This Article 
 Bibliographic References 
 Add to: 
Optimal Processor Mapping for Linear-Complement Communication on Hypercubes
May 2001 (vol. 12 no. 5)
pp. 514-527

Abstract—In this paper, we address the problem of minimizing channel contention of linear-complement communication on wormhole-routed hypercubes. Our research reveals that, for traditional routing algorithms, the degree of channel contention of a linear-complement communication can be quite large. To solve this problem, we propose an alternative approach, which applies processor reordering mapping at compile time. In this compiler approach, processors are logically reordered according to the given communication(s) so that the new communication(s) can be efficiently realized on the hypercube network. It is proved that, for any linear-complement communication, there exists a reordering mapping such that the new communication has minimum channel contention. An $O(n^3)$ algorithm is proposed to find such a mapping for an $n \hbox {-} {\rm dimensional}$ hypercube. An algorithm based on dynamic programming is also proposed to find an optimal reordering mapping for a set of linear-complement communications. Several computer simulations have been conducted and the results clearly show the advantage of the proposed approach.

[1] nCUBE 2 Supercomputers Manual. NCUBE Company, 1990.
[2] Origin™ Servers Technical Report. Silicon Graphics, Inc. 1998.
[3] S. Abraham and K. Padmanabhan, "Performance of the Direct Binary n-Cube Network for Multiprocessors," IEEE Trans. Computers, vol. 38, no. 7, pp. 1000-1011, July 1989.
[4] R. Boppana and C.S. Raghavendra, “Optimal Self-Routing of Linear-Complement Permutations in Hypercubes,” Proc Fifth Distributed Memory Computing Conf., pp. 800-808, Apr. 1990.
[5] H.L. Chen and H.S. Yihng, “Generalized Wormhole Routing Strategies in Hypercubes,” J. Information Science and Eng., vol. 10 pp. 387-341, 1994.
[6] A.A. Chien, “A Cost and Performance Model for$k$-ary$n$-cubes Wormhole Routers,” IEEE Trans. Parallel and Distributed Systems, vol. 9, no. 2, pp. 150-162, Feb. 1998.
[7] G.M. Chiu, S. Chalasani, and C.S. Raghavendra, “Flexible, Fault-Tolerant Routing Criteria for Circuit-Switched Hypercubes,” Proc. IEEE 11th Int'l Conf. Distributed Computing Systems, pp. 582-589, 1991.
[8] T.H. Cormen,C.E. Leiserson, and R.L. Rivest,Introduction to Algorithms.Cambridge, Mass.: MIT Press/McGraw-Hill, 1990.
[9] W.J. Dally, "Performance Analysis of k-ary n-Cube Interconnection Networks," IEEE Trans. Computers, vol. 39, no. 6, pp. 775-785, June 1992.
[10] W.J. Dally and C.L. Seitz, “Deadlock-Free Message Routing in Multiprocessor Interconnection Networks,” IEEE Trans. Computers, Vol. C-36, No. 5, May 1987, pp. 547-553.
[11] C.J. Glass and L.M. Ni, “The Turn Model for Adaptive Routing,” J. ACM, vol. 41, no. 5, pp. 874-902, Sept. 1994.
[12] Y. Hou, C.-M. Wang, and L.-S. Hsu, “Optimal Processor Mapping for Linear-Complement Communication on Hypercubes and Their Variations,” Technical Report TR-IIS-00-014, Inst. Information Science, Academia Sinica, Taiwan, R.O.C., Nov. 2000.
[13] R.C. Gonzalez and R.E. Woods, Digital Image Processing, Addison-Wesley, New York, 1993.
[14] F.T. Leighton,Introduction to Parallel Algorithms and Architectures: Arrays, Trees, Hypercubes.San Mateo, Calif.: Morgan Kaufmann, 1992.
[15] Q. Li, “Minimum Deadlock-Free Message Routing Restrictions in Binary Hypercubes,” J. Parallel and Distributed Computing, vol. 15, no. 2, pp. 153-159, 1992.
[16] F.C. Lin and F.H. Wang, “Minimum Deadlock-Free Message Routing Restrictions in Binary Hypercubes,” J. Parallel and Distributed Computing, vol. 29, no. 2, pp. 27-42, 1995.
[17] H. Masuyama, “Algorithms to Realize an Arbitrary BPC Permutation in Chordal Ring Networks and Mesh Connected Networks,” IEICE Trans. Inf. Syst. (Japan), vol. E77-D, no. 10, pp. 1118-1129, Oct. 1994.
[18] H. Masuyama, Y. Morita, and E. Masuyama, “A Realization of an Arbitrary BPC Permutation in Hypercube Connected Computer Networks,” IEICE Trans. Inf. Syst. (Japan), vol. E78-D, no. 4, pp. 428-435, Apr. 1995.
[19] R.J. McEliece, Finite Fields for Computer Scientists and Engineers. Kluwer Academic, 1987.
[20] D. Nassimi and S. Sahni,“An optimal routing algorithm for mesh-connected parallel computers,”J. ACM, pp. 6–29, 1980.
[21] D. Nassimi and S. Sahni, “Optimal BPC Permutations on a Cube Connected SIMD Computer,” IEEE Trans. Computers, vol. 31, no. 4, pp. 338-341, Apr. 1982.
[22] L.M. Ni and P.K. McKinley, "A Survey of Wormhole Routing Techniques in Direct Networks," Computer, vol. 26, no. 2, pp. 62-76, Feb. 1993.
[23] Y. Saad and M. Schultz, "Topological Properties of Hypercubes," IEEE Trans. Computers, vol. 37, no. 7, pp. 867-872, July 1988.

Index Terms:
Hypercubes, linear-complement communication, channel contention, processor mapping, wormhole routing.
Yomin Hou, Chien-Min Wang, Chiu-Yu Ku, Lih-Hsing Hsu, "Optimal Processor Mapping for Linear-Complement Communication on Hypercubes," IEEE Transactions on Parallel and Distributed Systems, vol. 12, no. 5, pp. 514-527, May 2001, doi:10.1109/71.926171
Usage of this product signifies your acceptance of the Terms of Use.