Accepted 14 March 2000
Abstract
The 3-stage Clos network is generally considered the most basic multistage interconnecting network (MIN). The nonblocking property of such network has been extensively studied in the past. However, there are only a few lower bound results regarding wide-sense nonblocking. We show that in the classical circuit switching environment, to guarantee wide-sense nonblocking for large r, 2n − 1 center switches are necessary, where r is the number of input switches and n is the number of inlets of each input switch. For the multirate environment, we show that for large r, any 3-stage Clos network needs atleast 3n − 2 center switches to guarantee wide-sense nonblocking. Our proof works even for the two-rate environment. c 2001 Elsevier Science
B.V. All rights reserved.
1. Introduction
The 3-stage Clos network is generally considered the most basic multistage intercon-necting network (MIN). It is symmetric with respect to the center stage. The 7rst stage, or the input stage, has rn × m crossbar switches; the center stage has mr × r crossbar switches and the 7nal stage, or the output stage, has rm × n crossbar switches. The n inlets (outlets) on each input (output) switch are the inputs (outputs) of the network. There is exactly one link between every center switch and every input (output) switch. We use C(n; m; r) to denote a 3-stage Clos network. An example of C(3; 3; 4) is shown in Fig. 1.
In the classical circuit switching environment, i.e., every link can only serve one con-nection request, three types of nonblocking properties have been extensively studied.
∗Corresponding author.
E-mail address: [email protected] (D.-W. Wang).
0304-3975/01/$ - see front matter c 2001 Elsevier Science B.V. All rights reserved.
Fig. 1. C(3; 3; 4).
They are strictly nonblocking (SNB), wide-sense nonblocking and rearrangeably non-blocking (RNB). The focus of this paper is to establish lower bounds for the number of center switches needed to guarantee wide-sense nonblocking. A network is wide-sense nonblocking (WSNB) if a new call is always routable as long as all previous requests were routed according to a given routing algorithm.
In the multirate switching environment, a request is a triple (u; v; w), where u is an inlet, v an outlet and w a weight which can be thought of as the bandwidth requirement (rate) of that request. We normalize the weights so that 1¿w¿0, and each link has capacity one; i.e., it can carry any number of calls as long as the sum of weights of these calls does not exceed one.
Clos [3] proved that for the classical model, 2n−1 center switches are necessary and suFcient to guarantee SNB for C(n; m; r). Benes [1, 2] proved that C(n; m; 2) is WSNB (using the packing routing) if and only if m¿3n=2, thus giving hope that WSNB can be achieved with fewer center switches than SNB in general. However recently, Du et al. [4] gave the surprising result that C(n; m; r) for r¿3 is WSNB under the packing routing if and only if m¿2n − 1; namely, it requires the same num-ber of center switches as SNB. In this paper, we further dash the hope by showing that for large r, C(n; m; r) is WSNB (under any routing algorithm) if and only if m¿2n − 1.
While WSNB, as commented above, plays a very restrictive role in the classical model, the multirate environment provides a fertile playground. This is because we have a new dimension, the rate, to design routing algorithm. For example, Gao and Hwang [5] gave a routing algorithm such that C(n; m; r) is WSNB if m¿5:75n. If there are only two diHerent rates, then the requirement reduced to 4n. In this paper we show that 3n − 2 is a lower bound of WSNB under any routing algorithm, and this bound is obtained by using only two diHerent rates. This lower bound provides a gauge to measure how good the algorithm of Gao and Hwang is and how much room is left for improvement. We will also talk about some impacts on repackable algorithms, a relatively new type of nonblocking.
is WSNB if and only if m¿2n − 1.
Proof. Since SNB implies WSNB, it suFces to prove the “only if” part. Suppose the number of center switches, m, is 2n − 2. We shall prove that if
r = (n − 1) 2n − 2 n − 1 + 1;
then the network is not WSNB. In order to prove it, let the network carry a set of initial requests. Then we show that there must exist a set of new requests such that atleast one of them is not routable.
Suppose there are r(n − 1) initial requests, which involve (n − 1) inlets from each input switch but not to a particular output switch O. Since the number of total outputs excluding output switch O is (r − 1)n and r¿n. This is a feasible set of requests. The network then routes these initial requests (according to any routing algorithm). Notice that no two requests from an input switch can share a common center switch. Hence, the n−1 connections from an input switch are routed by a set of n−1 center switches. We say that the set of these center switches is the routing set of that input switch. Since there are 2n − 2 center switches, and the size of each routing set is n − 1, the number of distinct routing set is
2n − 2
n − 1
:
If the number of input switches is large enough, i.e., r = (n − 1) 2n − 2 n − 1 + 1;
then by the pigeon-hole principle, there must exist X , a set of n input switches, which has the same routing set, Y .
Consider a new set of n requests {(x; o): x ∈ X; o ∈ O}. Each of the n requests must be routed through a distinct center switch, which is not in Y . Hence, atleast
|X | + |Y | = 2n − 1 center switches are needed (See Fig. 2).
Fig. 2. Figure for Theorem 1.
Theorem 2. C(n; m; r) is not WSNB for any two rates B; b; B¿b; satisfying B+b¿1 if n¿2; m6 min(k + 2n − 3; 3n − 3) and r is large enough; where k = 1=b. Proof. The condition B + b¿1 forces B¿1
2 and also the consequence that any link carrying a B-request cannot carry a second request, therefore we may assume B = 1. Without loss of generality, assume that b = 1=k for some integer k¿2. We mention that B is too large to share a link with other requests. Another fact is that the number of B-requests is limited as each inlet may generate atmost one B-request. On the contrary, b is small, and an inlet may generate up to k b-requests. To employ these special properties of two rate B; b, we devise a two-phase construction of requests. In Phase 1, only b-requests are generated, and B-requests appear in Phase 2. Since a B-request cannot share a link with a b-request, any link used in Phase 1 is no longer available in Phase 2.
The object of Phase 1 is to spread out some b-requests to some common center switches. On the other hand, we also hope that the input switch maintain the capability to generate the maximum number of B-requests in Phase 2. Hence, in our construc-tion, only the 7rst inlet of each input switch can generate requests in Phase 1. The construction in Phase 1 can be further divided into several steps. In each step, one b-request is generated in the 7rst inlet of the input switches. The exact number of steps is bounded by two inequalities, which depend on the relation between k and n, and will be examined later.
Fig. 3. Figure for step t of Theorem 2.
Step 1: Each input generates one request. Since r is large enough, there exists atleast a large set I1 of input switches whose requests are all routed through the same center switch M1.
Step 2: Each input in I1 generates a second request. Partition I1 into size k + 1 groups such that inputs in the same group all have their second requests going to the same fresh output switch O. This is possible since b(k + 1)6n, the capacity of a fresh output switch. But the total weights of each group, b(k + 1), require atleast a new center switch M2 to carry since the link between M1 and O can carry only bk. Therefore in each group atleast one input switch whose second request is carried by some center switch other than M1. Since I1 is large there exists a large set I2 of input switches whose requests are routed through the same set of center switches M1; M2.
Step t: Each input in It−1 generates a tth request. Partition It−1 into size (t −1)k +1 groups such that inputs in the same group all make the tth request to the same fresh output, this is possible if (t − 1)k + 16nk. But the total weights of each group, (t − 1)bk + b, is greater than t − 1, the total capacity between the speci7ed output switch and M1; M2; : : : ; Mt−1. Therefore, in each group atleast one input switch whose tth request is carried by some center switch other than M1; M2; : : : ; Mt−1. Since It−1 is large there exists atleast a large set It of input switches whose requests are routed through the same set of center switches (see Fig. 3).
The number of steps is controlled by the following two inequalities. Firstly, there is atmost k iterations, as the 7rst inlet can generate not more than k b-request. Secondly, these requests coming from the same group are directed to an output switch. Hence, the size of the group as well as the number of steps are bounded. Suppose that Step t is the last step. Then, t6k and (t − 1)k + 16nk. It can be veri7ed that t = min(k; n).
In Phase 2, Consider only the input switches in It. Since each switch is able to generate n − 1 B-requests going to a set of fresh output switches, which are able to receive n B-requests. By applying the same argument in Theorem 1, we may argue that 2n − 2 center switches are needed. Notice that M1; M2; : : : ; Mt−1 are not availabe in this phase as the links between them and It already carried a b-request, which was generated in Phase 1. Hence, a total of t + 2n − 2 or min(k; n) + 2n − 2 center switches are necessary.
Corollary 1. In a multirate environment; C(n; m; r) is not WSNB if m63n − 3 and r is large enough.
3. Some concluding remarks
A network is repackable if existing calls can be rearranged at any moment when a connection is deleted (e.g., a call hangs up). Repacking algorithms have been studied for both the classical model [6] and the multirate model [7, 8] to show that they can help to reduce the number of center switches needed for C(n; m; r) to be nonblocking. Theorems 1 and 2 imply that no repacking algorithm can be eHective when r is large. Since the request sequences we construct in these theorems have no deletion, these results apply to all repacking algorithms. Thus, our results solidly con7rm a major diHerence between {WSNB, Repackable} and {SNB, RNB}, namely, the numbers of center switches required for C(n; m; r) are dependent on r for the former pair, but independent from r for the latter pair. In other words, the hope of 7nding an eHective WSNB or repackable algorithm is restricted to small r.
References
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