Publication 1999 Issue No. 3 - March Abstract - Wide-Sense Nonblocking Clos Networks Under Packing Strategy
Wide-Sense Nonblocking Clos Networks Under Packing Strategy
March 1999 (vol. 48 no. 3)
pp. 265-284
 ASCII Text x 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.
 BibTex x @article{ 10.1109/12.754994,author = {Yuanyuan Yang and Jianchao Wang},title = {Wide-Sense Nonblocking Clos Networks Under Packing Strategy},journal ={IEEE Transactions on Computers},volume = {48},number = {3},issn = {0018-9340},year = {1999},pages = {265-284},doi = {http://doi.ieeecomputersociety.org/10.1109/12.754994},publisher = {IEEE Computer Society},address = {Los Alamitos, CA, USA},}
 RefWorks Procite/RefMan/Endnote x TY - JOURJO - IEEE Transactions on ComputersTI - Wide-Sense Nonblocking Clos Networks Under Packing StrategyIS - 3SN - 0018-9340SP265EP284EPD - 265-284A1 - Yuanyuan Yang, A1 - Jianchao Wang, PY - 1999KW - Interconnection networksKW - wide-sense nonblockingKW - routing control strategiesKW - packingKW - linear programmingKW - Fibonacci numbers.VL - 48JA - IEEE Transactions on ComputersER -

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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