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<p><b>Abstract</b>—In this paper, we study wide-sense nonblocking conditions under packing strategy for the three-stage Clos network, or <tmath>$v(m,n,r)$</tmath> 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 [<ref rid="bibT02652" type="bib">2</ref>]) proved that a <tmath>$v(m,n,2)$</tmath> network is nonblocking under packing strategy if the number of middle stage switches <tmath>$m \geq \left\lfloor{3 \over 2}n\right\rfloor$</tmath>. 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 <tmath>$v(m,n,r)$</tmath> networks in some papers, such as [<ref rid="bibT02657" type="bib">7</ref>]. In fact, it is still not known that whether the condition <tmath>$m \geq \left\lfloor {3 \over 2}n\right\rfloor$</tmath> holds for <tmath>$v(m,n,r)$</tmath> networks when <tmath>$r \geq 3$</tmath>. In this paper, we introduce a systematic approach to the analysis of wide-sense nonblocking conditions for general <tmath>$v(m,n,r)$</tmath> networks with any <tmath>$r$</tmath> value. We first translate the problem of finding the nonblocking condition under packing strategy for a <tmath>$v(m,n,r)$</tmath> 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 <tmath>$v(m,n,r)$</tmath> network to be nonblocking under packing strategy is <tmath>$ m \geq \left\lfloor\left(2 - \displaystyle{{1} \over {F_{2r-1}}}\right)n\right\rfloor$</tmath>, where <tmath>$F_{2r-1}$</tmath> is the Fibonacci number. In the case of <tmath>$n \leq F_{2r-1}$</tmath>, 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.</p>
Interconnection networks, wide-sense nonblocking, routing control strategies, packing, linear programming, Fibonacci numbers.

J. Wang and Y. Yang, "Wide-Sense Nonblocking Clos Networks Under Packing Strategy," in IEEE Transactions on Computers, vol. 48, no. , pp. 265-284, 1999.
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