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Wide-Sense Nonblocking Clos Networks Under Packing Strategy
March 1999 (vol. 48 no. 3)
pp. 265-284

Abstract—In this paper, we study wide-sense nonblocking conditions under packing strategy for the three-stage Clos network, or $v(m,n,r)$ network. Wide-sense nonblocking networks are generally believed to have lower network cost than strictly nonblocking networks. However, the analysis for the wide-sense nonblocking conditions is usually more difficult. Moore (cited in Benes' book [2]) proved that a $v(m,n,2)$ network is nonblocking under packing strategy if the number of middle stage switches $m \geq \left\lfloor{3 \over 2}n\right\rfloor$. This result has been widely cited in the literature, and is even considered as the wide-sense nonblocking condition under packing strategy for the general $v(m,n,r)$ networks in some papers, such as [7]. In fact, it is still not known that whether the condition $m \geq \left\lfloor {3 \over 2}n\right\rfloor$ holds for $v(m,n,r)$ networks when $r \geq 3$. In this paper, we introduce a systematic approach to the analysis of wide-sense nonblocking conditions for general $v(m,n,r)$ networks with any $r$ value. We first translate the problem of finding the nonblocking condition under packing strategy for a $v(m,n,r)$ network to a set of linear programming problems. We then solve this special type of linear programming problems and obtain a closed form optimum solution. We prove that the necessary condition for a $v(m,n,r)$ network to be nonblocking under packing strategy is $ m \geq \left\lfloor\left(2 - \displaystyle{{1} \over {F_{2r-1}}}\right)n\right\rfloor$, where $F_{2r-1}$ is the Fibonacci number. In the case of $n \leq F_{2r-1}$, this condition is also a sufficient nonblocking condition for packing strategy. We believe that the systematic approach developed in this paper can be used for analyzing other wide-sense nonblocking control strategies as well.

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Index Terms:
Interconnection networks, wide-sense nonblocking, routing control strategies, packing, linear programming, Fibonacci numbers.
Yuanyuan Yang, Jianchao Wang, "Wide-Sense Nonblocking Clos Networks Under Packing Strategy," IEEE Transactions on Computers, vol. 48, no. 3, pp. 265-284, March 1999, doi:10.1109/12.754994
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