Monotone routing in multirate rearrangeable clos networks

RESEARCH NOTE

Monotone Routing in Multirate Rearrangeable

Clos Networks

Xiao-Dong Hu2

Institute of Applied Mathematics, Chinese Academy of Sciences, Beijing 100080, China; and Department of Computer Science, City University of Hong Kong, Hong Kong

E-mail: xdhuamath3.amt.ac.ch

Xiao-Hua Jia

Department of Computer Science, City University of Hong Kong, Hong Kong E-mail: jiacs.city.edu.hk

Ding-Zhu Du1 , 2

Department of Computer Science, University of Minnesota, Minneapolis, Minnesota 55455; and Institute of Applied Mathematics, Chinese Academy of Sciences, Beijing 100080, China

E-mail: dzdcs.umn.edu

and Frank K. Hwang

Department of Applied Mathematics, Chiao Tung University, Hsinchu, Taiwan E-mail: fhwangmath.nctu.edu.tw

Received November 27, 1999; revised May 7, 2000; accepted July 20, 2000;

In this paper, we study monotone routing in the symmetric three-stage Clos network with general bandwidth, and propose a new approach to analyze the multirate rearrangeability. For networks with small size switches, we show that monotone routing is better than the previous methods.  2001 Academic Press

1382

0743-731501 35.00

Copyright  2001 by Academic Press All rights of reproduction in any form reserved.

1_{Supported in part by the National Science Foundation under Grant CCR-9530306.} 2_{Supported in part by National 973 Program of China.}

1. INTRODUCTION

The symmetric three-stage Clos network C(n, m, r) has been widely used in them design of telecommunication networks [1, 4]. C(n, m, r) consists of r (n_m)-crossbars (switches) in the first (or input) stage, m (r_r)-(n_m)-crossbars in the second (or central) stage, and r (m_n)-crossbars in the third (or output) stage. Every crossbar in the first stage has an outlet connected to an inlet of every crossbar in the second stage and every crossbar in the second stage has an outlet connected to an inlet of every crossbar in the third stage. There are rn inlets in total in the first stage, called inputs, and rn outlets in total in the third stage, called outputs. A C(2, 3, 4) is illustrated in Fig. 1.

A Clos network C(n, m, r) in classical circuit switching is rearrangeable if it can route every matching between inputs and outputs.

Since Melen and Turner [7] initiated the study on nonblocking properties in multirate interconnection networks, it has become one of the most important research topics in ATM networks with applications in computer networks, telecom-munications, and Internets. In the multirate Clos network C(n, m, r), the switch is more powerful. Each switch can realize an edge-weighted bipartite graph between inlets and outlets with the property that the total weight of edges at each inlet and outlet is at most one. Each edge still corresponds to a connection (call or request). A connection in the multirate network is denoted by a triple (i, j, w), where i and j are the input and output of the connection, respectively, while w is the weight of the connection representing the bandwidth required by the connection. A route is a path in the network joining an input switch (a switch in the first stage) to an out-put switch (a switch in the third stage). A route r realizes a connection (i, j, w) if the switch with input i and the switch with output j are connected by r with capacity w. Each link in the network is assumed to have unit capacity (after nor-malization). Thus the weight of any connection is in the interval [0, 1].

A set of connections is compatible if the sum of weights of all connections from any input and to any output are at most one. A request frame is a set of compatible connections. A configuration is a set of routes, and it is compatible if the total weight of routes passing through every link is at most one. A request frame is said to be realizable if there exists a compatible configuration which contains routes realizing all connections in the request frame. A multirate network is said to be (multirate) rearrangeable if every request frame is realizable.

In classical circuit switching, all connections are assumed to have the same rate one. Namely, a network is said to be rearrangeable in classical circuit switching if every compatible request frame of connections with weight one is realizable. It is well known [1] that the symmetric three-stage Clos network C(n, m, r) is rearrangeable in circuit switching if and only if mn. Now, since multirate is involved, more crossbars are needed in the center stage to reach the rearrangeability. Chung and Ross [3] showed that if m2n&1, the symmetric three-stage Clos network C(n, m, r) is multirate rearrangeable when each connection has weight chosen from a given set [1, p]. After proving this result (Corollary 3), they stated that ``It would be of interest to show that Corollary 3 holds for the general discrete bandwidth case with K distinct rates.'' For an easy reference, we call it the ChungRoss conjecture. ChungRoss conjecture. If m2n&1, the symmetric three-stage Clos network C(n, m, r) is multirate rearrangeable when each connection has weight chosen from a given finite set [ p₁, p₂, ..., pk] where 1 p1> p2> } } } > pk>0 and pi is an

integer multiple of pkfor 1ik&1.

Melen and Turner [7] gave a routing algorithm CAP and with CAP, it can be shown that the multirate three stage Clos network C(n, 2n&1, r) is rearrangeable when each connection has a weight at most 12. Using this fact, Lin et al. [6] recently showed that the ChungRoss conjecture is true for a restricted discrete bandwidth case where each connection has a weight chosen from a set [ p1,

p2, ..., pk] such that 1 p1> p2> } } } > pk>0 and piis an integer multiple of pi+1

for 2ik&1. In fact, the ChungRoss conjecture seems to be true not only in the discrete bandwidth case but also in the continuous bandwidth case. By using some coloring and partition arguments from graph theory, Du et al. [5] proved that C(n, m, r) for m41n16 is multirate rearrangeable in the general bandwidth case.

In this paper, we consider multirate rearrangeability in the Clos network C(n, m, r) with arbitrary rates. We study a monotone routing and establish a rela-tion between the multirate rearrangeability under monotone routing and a system of linear inequalities. From investigating the system, we obtain some properties of the monotone routing and improve some best known results for small n.

2. MAIN RESULTS We study a simple routing method as follows.

Monotone routing. Sort all requests in weight-nonincreasing order and realize each request one by one whenever a connection can be found.

Define a linear system I(n, k) with kn+1, consisting of k+n inequalities as follows. x1 1+x 1 2+ } } } +x 1 n+x 0 1>1 x2 1+x22+ } } } +x2n+x01>1

{

b b . . . b b b (1) xn 1+x n 2+ } } } +x n n+x 0 1<1 b b . . . b b b xk 1+x k 2+ } } } +x k n+x 0 1>1 x0 1+x 1 1+x 2 1+ } } } +x n 1+ } } } +x k 11 x1 1+x 2 2+ } } } +x n 2+ } } } +x k 21

{

x1 3+x 2 3+ } } } +x n 3+ } } } +x k 31 (2) b b . . . b . . . b b x1 n+x2n+ } } } +x n n+ } } } +x k n1

where real variables x0

10 and x j

i0 for 1in and 1 jk satisfy the

follow-ing constraint. xj i(x j i&x 0 1)0, for 1in, 1 jk (3)

The above inequality (3) means that if xj

i>0, then x j ix

0

1 for 1in and

1 jk.

The relationship between system I(n, k) and the multirate rearrangeability of the Clos network C(m, n, r) is shown in the following lemma.

Lemma 2.1. If I(n, k) has no solution, then C(m, 2k&1, r) is multirate rear-rangeable under monotone routing.

Proof. By contradiction, suppose C(m, 2k&1, r) is not multirate rearrangeable under monotone routing. Consider the first request which cannot be routed, and assume it is from input switch I to output switch J with weight w. Then for each center switch H, either the link from input switch I to H has a load greater than 1&w or the link from H to output switch J has a load greater than 1&w. There-fore, either there exist k center switches such that every link from I to them has a load greater than 1&w or there exist k center switches such that every link from them to J has a load greater than 1&w. Without loss of generality, assume the former case occurs. Note that I has n inlets, and without loss of generality, assume the request (I, J, w) is from the first inlet. Set xi

j to be the weight of request from

the jth inlet through the i-th center switch. Then we have

:

n

j=1

xi

By substituting w with x0

1we obtain the inequalities (1). In addition, the inequalities

(2) are satisfied under the constraint of connection capacity, and the inequalities (3) are satisfied, because monotone routing is applied. Therefore, system I(n, k) has a solution, a contradiction. K

Now we study the necessary and sufficient conditions that system I(n, k) has a solution.

Lemma 2.2. If I(n, k) has a solution, then 1

3x01> k&n k&1.

Proof. Summing up all inequalities in (1) will lead to (k&1) x0

1>k&n. Now

consider the left hand side of the inequality. Suppose, by contradiction, x0 1>

1 3.

Then there are at most two nonzero variables in [xj

i : 1 jk], for each i,

2in, and at most one nonzero variable in [xj

1: 1 jk]. Without loss of

generality, we assume x1

1>0 and x12>0. Moreover, we can further assume that

x2 2>0, x 2 3>0, and x 3 4>0, x 3

4>0, ..., and so on. It is clear that under the constraint

(2) at most n&1 inequalities in (1) can be satisfied at the same time, a contradic-tion. K

Corollary 2.1. I(n, n+wn

2x) has no solution.

Proof. It follows immediately from Lemma 2.2. K Lemma 2.3. If x0

1 k&n k&2, then

(a) each inequality in (1) has at least two nonzero variables, and (b) each set [xj

i : 1 jk], for 1in, has at least two non zero variables.

Proof. (a) Suppose, by contradiction (and without loss of generality), that x1

2x01>0 and x1i=0 for i=1 and 3in. Note that if x j 2{0, then x j 2x01, and x1 2+x j 2x 1 2+x 0 1>1, a contradiction. Hence, x j

2=0 for 2ik. The sum of

nonzero variables in the last (k&1) inequalities of (1) is at most n&1+ (k&2) x0

1(n&1)+(k&2) k&n

k&2=k&1, a contradiction.

(b) Suppose, again by contradiction (and without lose of generality), that there exists x1

i₀>0 such that x j

i₀=0 for 2 jk. Then the sum of nonzero variables

in the last (k&1) inequalities of (1) is at most n&1+(k&2) x0

1k&1, a

con-tradiction. K

Lemma 2.4. I(4, 5) has no solution.

Proof. Suppose, by contradiction, that it has a solution. Then by Lemma 2.2, we can assume that x0

1= 1

4+=, where =

1

12. We can further assume, by

Lem-ma 2.3(b), that x2 1x 0 1and x 0 1x 1 1 1 2(1&x 0 1)= 3 8& =

2. In the following we consider

two cases separately. Case 1. x1 2{0 and x 1 3=x 1 4=0. In this case, x 1 2> 3 8& =

2. If there are two nonzero

variables in [x2 2, x

2 3, x

2

4], then the sum of nonzero variables in the last three

inequalities of (1) is at most 27 8+ 3 2=3, a contradiction. Subcase 1.1. x2 3{0 and x 2 2=x 2

4=0. In this special case, we can assume, by

Lemma 2.3, that x3

2{0 and at least one variable in [x 3 3, x

3

them are nonzero (note that x1 2+x 3 2+x 2 3+x 3 3+x 3 4>1+2x 0

1), then the sum of

non-zero variables in the last two inequalities of (1) is less than 3&(1+2x0

1)+2x01=2, a contradiction. If x3 3{0 and x 3 4=0 (note that x 1 2+x 3 2+x 2 3+x 3 3>2&2x 0 1), then

at most one variable in [x4

2, x43, x52, x53] is nonzero. This means that at least one

of the last two inequalities of (1) has just one nonzero variable, contradicting Lemma 2.3(a). Thus x3

3{0 and x 3 4{0. If x 4 2=x 5 2=0, then x 4 3+x 4 4+x 5 3+x 5 4+2x 0 1 2&2x0 1+2x 0

1=2, a contradiction. Hence, without loss of generality, let x 4 2{0 and x4 4{0. Since x 3 4+x 4 4>2&2x 0 1&(1& 3 8+ = 2)= 7 8& 5 2=1&x 0 1, then x 5 4=0 (and x5

1=x52=0), again contradicting Lemma 2.3(a).

Subcase 1.2. x2

4{0 and x

2 3=x

2

2=0. This special case is the symmetry of

Case 1.1. Case 2. x1

2{0 and x13 or x14 is nonzero. In this case, we know, from

Lem-ma 2.3(a), that at least one variable in [x2 2, x

2 3, x

2

4] is nonzero. Thus the sum of

nonzero variables in the last three inequalities of (1) is at most 3&3x0

1+3x01=3,

a contradiction. K

Lemma 2.5. I(6, 8) has no solution.

Proof. Suppose, by contradiction, that I(6, 8) has a solution. Then due to

Lemma 2.3 we can assume that x0

1= 2

7+=, where 0<= 1

21. According to

Lem-ma 2.3(b), we can further assume (without lose of generality), that x1

1>0 and

x2

1>0. If there are more than two nonzero variables in [x j

i: j=1, 2, 2i6], then

the sum of nonzero variables in the last six inequalities of (1) is at most 5&3x0

1+6x016, a contradiction. Hence, for the first two inequalities of (1), each

of them has exactly one nonzero variable besides xj

1, j=0, 1, 2. If there exists an i0

such that xj

i₀{0, j=1, 2, then x j

i₀=0 for 3 j8, since x1i₀+x2i₀>1&x01. Hence,

the sum of the nonzero variables in the last six inequalities of (1) is at most 4+6x0

16, a contradiction. So without loss of generality, we can assume that

x1

1>0, x12>0, and x1i=0, for i{1, 2;

x2 1>0, x 2 3>0, and x 2 i=0, for i{1, 3.

If one of the last six inequalities of (1) has four nonzero variables, then for the rest of the five inequalities, the sum of nonzero variables is at most 5&5x0

1+5x 0 1=5,

a contradiction. Thus, there must exist a nonzero variable xi₀ that appears in

exactly two inequalities. If i0{2, 3, then among the last six inequalities of (1) four

of them do not have nonzero variable xi₀. Thus the sum of nonzero variables is at

most 4&4x0 1+4x

0

1=4, a contradiction. Therefore, without loss of generality, we

assume i0=2, i.e., x32{0 and x j

2=0 for j{1, 3. Note that the sum of nonzero

variables in the last five inequalities of (1) is at most 4&2x0

1+5x015, a

contradic-tion. K

Theorem 2.1. C(n, m, r) is multirate rearrangeable under monotone routing when

(a) 2n4 and m2n+1, or

Proof. It follows from Lemma 1 and above results on system I(n, k). K When comparing this theorem with the best known results (Theorem 5 in [5]), we find that monotone routing shows better performance when n=3, 4 and equal performance when n=2, 5, 6.

3. CONCLUSION

In this paper, we have established a relationship between the rearrangeability of the multirate Clos network C(n, m, r) and a linear system I(n, k) of inequalities through studying monotone routing. This gives an improvement for the upper bound of m for small n. Since small Clos networks are used to be fundamental recursive structure of large networks, this improvement is significant in the hardware optimization of switching networks. For future research, it would be interesting to analyze monotone routing combined with other routings. In addition, as monotone routing does not require any assumption on bandwidth and structures of interconnection networks, the proposed approach used in the Clos network may be extended in other multirate rearrangeable switching networks such as the Benes$ network [1] and the Cantor network [2].

ACKNOWLEDGMENT

The authors thank three anonymous referees for their helpful and insightful comments.

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4. C. Clos, A study of non-blocking switching networks, Bell Systems Technol. J. 32 (1953), 406424. 5. D.-Z. Du, B. Gao, F. K. Hwang, and J. H. Kim, On multirate rearrangeable Clos networks, SIAM

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