ON SELF-ROUTING IN CLOS CONNECTION NETWORKS. BARRY G. DOUGLASS Electrical Engineering Department Texas A&M University College Station, TX

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1 ON SELF-ROUTING IN CLOS CONNECTION NETWORKS BARRY G. DOUGLASS Electrical Engineering Departent Texas A&M University College Station, TX A. YAVUZ ORUÇ Electrical Engineering Departent and Institute of Advanced Coputer Studies University of Maryland College Park, MD 07 ABSTRACT A self routing connection network is a switching device where the routing of each switch can be deterined in ters of the destination addresses of its inputs alone, i.e., independent of the routing inforation regarding the other switches in the network. One faily of connection networks that were considered in the literature for self routing are Clos networks. Earlier studies indicate that soe Clos networks can be self routed for certain perutations. This paper proves that the only category of Clos networks that can be self routed for all perutations are those with at ost two switches in their outer stages. Index Ters: Benes network, Clos network, connection network, parallel coputing, perutation switching, self routing. This work is supported in part by the National Science Foundation under Grant No: CCR

2 Introduction This paper considers the self routability of Clos networks. Such networks find applications as connectors in telephone switching and interprocessor counications in parallel coputers [PI8, MGN79, FE8,MA77] and have been the focus of uch research. An n-input Clos network [CL], where n = k for soe positive integers and k, consists of three stages as depicted in Figure. The first and third stages of the network, each, consist of n switches which ay be ipleented as crossbars, or by other saller Clos networks and this ay be recursively repeated. The center stage consists of n n switches which are siilarly ipleented. The network has exactly one link between every two switches in its consecutive stages. Throughout the paper the switches in the first stage will be denoted IS i, those in the third stage will be denoted OS i where i n, and the switches in the center stage will be denoted MS i where i. The network itself will be denoted CL(n, ). The networks we consider in this paper are all Clos networks, but possibly with different values of. Two Clos networks will be of particular interest: the Clos network with = which is widely referred to as the Benes network in literature [BE], and that with = n/, which is called the copleentary Benes network [PA80, CO8, KO89]. Let Σ n denote the set of all n! perutation aps over a set of n eleents. It is known that an n-input Clos network can realize each of these n! perutations [BE]. This eans that, for each perutation p Σ n, the network contains a set of disjoint paths between its inputs and outputs so as to connect input i to output p(i); i =,,...,n. While this fact guarantees that a Clos network exhibits a set of disjoint paths corresponding to each perutation in Σ n, it does not show how such a set of vertex disjoint paths can be fored. This proble is

3 { { } IS IS { } IS n MS MS MS OS OS OS n } Figure : An n-input Clos network. coonly referred to as the network control, ornetwork routing proble, and has been posed in two different ways in the literature. The first approach, called global routing, assues that there is a single controller (soe kind of progra or procedure) which receives the entire perutation p as input, and coputes the settings for the switches in the network directly fro this inforation. In contrast, the second approach, which is called self routing, divides the inforation about the perutation p over the switches in soe particular way, and this inforation can be transitted only over the paths which exist between the switches. More precisely, a switching network is called self routing if each of its switches can deterine its setting only fro the destination addresses of its own inputs, regardless of the destination addresses of the inputs of other switches in the network. Much work has been reported on global routing schees for Clos networks, both in sequential and parallel algorith doains [see, for exaple, OT7, NS8, LPV8]. We shall not deal with such routing schees in this paper. It suffices

4 to say that global routing either requires too uch tie or too uch hardware. For exaple, on a single processor, the n-input Benes network needs O(n log n) steps to progra and this is incopatible with the network s O(log n) path length [WA8, OT7]. The prograing tie can be reduced to O(log n), by using an n-processor parallel coputer, but this requires interconnection networks which cost ore than the Benes network itself [NS8, LPV8, CO87]. These probles with global routing have propted soe research on the self routing aspect of Clos networks. Nassii and Sahni established that any perutations, frequently used in parallel coputations, and which they naed class F, can be self routed through the Benes network [NS8]. Recently, Boppana and Raghevandra showed that any ore perutations, which they naed class L, can also be self routed by the sae network [BR88]. Most recently, it was shown in [KO89] that the copleentary Benes network can be self routed. Given these results, a question naturally arises as to whether Clos networks can self route all perutations. The ain result of this paper is the answer to this question in the negative. In Section, it is shown that all Benes networks with ore than five inputs are not self routing. More generally, in Section, it is shown that no Clos network whose first stage contains ore than two switches is self routing. The paper is concluded in Section. Self-routability of Benes Networks First, consider the self-routability of the Benes network which is depicted in Figure. It is obvious that the Benes networks with one and two inputs are both self routing. Soe additional thought reveals that the Benes networks with three and four inputs are also self routing. Furtherore, given that the

5 MS n- MS n- n n Figure : The Benes network. -input Benes network is self routing, it is easy to see that the -input Benes network is also self routing if its fifth input and output are connected to both switches in the center stage. We prove that these are the only Benes networks which can be self routed. The proof is carried out by first observing that the nuber of inputs which are apped to each half of outputs in such a network is constant over all perutations in Σ n. We then classify the types of switches in the first stage according to how their inputs are apped to each half of outputs under perutations in Σ n. Next we prove that for all even n 8, there exist perutations in Σ n which, when odified in a certain way, force the nuber of inputs apped to the two halves of outputs through a center-stage switch to change. This then leads to the proof that the Benes network is not self routing for all even n 8. What reains to be considered is the case for n =, and networks with odd nubers of inputs both of which we will handle separately. Let R denote the set of outputs of the first n/ switches and R denote the

6 set of outputs of the next n/ switches in the third stage of an n-input Benes network. Let µ p i,j denote the nuber of inputs which are apped into R j by perutation p through center stage switch MS i where i, j. Proposition : µ p i, = n/ and µ p i, = n/ for all p Σ n,neven, and i. Proof: Each center stage switch has n/ output links, one to each of the n/ third stage switches. Exactly n/ of these are connected to OS,OS,..., OS n/ whose outputs define R, and the rest n/ to OS n/ +,OS n/ +,...,OS n/ whose outputs define R. Since all n output links between the center and third stage switches ust be occupied to realize any perutation in Σ n, each p Σ n ust ap n/ inputs into R and n/ inputs into R through the n/ output links of each of the center stage switches. Let S p i,j denote the set of switches in the first stage of an n-input Benes network one input of which is apped to R i and the other input of which is apped to R j by p Σ n where i j. Call the switches for which i = j =, switches of type, those for which i = j =, switches of type, and those for which i =,j =, switches of type. Then the following stateents hold. Proposition : For all even n 8 there exists a perutation p Σ n for which S, p. Proof: For n 8, the first stage contains at least four switches. Choose p so that at least three of these switches belong to S,. p Obviously, p is not unique. Proposition : For all even n 8 there exists a perutation in Σ n which aps one input of each of at least two first stage switches of type to R through the sae center stage switch, and the other inputs of these two switches to R through the other center stage switch.

7 To R To R To R MS R To R To R To R MS R Figure : Type switches in a Benes network. Proof: Fro Proposition, there exists at least four first stage-switches any subset of which can be fixed as type switches by choosing an appropriate perutation in Σ n. If at least three are fixed as type switches, then, obviously, that perutation ust ap one input of each of at least two of these switches into R through the sae center stage switch, and their other inputs into R through the other center stage switch. A graphical construction of this proposition is depicted in Figure. It is seen that the first three switches in the first stage are fixed as type, and the first two ap one of their inputs to R through the upper center stage switch, and their other inputs to R through the lower center stage switch. The third switch routes one of its inputs to R through the lower center stage switch and its other input to R though the upper center stage switch. The fourth switch is left unspecified, even though it should also be of type as iplied by the following proposition. Proposition : R i = Si,i p + S, ; p i, and S, S p, p = n/ 7

8 p(x) x x y y ISx ISy MS MS p(y) p(x) p(y) R R n- n Figure : The partial setting of a Benes network under perutation p. n/ for all p Σ n. Proof: That R i = Si,i p + S, ; p i is obvious since each first stage switch of type as well as type contribute two inputs, and each first stage switch of type contributes one input to R i. That S, S p, p = n/ n/ follows iediately fro the fact that R i R = n/, and R = n/. = S p i,i + S p, ; i, and We now state the ain result of this section. Theore : No Benes network with an even nuber of inputs equal to, or greater than eight can be self routed. Proof: First, by Proposition, construct a perutation p for which S, p. Then by Proposition, we can find two first stage switches of type, say IS x and IS y such that p aps one input fro each to R through the upper center stage switch, and the other input fro each to R through the lower center stage switch as shown in Figure. Call the inputs to the first of these two 8

9 switches x,x, and the inputs to the second y,y, so that p(x ),p(y ) R, and p(x ),p(y ) R. Now construct another perutation, say q Σ n such that q(x )=p(y ) and q(y )=p(x ), and q(x) =p(x) for all other inputs x over which p is defined, i.e., the reaining inputs of the network. Since only the outputs of x and y have changed under q, the network can accoodate this change only by redefining the states of the two switches to which these inputs are connected if it is to be self routing. However, regardless of how the two switches are set, now both inputs to both switches are to be apped to the sae set of outputs, i.e., both inputs of IS x to R, and both inputs of IS y to R. Therefore, the nuber of inputs which are routed to R through the upper center-stage switch decreases by one, and the nuber of inputs which are routed to R through the lower center-stage increases by one. A siilar change occurs in the nuber of inputs which are routed to R. But this contradicts Proposition which states that the nuber of inputs which are apped to each half of outputs of a Benes network through each of its center-stage switches is constant for all p Σ n. Hence the stateent follows. We now consider the reaining cases. For n =, the Benes network has three switches in each of its outer stages and two switches in the center stage. In order for the network to be self routing, the setting of each switch ust be fixed for each pattern of inputs it receives. In Figure we list a sequence of perutations and their realizations on a -input Benes network. In Figure (a), switch IS is arbitrarily fixed so that input is routed to output through switch MS, and input is routed to output through MS. The settings of switches IS and IS are then deterined. (In fact, IS can be set either way, but this does not alter our arguent.) In Figure (b), the destinations of IS and IS have changed, and the inputs of IS are kept the sae as in Figure (a) as 9

10 highlighted by shading. Therefore, the setting of IS reains the sae and the settings of the other switches are then deterined. Continuing in the sae way, we arrive at Figure (d), and find that the setting of IS disagrees with that in Figure (a) even though its pattern of inputs reains the sae. We conclude that the network is not self routing. As for odd values of n 7, it is easily seen that a Benes network with an odd nuber of inputs can be obtained by adding an input and output to the Benes network with n inputs, and connecting the to both of the center stage switches. The proof that a network with odd nuber of inputs is not self routing is then an iediate consequence of the proof that the network with one less inputs is not self routing. Thus, we have established: Theore : An n-input Benes network is self routing if and only if n. Self routability of Clos Networks Theore can be extended to Clos networks as follows. First, we divide the outputs into two halves R and R as before, and extend the definition of a type switch as one which sends exactly / inputs to R, and / inputs to R. Then we note that Proposition still holds for the extended type switch. Therefore, for n, we can construct a perutation, say p, so as to have at least three type switches. Furtherore, along the lines of Proposition, we can find a switch, say MS i, in the center stage such that p sends one input of each of at least two type switches to R through MS i. With these stateents, we can now prove the following stateent. Theore : The CL(n, ) network with n, and even isnot self routing. 0

11 Inconsistent Routing (b) (a) (c) (d) Figure : A sequence of routings for the -input Benes network.

12 Proof: As specified above, choose a perutation, p so that MS i receives one input fro a first stage switch IS x of type and one input fro another first stage switch IS y of type both going to R. Now, by interchanging the inputs of these two switches only, construct a perutation, q, under which all inputs of IS x go to R, and all inputs of IS y go to R, and which, otherwise, is identical to p. With this change, regardless of how the inputs of the two switches are routed, MS i ust send one input fro IS x to R, and one input fro IS y to R. But, since the destinations of the inputs of all switches except those of IS x and IS y reain unchanged under q, this requires that the nuber of inputs which are apped to R through MS i decrease by one, and the nuber of inputs which are apped to R through MS i increase by one. However, this contradicts the fact that the nuber of paths fro MS i to R and R is fixed, and hence the stateent. There are two cases which reain to be considered. That Clos networks with odd values of cannot be self routed iediately follows fro the above theore. The case for n/ = can be proven by constructing a counter exaple which is analogous to that in Figure, and is oitted here. Cobining the above stateents we now have Theore : The CL(n, ) network is self routing if and only if n, or =. Conclusions This paper has established that an n-input Benes network is self routing if and only if n. More generally, it has been shown that an n-input Clos network, with -input switches in its outer stages, is self routing if and only if n, or =. On a ore positive note, the authors showed in another paper [BO90]

13 ( that the sae network can self route at least (!) n ( n )! ) perutations. For = n, this gives a network with O(n. ) cost and O() delay that can self route at least (πn ) n ( n ) n perutations. It reains to be shown if this e lower bound is tight, i.e., whether or not ore perutations can be self routed by the sae network. Along a different direction, in that sae paper [BO90] the authors introduced a weaker for of self routing which relies on balancing routes in Clos networks. In particular, it was shown that an n-input Benes network can be odified so as to have a network that can realize all perutations with O(n log n) gates and in O(log n) constant fan-in gate delays. Acknowledgeent The authors thank the anonyous referees for their constructive rearks and suggestions.

14 REFERENCES [BE ] V.E. Benes, Matheatical Theory of Connecting Networks and Telephone Traffic, Acadeic Press Pub. Copany, New York, 9. [BR88 ] R. Boppana and C. S. Raghevandra, On self routing in Benes and shuffle-exchange networks, in Proceedings of International Conference on Parallel Processing,, Aug. 988, St. Charles, IL., Vol., pp [CO87 ] J.D. Carpinelli and A. Y. Oruç Parallel set-up algoriths for Clos networks, in Proceedings of nd International Conference on Supercoputing. May 987, Santa Clara, CA, pp. -7. [CL ] C. Clos, A study of non blocking switching networks, The Bell Syste Technical Journal, March 9, pp. 0. [FE8 ] T Y Feng, A survey of interconnection networks, IEEE Coputer, Dec. 98, pp. 7. [BO90 ] B. G. Douglass and A. Y. Oruç Self routing and route balancing in connection networks, in Proceedings of International Conference on Parallel Processing, May 990, St. Charles, IL, pp. -7. [KO89 ] D. M. Koppelan and A. Yavuz Oruç, A self routing perutation network, in Proceedings of International Conference on Parallel Processing, 989, St. Charles, IL, Vol, pp [LPV8 ] G.F. Lev, N. Pippenger and L. Valiant, A fast parallel algorith for routing in perutation networks, IEEE Transactions on Coputers, Vol. C 0, No., Feb. 98, pp

15 [MA77 ] M.J. Marcus, The theory of connecting networks and their coplexity: a review, Proceedings of the IEEE, Vol., No. 9, Septeber 977, pp. 7. [MGN79 ] G.M. Masson, G.C. Gingher, and S. Nakaura, A sapler of circuit switching networks, IEEE Coputer, June 979, pp. 8. [NS8 ] D. Nassii and S. Sahni, Parallel algoriths to set up the Benes perutation network, IEEE Transactions on Coputers, Vol. C-, No., Feb. 98, pp. 8. [NS8 ] D. Nassii and S. Sahni, A self routing Benes network, and parallel perutation algoriths, IEEE Transactions on Coputers, Vol. C-0, No., May 98, pp. 7. [OT7 ] D.C. Opferan and N.T. Tsao Wu, On a class of rearrangeable switching networks, The Bell Syste Technical Journal, May-June 97, pp [PA80 ] D. S. Parker Jr, New points of view on three-stage rearrangeable switching networks, in Proceedings of the Workshop on Interconnection Networks, Coputer Society Pub., 980, pp.. [PI8 ] N. Pippenger, Telephone switching networks, in Proc. of Syposia of Applied Matheatics, Vol., May 98, pp. 0.

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