The art of computer programming, volume 1 (3rd ed.): fundamental algorithms
The art of computer programming, volume 1 (3rd ed.): fundamental algorithms
ISCA '85 Proceedings of the 12th annual international symposium on Computer architecture
Interconnection Networks for Multiprocessors and Multicomputers: Theory and Practice
Interconnection Networks for Multiprocessors and Multicomputers: Theory and Practice
An Analytical Model on the Blocking Probability of a Fault-Tolerant Network
IEEE Transactions on Parallel and Distributed Systems
A "Single-Box" Re-routing Architecture for a 3-Stage Rearrangeable CLOS Interconnection Networks
PARA '02 Proceedings of the 6th International Conference on Applied Parallel Computing Advanced Scientific Computing
IEEE Transactions on Parallel and Distributed Systems
Blocking behaviors of crosstalk-free optical Banyan networks on vertical stacking
IEEE/ACM Transactions on Networking (TON)
A Fault-Tolerant Rearrangeable Permutation Network
IEEE Transactions on Computers
Theoretical Computer Science
Fast simulation of wavelength continuous WDM networks
IEEE/ACM Transactions on Networking (TON)
Wide-sense nonblocking for multi-logd N networks under various routing strategies
Theoretical Computer Science
On the routing algorithms for optical multi-log2N networks
NPC'07 Proceedings of the 2007 IFIP international conference on Network and parallel computing
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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.