This Article 
   
 Share 
   
 Bibliographic References 
   
 Add to: 
 
Digg
Furl
Spurl
Blink
Simpy
Google
Del.icio.us
Y!MyWeb
 
 Search 
   
The KR-Benes Network: A Control-Optimal Rearrangeable Permutation Network
May 2005 (vol. 54 no. 5)
pp. 534-544
Web Extra: View supplemental material
The Benes network has been used as a rearrangeable network for over 40 years, yet the uniform N(2 \log N-1) control complexity of the N \times N Benes is not optimal for many permutations. In this paper, we present a novel O(\log N) depth rearrangeable network, called KR-Benes, that is permutation-specific control-optimal. The KR-Benes routes every permutation with the minimal control complexity specific to that permutation and its worst-case complexity for arbitrary permutations is bounded by the Benes; thus, it replaces the Benes when considering control complexity/latency. We design the KR-Benes by first constructing a restricted 2 \log K +2 depth rearrangeable network called K{\hbox{-}}{\rm Benes} for routing K{\hbox{-}}{\rm bounded} permutations with control 2N \log K, 0 \leq K \leq N/4. We then show that the N \times N Benes network itself (with one additional stage) contains every K{\hbox{-}}{\rm Benes} network as a subgraph and use this property to construct the KR-Benes network. With regard to the control-optimality of the KR-Benes, we show that any optimal network for rearrangeably routing K{\hbox{-}}{\rm bounded} permutations must have depth 2 \log K + 2 and, therefore, the K{\hbox{-}}{\rm Benes} (and, hence, the KR-Benes) is optimal.

[1] V.E. Benes, “Optimal Rearrangeable Multistage Connecting Networks,” Bell System Technical J., vol. 43, pp. 1641-1656, 1964.
[2] F.T. Leighton, Introduction to Parallel Algorithms and Architectures: Arrays, Trees, Hypercubes. San Mateo, Calif.: Morgan Kaufmann, 1992.
[3] D. Nassimi and S. Sahni, “A Self-Routing Benes Network and Parallel Permutation Algorithms,” IEEE Trans. Computers, vol. 30, no. 5, pp. 332-340, May 1981.
[4] S. Keshav, An Engineering Approach to Compter Networking. Addison-Wesley, 1997.
[5] B. Prabhakar and N. McKeown, “On the Speedup Required for Combined Input and Output Queued Switching,” Automatica, vol. 35, no. 12, pp. 1909-1920, 1999.
[6] J. Duato and L. Ni, Interconnection Networks. Los Alamitos, Calif.: IEEE CS Press, 1999.
[7] D. Nassimi and S. Sahni, “Parallel Algorithms to Set Up the Benes Permutation Network,” IEEE Trans. Computers, vol. 31, no. 2, pp. 148-154, Feb. 1982.
[8] A. Pattavina and G. Maier, “Photonic Rearrangeable Networks with Zero Switching-Element Crosstalk,” Proc. IEEE INFOCOM 1999, pp. 337-344, Apr. 1999.
[9] Y. Yang, J. Wang, and Y. Pan, “Permutation Capability of Optical Multistage Interconnection Networks,” J. Parallel and Distributed Computing, vol. 60, no. 1, pp. 72-91, Jan. 2000.
[10] A. Waksman, “A Permutation Network,” J. ACM, vol. 15, no. 1, pp. 159-163, Jan. 1968.
[11] K.Y. Lee, “On the Rearrangeability of ($2\log N - 1$ ) Stage Permutation Networks, IEEE Trans. Computers, vol. 34, no. 5, pp. 412-425, May 1985.
[12] D.C. Opferman and N.T. Tsao-Wu, “On a Class of Rearrangeable Switching Networks, Part I: Control Algorithm,” Bell System Technical J., vol. 50, pp. 1579-1600, 1971.
[13] M.K. Kim, H. Yoon, and S.R. Maeng, “Bit-Permute Multistage Interconnection Networks,” Microprocessing and Microprogramming, vol. 41, pp. 449-468, 1995.
[14] Y.-M. Yeh and T.-Y. Feng, “On a Class of Rearrangeable Networks,” IEEE Trans. Computers, vol. 41, no. 11, pp. 1361-1379, Nov. 1992.
[15] D.M. Koppelman and A.Y. Oruc, “A Self-Routing Permutation Network,” J. Parallel and Distributed Computing, vol. 10, no. 2, pp. 140-151, 1990.
[16] C. Jan and Y. Oruc, “Fast Self-Routing Permutation Switching on an Asymptotically Minimum Cost Network,” IEEE Trans. Computers, vol. 42, no. 12, pp. 1369-1379, Dec. 1993.
[17] H. Cam and J.A.B. Fortes, “A Fast VLSI-Efficient Self-Routing Permutation Network,” IEEE Trans. Computers, vol. 44, no. 3, pp. 448-453, Mar. 1995.
[18] A. Samsudin and K.Y. Lee, “nD-dBPN: New Self-Routing Permutation Networks Based on the de Bruijn Digraphs,” Proc. Int'l Conf. Parallel Processing, pp. 604-611, 1998.
[19] E. Lu and S.Q. Zheng, “A Fast Parallel Routing Algorithm for Benes Group Switches,” Proc. 14th IASTED Int'l Conf. Parallel and Distributed Computing and Systems, pp. 67-72, Nov. 2002.
[20] S.-W. Seo, T.-Y. Feng, and H.-I. Lee, “Permutation Realizability and Fault Tolerance Property of the Inside-Out Routing Algorithm,” IEEE Trans. Parallel and Distributed Systems, vol. 10, no. 9, pp. 946-957, 1999.
[21] T.-Y. Feng and S.-W. Seo, “A New Routing Algorithm for a Class of Rearrangeable Networks,” IEEE Trans. Computers, vol. 43, no. 11, pp. 1270-1280, 1994.
[22] M.K. Kim, H. Yoon, and S.R. Maeng, “On the Correctness of Inside-Out Routing Algorithm,” IEEE Trans. Computers, vol. 46, no. 7, pp. 820-823, July 1997.

Index Terms:
Benes network, rearrangeability, optimal control algorithm.
Citation:
Rajgopal Kannan, "The KR-Benes Network: A Control-Optimal Rearrangeable Permutation Network," IEEE Transactions on Computers, vol. 54, no. 5, pp. 534-544, May 2005, doi:10.1109/TC.2005.84
Usage of this product signifies your acceptance of the Terms of Use.