454 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 46, NO. 4, APRIL 1998
Choosing the Best
k
(N, m, P) Strictly Nonblocking Networks
F. K. Hwang
Abstract—We extend thelog2(N; m; P ) network proposed by Shyy and Lea to basek. We give a unifying proof (instead of three separate cases as done by Shyy and Lea) for the condition of being strictly nonblocking, and a simpler expression of the result. We compare the number of crosspoints forlogk(N; m; p) over various k.
Index Terms— Banyan network, Clos network, strictly non-blocking network.
I. INTRODUCTION
T
HE NOTION of a network for designing photonic switching systems was introduced by Lea in [1] and by Shyy and Lea in [2]. Following [2], we use theBanyan network as representative. An -extra-stage Banyan
network is a cascade of the Banyan network with extra stages which are the mirror image of the first stages of the Banyan network. A network can be treated as a symmetrical three-stage Clos network with inlets, while the middle stage consists of copies of an -extra-stage Banyan network, and the first stage consists of copies of a crossbar.
While a Banyan network usually uses crossbars as components, it can be easily extended to a -nary Banyan network using crossbars. Fig. 1 shows a ternary Banyan network with inlets.
By using the -nary Banyan networks in the middle stage, we can define a network with inlets, where the input stage consists of crossbars and the output stage crossbars.
The value of to guarantee strict nonblockingness of the network was given in [1] and [2]. Their arguments, divided into three cases also hold for the network by simply replacing base 2 with base . We give a simpler proof by unifying the three cases (also a simpler expression for ).
Theorem 1: A network is strictly nonblock-ing if
for even for odd
Proof: Suppose . Define if is even and otherwise. We use an argument analogous to the one given in [1] for the case.
Paper approved by N. McKeown, the Editor for Switching and Routing of the IEEE Communications Society. Manuscript received October 16, 1996; revised September 12, 1997.
The author is with the Department of Applied Mathematics, Chiao-Tung University, Hsin-Chu, Taiwan 30050, R.O.C.
Publisher Item Identifier S 0090-6778(98)03136-5.
Consider the channel graph between an input and an output . From the structure of and the pattern , it is easily verified that is a symmetric series-parallel channel graph with branching at the outer shells. Let denote the number of paths at shell . Then
for for
A stage- link may also be seized by a connection where and . We call such a connection an
intersecting connection. To avoid counting twice, we must
assign such an intersecting connection either to or to . We assign it to the input side of inputs (outputs) which can generate an intersecting connection seizing a shell- link. Then
for except
Assuming the worst case that the and intersecting connections are all disjoint, then a portion
of the paths in is unavailable to . Therefore, the condition of SNB is
Theorem 1 follows immediately.
II. MINIMIZING THE NUMBER OF CROSSPOINTS
Let # denote the number of crosspoints. We first compare strictly nonblocking over various for # by keeping invariant. It is tacidly assumed that can be approximated by a power of so that the formula in Theorem 1 applies. We will also ignore the integrality of .
An -extra-stage Banyan network has stages each consisting of crossbars; therefore, it has
crosspoints. Thus, has crosspoints in the middle and crossbars in each of its input and output stages. Thus
#
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 46, NO. 4, APRIL 1998 455
Fig. 1. A ternary Banyan network with 27 inlets.
Case 1: is even. Then #
Let denote a positive integer much smaller than . If , then
#
which is essentially increasing in for or and decreasing in for , where “essentially increasing” means exceptions are allowed for very small . If
then #
which is essentially increasing in for all .
Case 2: is odd. Then #
If , then #
which is essentially increasing in for but decreasing in for . If , then
#
The analysis is the same as in case 1.
When # is essentially increasing in , the optimal is a small which can be determined by standard method. When # is decreasing in , the optimal should be as large as prac-ticality allows. Note that the optimal is independent of . Also note that requires crosspoints for but only crosspoints for
( yields a variation of the Cantor network). Sometimes, for a technology or performance reason, it is necessary to keep constant. Then varies with and will be denoted by . In this case one should compare # , the number of crosspoints per input (or output). We have previously shown
#
It is easily verified that as given in Theorem 1 is increasing in . Hence, # is increasing in , and the optimal choice
of is .
Since there is no a priori reason to argue for being the optimal choice, one expects # to consist of two factors, one increasing in and the other decreasing, and the optimal is determined by balancing these two factors. It is surprising to find both factors in # increasing in .
REFERENCES
[1] C.-T. Lea, “Multi-Log2N networks and their applications in high-speed electronic and photonic switching systems,” IEEE Trans. Commun., vol. 38, pp. 1740–1749, Oct. 1990.
[2] D.-J. Shyy and C.-T. Lea, “Log2(N; m; p) strictly nonblocking net-works,” IEEE Trans. Commun., vol. 39, pp. 1502–1510, Oct. 1991.
