Rearrangeable networks are switching systems capable of establishing simultaneous independent communication paths in accordance with any one-to-one correspondence between their n inputs and n outputs. Classical results show that Ω( n log n ) switches are necessary and that O( n log n ) switches are sufficient for such networks. We are interested in the minimum possible number of switches in rearrangeable networks in which the depth (the length of the longest path from an input to an output) is at most k, where k is fixed as n increases. We show that Ω( n1 + 1/k ) switches are necessary and that O( n1 + 1/k ( log n )1/k ) switches are sufficient for such networks

Pippenger, Nicholas

Yao, Andrew C.-C.

Scholarship@Claremont

Claremont CollegesScholarship @ ClaremontAll HMC Faculty Publications and Research HMC Faculty Scholarship1-1-1982Rearrangeable Networks with Limited DepthNicholas PippengerHarvey Mudd CollegeAndrew C.-C. YaoUniversity of California - BerkeleyThis Article is brought to you for free and open access by the HMC Faculty Scholarship at Scholarship @ Claremont. It has been accepted for inclusionin All HMC Faculty Publications and Research by an authorized administrator of Scholarship @ Claremont. For more information, please contactscholarship@cuc.claremont.edu.Recommended CitationPippenger, Nicholas, and Andrew C.-C. Yao. “Rearrangeable Networks with Limited Depth.” SIAM Journal on Algebraic and DiscreteMethods 3, no. 4 (December 1982): 411-417.SIAM J. ALl3. DISC. METH.Vol. 3, No. 4, December 19821982 Society for Industrial and Applied Mathematics0196-5212/82/0304-0001 $01.00/0REARRANGEABLE NETWORKS WITH LIMITED DEPTH*NICHOLAS PIPPENGERt AND ANDREW C.-C. YAOAbstract. Rearrangeable networks are switching systems capable of establishing simultaneous indepen-dent communication paths in accordance with any one-to-one correspondence between their n inputs andn outputs. Classical results show that f(n log n) switches are necessary and that O(n log n) switches aresufficient for such networks. We are interested in the minimum possible number of switches in rearrangeablenetworks in which the depth (the length of the longest path from an input to an output) is at most k, wherek is fixed as n increases. We show that I(n 1+1/k) switches are necessary and that O(nl+I/k(log n)Ilk)switches are sufficient for such networks.1. Introduction. An (m, n )-network G V, E, A, B) comprises an acyclic direc-ted graph with vertices V and edges E, a set of m distinguished vertices A calledinputs and a set of n other distinguished vertices B called outputs.A request is an ordered pair (a, b) comprising an input a and an oatput h. Aroute is a directed path from an input to an output. A route satisfies a request (a, b)if it is from a to b.An l-assignment is a set of requests, no two of which have an input or outputin common. An l-state is a set of routes, no two of which have a vertex in common.An/-state satisfies an/-assignment if it contains a route satisfying each request in theassignment.An n-connector (also known as a rearrangeable n-network) is an (n, n)-networkthat has an n-state satisfying each of the n! n-assignments. The size of a network isthe number of edges in it. The depth of a network is the maximum number of edgesin any route in it.Let f(n) denote the minimum possible size of an n-connector. An information-theoretic argument (due to C. E. Shannon) shows that f(n) f(n log n) (see Pippenger[4]; f(...) means "some function bounded below by a strictly positive constanttimes. "). A classical construction (due to D. Slepian, A. M. Duguid and J. LeCorre)shows that f(n)=O(n log n) (see Pippenger [3]; O(...) means "some functionbounded above in absolute value by a constanttimes... ").Let fk (n) denote the minimum possible size of an n-connector having depth atmost k. We shall be interested in the behavior of fk(n) as n grows while k remainsfixed. The case k 1 is trivial: fl(n)= n 2. For k 2, a probabilistic argument (usedby de Bruijn, Erd6s and Spencer [1] to solve a problem of van Lint [2]) shows thatf2(n) O(n3/E(log n)1/2). For odd k _-> 3, the classical construction referred to aboveshows that fk(n) O(rtl+E/(k+l)).In 2 we shall show (by adapting an argument due to Pippenger and Valiant [5])that fk(n)= I(nXX/k). In 3 and 4 we shall show (by a probabilistic argument) thatfk(n)= O(n+/k(log n)I/k).2. Lower bound. An n-tree is a (1, n)-network with inputs A={a}, outputsB {b,..., b,} and, for 1 <_-]-<_n, a unique route Rj satisfying the request (a, bj).If T is an n-tree, letA(T)= E E do,_j<n vR* Received by the editors June 6, 1981.f Computer Science Department, IBM Research Laboratory, San Jose, California 95193.Department of Electrical Engineering and Computer Sciences, University of California, Berkeley,California 94720.411412 NICHOLAS PIPPENGER AND ANDREW C.-C. YAOwhere do denotes the number of edges directed out of the vertex v.PROPOSITION 2.1. ff T is an n-tree of depth at most k >- 1, thenA(T)>=kn 1+1/k.Proof. The proof is by induction on k. The case k 1, A(T)-n 2 is trivial. Ifk >_-2, let d be the number of edges directed out of the input and let T1,. , Td (withn 1,""", nd outputs, respectively) be the subtrees into which these edges are directed.We haveA(T) dn + , A(Th) >=dn + , (k 1)n+l/(k-1),l<__h<=d l<_h<__dby inductive hypothesis. Since nl +...+nd =n and (k-1)O 1+1/(k-1) is convex in 0,we haveEl<:h<=dStraightforward calculus shows that1+1/(k-1)which completes the induction.An n-shifter is an (n, n)-network with inputs A={al,..., aN}, outputs B{b 1, , b, } and, for 1 -</’ <- n, a state satisfying the assignment{(al, bj/l),’’’, (aN, b./,)} (addition is modulo n).THEOREM 2.1. Any n-shifter of depth at most k has size at least kn 1/1/k.Proof. Let G (V, E, A, B) be an n-shifter of depth at most k. Let Ri, be theroute from input ai to output b./i in the state that satisfies the assignment{(al, bi/),...,(a,,bi/,)}. By identifying common initial segments, the routesR.I, , Rg., can be assembled into an n-tree Ti of depth at most k, for whichA(Ti) > kn l+l/kby Proposition 2.1 For 1 <= i-< n, 1 =</" =< n and e e E, let Ix(i,/’, e) be 1 if the edge eis directed out of a vertex on R,. and 0 otherwise. For any and/’, we haveand by summing over j, we haveZ (i,],e)>= Z d,eE vRi,iZ Z tx(i,], e) >-_ Z Z do >-A(T)>-_kn +l/k.<=i<--n eeE <--i_n vRi,Y Z Z Ix(i,j,e)>=kn 2+1/k.l<--i<--n l<-j<=n eEOn the other hand, since the routes R 1,j, , R,,i have no vertex in common, anedge e can be directed out of a vertex on at most one of them. Thus, for any f ande, we haveE Ix(i,],e) <-1.REARRANGEABLE NETWORKS 413By summing over/’ we have, lz(i,i,e)<--n,l]<--_n l<--_i<=nand by summing over e we have., Y ., p(i,j,e)<--n #(E)eeE <=]<--n# (" ’) means "the cardinality of. "). Comparing this with (2.1) gives@(E)>kn x+l/kas claimed. I3COROLLARY 2.1. Any n-connector of depth at most k has size at least kn 1+Ilk.Proof. An n-connector is an n-shifter. D3. Couplers. A set {Xx,. , Xr} is an x-packing of a set A if X1, ’, Xr aremutually disjoint x-element subsets of A. An x-packing@(A)/16 (or, equivalently, #()-> @(A)/16x).If G is a network and X a set of inputs of G, let G(X) denote the set of outputsof G reachable through routes from inputs in X.An (/,/)-networks G=(V,E,A,B) is an (l, x, y )-coupler if, for every tight x-packing {Xx,..., X} of A, there exists a tight y-packing {Y1, Ys} of Bsuch that, for every 1 _-</" -< s, there exists 1 <- <- r such that Y. _c G (Xi).If G is an (/, m )-network and H is an (m, n )-network, let GoH denote an(l, n)-network obtained by identifying the outputs of G with the inputs of H in anyone-to-one fashion (to become vertices that are neither inputs nor outputs of G oH).LEMMA 3.1. If G is an (l, x, y)-coupler and H is an (l, y, z)-coupler, then G oHis an (l, x, z )-coupler.Proof. The proof is immediate.An (m, m)-network G is a strong (m, x, y)-coupler if, for every m/2 <- <- m, each(l, /)-network obtained from G by deleting m -l vertex-disjoint routes (together withall edges incident with vertices on these routes) is an (l, x, y)-coupler.LEMMA 3.2. If G is a strong (m, x, y )-coupler and H is a strong (m, y, z )-coupler,then G H is a strong (m, x, z)-coupler.Proof. The proof follows from Lemma 3.1. [21LEMMA 3.3. Let X denote the number of successes among n trials that succeedindependently with probability p. Then(3.1) (X> 2np) <_- (1/4) "pand(3.2)((" ") means "the probability of... ").Proof. For (3.1), we may assume p < 1/2, for otherwise (X> 2np) 0. IfJ0 ifX=<2np,Y / 1 if X > 2np,then (X>2np)= g’(). If Z Tx-2"p (where T > 1 is a parameter to be chosenlater), then <- Z and so 8’() <- g" (Z). Thus it will suffice to estimate g’(Z).414 NICHOLAS PIPPENGER AND ANDREW C.-C. YAOSince X is the sum of n independent random variables that assume the value 1with probability p and the value 0 with probability 1-p, Tx is the product of nindependent random variables that have expected value pT + 1-p. Thus,g’ (Z) (pT + 1 p)nT-2nr’.Choosing T 2(1 -p)/(1 2p) and using the inequality 1 + 0 <=e yields (3.1). A similarargument yields (3.2). 13PRoeosi:orq 3.1. Iftn512x In rn <= y =<-,then there exists a strong (m, x, y)-coupler of depth 1 and size at most 32my/x.Proof. Let16yXand let G (V, E, A, B) be the random (m, m)-network K",,,(p) (an (m, m)-networkof depth 1 in which each of the m2 potential edges is independently present withprobability p). We expect m2p 16my/x edges in E, so4(E)> < -<--4’by Lemma 3.3. Thus, it will suffice to show that(G not a strong (m, x, y)-coupler)-<,1-.There are at most 4" (/,/)-networks F obtained from G by deleting m-l vertex-disjoint routes (together with all edges incident with vertices on these routes). It willthus suffice to show for m/2 <= <= m and F K,t(p) that1(F not an (1, x, y)-coupler)=<4"+There are at most <-m" minimal tight x-packings {X1,... ,Xr} (wherer Ill 16x of the inputs of F. Thus, it will suffice to show that1(F not an (I, x, y)-coupler for ) =<We shall consider each set Xi in turn. For each set Xi, we shall attempt to constructa set Y containing y outputs, disjoint from all previously constructed sets Y1, , Y-and satisfying Y F(Xi). If we show that(no Y. for X) =< (e2-)then the probability of fewer than s [l/16y] successes among r trials will be at most2r[ (e2-)Y] < 2[ (e2-) Y] /2 =< [4 (e2-)Y]//32x -< [4 (e2-) y]<= 4" <= 4" e -"y/a56x < 4" e"/64x1--2" In--4"+ira".REARRANGEABLE NETWORKS 415To construct Y., we shall consider each output of F that is not in Y1U. U Y_Iin turn (there are at least l-sy =>//2 >-m/4 such outputs). For each such output, weshall attempt to find an edge joining it to an input in Xi. The probability of findingsuch an edge is1_ (l_p)X => l_e_pX =>p__x2(using 1-0-<e-<--1-0/2 for 0<-0 <-- 1). Thus, we expect at least (m/4)(px/2)=2ysuccesses and the probability of fewer than y successes is at most (2/e)y, by Lemma3.3. ECOROLLARY 3.1. Ifand5121nm <-xthen there exists a strong (m, x, xk-)-coupler ofdepth k -2 and size atmost 32(k -2)rex.Proof. The proof follows from Lemma 3.2 and Proposition 3.1. E4. Upper bound. An (n, n)-network is an (a, b)-partial n-connector if, for everya-assignment P, there exists an (a- b)-assignment Q_P and a state satisfying Q.If Q and H are (n, n)-networks, let GIIH denote an (n, n)-network obtained byidentifying the inputs of G with the inputs of H in any one-to-one fashion (to becomethe inputs of G[IH) and identifying the outputs of G with the outputs of H in anyone-to-one fashion (to become the outputs of GI[H).LEMMA 4.1. If G is an (a, b)-partial n-connector and H is a (b, c)-partial n-connector, then GIIH is an (a, c )-partial n-connector.Proof. The proof is immediate. ELEMMA 4.2. Let L be an l-element set and let be a collection of subsets of Lsuch that if Y andX_Y, then X . If contains more than2-2 (2/x)(2x)-element subsets ofL, then it contains a tight x-packing ofL.Proof. Let Y be a random uniformly distributed (2x)-element subset of L, then(Y )> 2-2.Let be a maximal x-packing contained in . If is not tight, then<=16and(4( Ugh) >x) <-2 16 <__2-2x.Thus, there exists a (2x)-element set Y @ for which 4 (Y ’1U) <- x. Then Y Ucontains an x-element set that can be added to g to yield a larger x-packing, contra-dicting the maximalityPROPOSITION 4.1. If512k(2 In 2n)k-1 <-m <-n,416 NICHOLAS PIPPENGER AND ANDREW C.-C. YAOthen there exists an (m, m/2)-partial n-connector of depth k and size at most 64(k 1).n(2m In 2n)1/k.Proof. Setx [(2m In 2n)X/k].Then 512 Into =<x and Xk-l<=m/16, and by Corollary 3.1, there exists a strong(m, x, x k-1)-coupler G of depth k 2 and size 32(k- 2)mx.Let8xand let H=(V, E,A,B) be the random (n, n)-network K.,m(q)oGoK"*,.(q). Weexpect 2nmq 16nx edges in K..,. (q) and K"*,. (q) together, so(=(E)-> 32(k 1)nx) --< (1/4) 16nxby Lemma 3.3. It will thus suffice to show that(I-I not an (m, m/2)-partial n-connector)<-1/4.There are at most nZ"*/4 m-assignments P. Thus, it will suffice to show that(I-I not an (m, m/2)-partial n-connector for P) <- n -z"*.We shall consider each of the m requests in P in turn. For each request (a, b),we shall attempt to construct a route, vertex-disjoint from all previously constructedroutes and satisfying the request (a, b). If we show that1(no route for (a, b))<- 4,4nthen the probability of fewer than m/2 successes among m trials will be at most2"*(4) "*/2 -zm--nThe probability that there is no route for (a, b) is the probability that there is noroute in the random network l=Kx,(q)oFoKt,x(q), where m/2<-l <-m and F is an(1, x, xk-1)-coupler. Let denote the random number of outputs of Kl,(q) reachablethrough routes from the input of K,(q), let rt denote the random number of outputsof Kl,l(q)oF reachable through routes from the input of K,t(q)oF and let"denotethe random number of inputs of K,x(q) from which the output of Igl,(q) is reachable.Then(no route) <- (no route It/>-x k-x,">- 2x) + (rt <xk-1)+( <2X)_-<(no route [rt ->x -a,"> 2x) +(rt= <x’-alc>2x)+(<2x)=+(" <2x).The random variables and"have expected value lq >= mq/2 4x, so(<2x) (" < 2x) <=16n 4REARRANGEABLE NETWORKS 417by Lemma 3.3. Furthermore,(noroute It/>x k-x, >2X)<(I--xk-)/( )\//\/2X21<- e-2xt/l e-2X k/, ____< e-4 In 2n 16n 4"Thus, it will suffice to show that (r/<xk-ll2x)<= 1/16n 4.Let c6’ be the collection of subsets X of the inputs of F for which (F(X)) < x k-.Since F is an (l, x, xk-)-coupler, contains no tight x-packing and thus, by Lemma4.2, contains at most(2x)-element subsets. Thus,1(r <xk-I>2X)<2-2’--16n 4,as was to be shown. [3COROLLARY 4.1. /fb 512k (2 In 2n)k-1,then there exists an (n, b )-partial n-connector of depth k and size at most256k(k 1)n (2n In 2n)/k.Proof. The result follows from Lemma 4.1, Proposition 4.1 and., 2_/k 1 <-4k1<oo (1-2/k)(using e- __< 1 0/2 for 0 <_- O <_-- 1). [3LEMMA 4.3. There exists an (a, O)-partial n-connector of depth 2 and size 2an.Proof. Consider K,,, oKa,,. [3THEOREM 4.1. There exists an n-connector of depth k and size at most256k (k 1)n (2n In 2n)Ilk + 2(512)kn (2 In n)k-.Proof. The result follows from Lemma 4.1, Corollary 4.1 and Lemma 4.3; an(n, 0)-partial n-connector is an n-connector. [3REFERENCES[1] N. G. DE BRUIJN, P. ERDS AND J. SPENCER, Solution 350, Nieuw Archief voor Wiskunde, 22(1974), p. 94-109.[2] J. H. VAN LINT, Problem 350, Nieuw Archief voor Wiskunde, 21 (1973), p. 179.[3] N. PIPPENGER, On rearrangeable and non-blocking switching networks, J. Comput. System Sci., 17,(1978), pp. 145-162.[4] .,A new lower bound for the number of switches in rearrangeable networks, this Journal, 1 (1980),pp. 164-167.[5] N. PIPPENGER AND L. G. VALIANT, Shifting graphs and their applications, J. Assoc. Comput. Mach.,23 (1976), pp. 423-432.

Rearrangeable Networks with Limited Depth

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Rearrangeable Networks with Limited Depth

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