (n) denotethe minimum size of a multirate RNB, WSNB, and SNBn-distributor, respectively. In the limited-depth case, letmrdo[b,B] (n, k), mwdo[b,B] (n, k), and msdo[b,B] (n, k) denotethe minimum size of an RNB, WSNB, and SNB n-distributorwith depth k, respectively.

In the special case when b = 0, B = Q = 1, i.e.the case when there is no internal speedup and no requestrate restriction, we will drop the subscripts Q [0, B] and usemrd(.), mwd(.), msd(.) to denote the corresponding func-tions.

B. Classical networks

In constructing multirate distributors, we will also need thenotions of (classical) concentrators and super-concentrators.Recall that an (n, m)-concentrator is an (n, m)-network wheren > m, such that for any subset S of m inputs there existsa set of m vertex disjoint paths connecting S to the outputs.Let c(n, m) and c(n, m, k) denote the minimum sizes of an(n, m)-concentrator and an (n, m)-concentrator of depth k,respectively.An n-superconcentrator is an n-network with inputs X and

outputs Y such that for any S C X and T C Y with S =

ITI = c, there exist a set of c vertex disjoint paths connectingvertices in S to vertices in T. Let s(n) and s(n, k) denote theminimum sizes of an (n, m)-superconcentrator and an (n, m)-superconcentrator of depth k, respectively.

For n > m, an (n, m)-superconcentrator is a networkobtained by removing any (n -m) outputs from an n-superconcentrator. Obviously, an (n, m)-superconcentrator isan (n, m)-concentrator. Hence,

c(n, m) <sK(n)c(n, m, k) < s(n, k).

(1)(2)

Note that the concentrators and superconcentrators describedabove operate in the space domain or classical environment,namely no two paths can share a vertex. It has been knownfor more than 3 decades that there are concentrators andsuperconcentrators of linear size [27], [28]. The constructionswere based on expanders, whose applications in mathematicsand computer science are numerous.

For the fixed depth case, the asymptotic behaviors of allthe s(n, k) were only completely devised recently. Table Isummarizes the results. The function A(d, n) is the inverseof functions in the Ackerman hierarchy: they are increasing

extremely slowly. They can be defined as follows. Let

log* n := min{l > 0 log ...logn < 1}

where the logarithms are to base 2. By induction on k, define

d-I

A(d, n) := log* n :d-2 d-2

min{l > O log* * .. .log* * n <1}

The reader is referred to [32] for the definition of a(n) (whichis actually called p3(n) in their paper, but we change its nameto avoid confusion with our speedup factor 03).

C. Basic compositions of networksLet JV1 and JV2 be any two (n,m)-networks. We use

VLiFV2 to denote an (n,m)-network XV obtained by iden-tifying the inputs of JV1 and JV2 in any one-to-one manner,and identifying the outputs of JV1 and JV2 in any one-to-onemanner. We refer to JV1 V2 as the stacking of JV1 and A/2.When stacking k copies of a network AV, denote the result bykV.Given any k (n, m)-networks AV1, . .. , JVk, let

F (jV1,...,JV) denote the (n, mk)-network obtainedby identifying the inputs of A\1,...., fk in any one-to-one fashion. (In effect, we "paste" together the inputs ofA.,... ,JVk.) When the JVF are identical copies of the same(n, m)-network AV, we use _kkAV to denote the result insteadof writing F- (J,, . . V, AF). Given an (n, m)-network M anda (m, I)-network AV, let M o AV be the network obtained byidentifying the outputs of M and the inputs of AV in anyone-to-one fashion.

III. MULTIRATE CONCENTRATORS

There are several obvious ways to generalize the notionof classical concentrators to multirate concentrators. To avoidcumbersome notations, we will define here only a particulartype of multirate concentrators which are used in later sectionsto construct good multirate distributors.

Given integers n > m > 0. Consider an (n, m)-networkC = (V, E; X, Y) A concentration request is a pair (x, w),where x is an input and w < 1 is the weight of the request.A path from x to some output is called a route realizing thisrequest. A set of routes are compatible if the total weight ofroutes containing any vertex is at most 1. A concentrationassignment is a set of concentration requests such that eachinput generates requests with total weight at most 1, and thatthe total weight of all requests is at most m/2. The networkC is called an (n, m)-multirate concentrator if and only if,for each concentration assignment D there exists a set ofcompatible routes realizing requests in the assignment.

Lemma 111.1. Let C be any (n, m)-concentrator and S beany (n, m)-superconcentrator. Then, C(n, m) = CES is an(n, m)-multirate concentrator

(n, m)-concentrator

-NI ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~

(n, m)-superconcentrator

Fig. 1. Construction of an (n, m)-multirate concentrator C(n, m).

Proof The reader is referred to Figure 1 for an illustration ofC (n, m). To prove this lemma, we will use a routing algorithmadapted from the CAP algorithm proposed in [2].

Let D be any concentration assignment. Note that theinputs of these requests are not necessarily different from one

another. As long as there are still two requests (x, wi) and(x, w2) coming from the same input x, replace them by a

new request (x, w1 + w2). The new set of requests is still a

valid concentration assignment. Moreover, a valid route for(x, w1 + w2) can be "decomposed" back into two routes withweights w1 and w2 to satisfy the requests (x, wi) and (X, w2).Consequently, we can assume that the inputs of these requestsare distinct.

Partition D into two subsets D1 and D2, where D1 consistsof all requests with weights > 1/2. Let x = .Dl1, Y = 1D21For i = 1, 2, let Wi be the total weight of requests in Di.Then, because D is a concentration assignment,

m/2 > W1 + W2 > x/2 +W2. (3)

The set of requests in D1 can be routed through the (n, m)-concentrator C so that no two routes share a vertex. Thus, thevertex capacity constraint is satisfied.

Next, we route the requests in D2 through S to the otherm -x outputs that are unused after routing D1. These requestsare routed using the CAP algorithm.

First divide the y requests into t = Fy/ls groups of sizes = m- x each, with possibly the last group having size< s. Assume the weights for these requests are w, > w2 >*.. >wY. Then, the first group consists of s largest weightswi, .w..,ws; the second group consists of the next s largestweights Ws+1, . . ., W2s; and so forth.

Because s < m, for each group of requests, in the (n, m)-superconcentrator S there are s vertex disjoint paths joiningthe inputs of the requests to some s outputs. We will usethese paths as routes realizing these requests in the group. Thisensures that no two routes for requests in the same group share

any vertex.To this end, we need to show that no vertex of S carries

routes with total weight exceeding 1. In the worst case, a vertexcarries one request from each group. Thus, the maximumweight a vertex might carry is at most

W1+ Ws+±+ * * * + W(t-l)s+l

< 1 + Wi+...+Ws + + W(t-2)s+1+ + W(t-l)s

12+ < + m/2 /2= 1.2 s -2 mx-

The last inequality follows from (3). F-

Corollary 111.2. A multirate (n, m)-concentrator of depth kcan be constructed with the same asymptotic complexity ass(n, k) shown in Table I.

Proof This follows directly from the fact that, removingany n -m outputs from a classical n-superconcentratoryields an (n, m)-superconcentrator, which is also an (n, m)-concentrator. D

IV. REARRANGEABLE MULTIRATE DISTRIBUTORS

We shall use the concentrator C(n, n/2) constructed in theprevious section recursively construct rearrangeable distribu-tors.

A. Distributors for the case B < 0 < 1/2In this subsection, we construct distributors under the condi-

tion B < / < 1/2. In fact, we will construct a slightly strongerdistributor, where the capacity of input nodes are allowed tobe 1. Obviously, any distributor with capacity-i inputs is alsoa distributor with capacity-Q inputs. The outputs' capacitiesremain equal to < 1/2.

C(n, m) rm-distributor M

/ ~~~~~m-distributor AACC(n, m)

Fig. 2. Recursive construction of distributors with capacity-I inputs.

In the following lemma, we ignore the issue of divisibilityfor the sake of clarity. It is simple but tedious to deal directlywith divisibility. The following construction is the multirateversion of Pippenger's network [21].

Lemma IV.. Let m be a factor of n. Let C(n, m) be theconcentrator constructed in the previous section. Let M be

any multirate mT-distributor with capacity-i inputs. Then, thenetwork

g\ = -n/m (C oM)

is an n-distributor with capacity-i inputs. Note that, we onlyconsider b < B <Q < 1/2.

Proof The reader is referred to Figure 2 for an illustrationof JV. Consider a distribution assignment D. Partition D inton/rn subsets D = D1 U D2 U ... U Dfn/m as follows. For eachrequest D = (x, T, w) C D and i C { , ...., n/m}, let

T, =Tn{(i -)m+,(i- 1)m+2,...,im}Then, add (x, Ti, w) into Di, unless Ti = 0. Note that, if wecan find routes realizing all of .l,. .., fn/m, then a naturalunion of those routes will realize D. For example, to realizethe request D above, take the union of the routes realizing(,~Tl,W),...., (XzTn/m:W)The idea is to use the first concentrator and distributor to

realize D1, the second concentrator and distributor for D2, andso on. Since the construction is symmetric, we only need toconstruct routes realizing D1.

Firstly, notice that the total weight of requests from D1 isat most m/2, because there are at most m outputs involved inthese requests, each with capacity Q < 1/2. Thus, there arecompatible routes in C(n, m) joining each input x of a request(x, T1, w) in D1 to an output f (x) of C(n, m). For two inputsx and x', f (x) and f (x') might be the same, though.Now, construct a distribution assignment D' for the corre-

sponding m-distributor as follows. For each request (x, T1, w)in D1, add (f(x), T1, w) to D'. By definition of compatibility,the total weight of compatible routes to any output c ofC(n, m) is at most 1. Consequently, D' is a valid distributionassignment, which can be realized by some network state R'of D'.

Finally, each request (x, T1, w) in D1 can be realized by theconcatenation of the route in C(n, m) and the correspondingroute in 7Z' F

The following theorem can be made slightly better withmore careful calculus. We state a somewhat weaker versionfor the sake of clarity.

Theorem IV.2. For /3 < 1/2, we can construct n-distributorsof(a) depth k = 3 and size O (n3/2 g)n(b) depth k = 4 and size 0 (n3 /2 glgn).(c) depth k = 5 and size O (n4/3 Ig+ /3)n(d) depth k = 6 and size 0 (n4/3 (lg n)2/3) .(e) any depth k > 3 and size 0 (n1 ( gIg)1 ) where

i= Fk/21.(f) size O(nlg n).Proof The reader is referred to Table I and Corollary 111.2when examining the following reasoning.(a) Let m = i+/,;.9 Choose C(n,m) of depth 2 and

size 0 (nlglgnj) . Chose M to be the complete m x mbipartite graph.

(b) Let m = /n-lglgn. Choose C(n,um) of depth 3 andsize O (n lglgn). Chose M to be the complete m x mbipartite graph.

(c) Let m = n3 (Ig gn))31 Choose C(n, m) of depth 2, andM the depth-3 m-distributor constructed in part (a).

(d) Let m = n gig '.. Choose C(n,rm) of depth 3, and(lgn)2 3

M the depth-3 m-distributor constructed in part (a).(e) We induct on k. For 2 < k < 6, the previous cases serve

as the bases for our induction hypothesis. When, k =

2j with j > 4, choose m = n1-1/i(1ggn)(2j- 1)/ (j+1)j- I I~~ ~~~~~~gnC(n, m) of depth 3 and size O (nlgggn), and M to bethe depth-(k -3) m-distributor inductively constructed.The case when k = 2j 1 is similar.

(f) In this case, we choose m = n/2, C(n, m) to be the linearsize multirate concentrator (with depth p3(n) as in TableI). The network M is recursively constructed this way.Suppose the C(n, m) are of size cn for some constant c.The total size is thus

2 cn+4.cn+ ... + 2IgnC n

2 21gn 1O(nlgn).

B. Distributors for the general case

Lemma IV.3. Let S be a set of k positive real numbers{Wl...,w1}, where wi < 1/2, Vi C [k], and Z 1 wi <Then, S can be partitioned into at most 4 subsets, each ofwhose sums is at most 1/2.

Proof Let Si {wi} for each i C [k]. We will graduallymerge these Si until there are only at most four sets left withthe desired property. For each set X, let w(X) denote the sumof elements in X. Call X a type-j set if 1/2j+1 < w(X) <1/23.Now, consider the sets Si, i C [k]. For any j > 1, as long

as there are two sets Si, Si, of type j, merge Si and Si,.The merge results in a type-(j 1) set. When it is no longerpossible to merge, we have at most 3 sets of type-1, and atmost 1 set of type-j for each j > 1. To this end, merge allsets of type-j for all j > 1. Because 1/4 + 1/8 + = 1/2,the result of this last merge has sum at most 1/2. D

Lemma IV.4. LetM be any classical n-distributor Let JV/ bethe distributor constructed in Lemma IV]. Then, the stackingofM and 4 copies ofAV is a multirate n-distributor for any0 K b < B <K < 1.

Proof Consider any distribution assignment D. For each out-put vertex y, consider the set of weights of requests involvingthis vertex. Partition this weight set into at most 5 classes.Class 0 consists of (at most) one weight which is > 1/2.Partition all the weights < 1/2 into 4 sets using Lemma IV.3,then label the sets classes 1 to 4.

For each request (X, T, w) C D, partition T into at most 5classes To T1,. , T4, where y C Ti iff the weight w belongsto class i of output vertex y. In effect, we decompose therequest (X, T, w) into 5 separate requests (x, Ti, w).

The idea is to route the set of all (x, To, w) using theclassical n-distributor. The routes in the classical distributor

are vertex disjoint, hence they will certainly satisfy the vertexcapacity constraint. Moreover, each output has at most onerequest with weight > 1/2, implying that the set of requests(x, To, w) is valid for the distributor.

Then, route all requests (x, Ti, w) using the ith copy of JA.Note that the requests that a copy of A/' is responsible for werechosen so that each output has total requested weight at most1/2. Hence, AV can handle them easily by Lemma IV.I. D

Note that our construction works regardless of the valuesof Q, B, and b. If Q < 1/2 then we do not need the classicaldistributor in the stacking. However, asymptotically this factdoes not reduce the size of the multirate distributor.Theorem IV.2 and Lemma IV.4 give the key result of this

paper.

Theorem IV.5. For any b < B < Q, and for any k > 3,we can construct a depth-k multirate n-distributor of sizeO (n1+l/± (1gn)' /j, where j Fk/21. This meanskj (igig n)1-1/J

mrdo[b,B] (n, k) = O n /i (1gln)1 l/j ) Fk/21.I (4)

Furthermore, we can also construct a multirate n-distributorof size O(n lg n). Thus,

mrdo[b,B] (n) = O(n lg n). (5)

Proof Consider first the fixed-depth case. For the 4 copiesof AV in Lemma IV.4, we use part (e) of Theorem IV.4. Forclassical n-distributor of depth k, we can use the constructionsin [33] (see also [34]) with size O(nl+3 (log n) 2 ), whichis asymptotically slightly smaller than our depth-k distributorsfrom part (e) of Theorem IV.4.

If there is no restriction in the network depth, we use part(f) of Theorem IV.4 for JV. The classical n-distributor of sizeO(n Ig n) has was constructed in [35]. D

C. On the optimality of our distributorFor classical distributors, it has been known for a long time

that every n-distributor must have size Q(nlgn) [36]. Is itpossible that, due to the internal speedup factor of 1/3, onecan construct multirate n-distributors with size asymptoticallybetter than 0(nlgn)? For example, when 1/3 is extremelylarge (compared to n) it is easy to see that one internal node issufficient because this node's capacity can handle all requests.

In the following theorem, we show that, when B is aconstant, we cannot do better than 0(nlgn), implying thatour result in Theorem IV.5 is optimal!

Theorem IV.6. Suppose 0 = b < B = K< 1, where B isa constant. Given any multirate n-distributor of size s(n), wecan construct a classical n-distributor ofsize O(s(n)) with thesame depth. Thus, any asymptotic lowerbound for classicaldistributors is also an asymptotic lowerbound for multiratedistributors, whether or not the depth is specified.

In particular, a multirate n-distributor must have sizeQ(n log n); that is,

mrda[b,B] = Q(n log n).

Proof Let c = Li/Bi, which is a constant. Let AV be any

multirate n-distributor of size s(n). We will construct a clas-sical n-distributor M of size c2s(n) = 0(s(n)) and the same

depth. By the aforementioned result, c2s(n) = Q(n Ig n); thus,s(n) = Q(n Ig n), completing the proof.

The network M is constructed as follows. Replace eachinternal vertex v of A` by c copies vl, . . ., v,. For each edge(U, v) of JV, do the following:

. if u is an input and v is an output of AV, add the edge(U, v) to A4;if u is an input and v is an internal vertex of JV, createc new edges (u, v1), . . ., (u, v,) in M;if u is an internal vertex and v is an output of JV, createc new edges (u1, v), . . ., (ua, v) in M;and lastly, if both u and v are internal vertices, then createc2 new edges (ui, vj) in M, for all i,j C [c] .

We need to show that M is indeed a classical n-distributor.Consider any distribution assignment D in the space domain.This assignment consists of requests of the form (x, T), whereT is a subset of the outputs. Each output can only be requestedat most once. Now, create a distribution assignment D' for AVas follows. For each request (x, T) in D, create a request(x, T, B) and add to D'. Let R' be a network state of AVrealizing D'. Obviously each internal node of A/' belongs toat most c routes in T'. Thus, from the routes in T' we can

construct a set of routes realizing D for M easily becauseeach vertex v of AV has c copies in M.

V. DISCUSSIONS

Just like in the classical case, there are still small gaps

between the upper and lower bounds of depth-k distributors.These are still open problems. The reader is referred to [23]for more details. With more careful computation, the resultsof Theorem IV.5 for given depths can be made better. Anotheropen problem is the asymptotic sizes of multirate distributorswhen is not a constant.

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