This Article 
 Bibliographic References 
 Add to: 
Multinode Broadcast in Hypercubes and Rings with Randomly Distributed Length of Packets
February 1993 (vol. 4 no. 2)
pp. 144-154

Multinode broadcast (MNB) in a hypercube and in a ring network of processors isconsidered. It is assumed that the lengths of the packets that are broadcast are notfixed, but are distributed according to some probabilistic rule, and the optimal timesrequired to execute the MNB are compared for variable and for fixed packet lengths. Forlarge hypercubes, it is shown, under very general probabilistic assumptions on the packetlengths, that the MNB is completed in essentially the same time as when the packetlengths are fixed. In particular, the MNB is completed by time (1+ delta )T/sub s/ withprobability at least 1- epsilon , for any positive epsilon and delta , where T/sub s /is theoptimal time required to execute the MNB when the packet lengths are fixed at theirmean, provided that the size of the hypercube is large enough. In the case of the ring, itis proved that the average time required to execute a MNB when the packet lengths areexponentially distributed exceeds by a factor of ln n the corresponding time for the casethere the packet lengths are fixed at their mean, where n is the number of nodes of thering.

[1] D. P. Bertsekas and J. N. Tsitsiklis,Parallel and Distributed Computations. Englewood Cliffs, NJ: Prentice-Hall, 1989.
[2] D. Bertsekas, C. Ozveren, G. Stamoulis, P. Tseng, and J. Tsitsiklis, "Optimal communication algorithms for hypercubes,"J. Parallel Distributed Comput., vol. 11, pp. 263-275, 1991.
[3] S. N. Bhatt and I. C. F. Ipsen, "How to embed trees in hypercubes," Yale Univ., Dep. Comput. Sci., Res. Rep. YALEU/DCS/RR-443, 1985.
[4] 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.
[5] E. Dekel, D. Nassimi, and S. Sahni, "Parallel matrix and graph algorithms,"SIAM J. Comput., vol. 10, pp. 657-673, 1981.
[6] I. S. Gradshteyn and I. M. Ryzhik,Table of Integrals. Series, and Products. Orlando, FL, Academic, 1980.
[7] S. M. Hedetniemi, S. T. Hedetniemi, and A. L. Liestman, "A survey of gossiping and broadcasting in communication networks,"Networks, vol. 18, pp. 319-349, 1988.
[8] S. L. Johnsson, "Communication efficient basic linear algebra computations on hypercube architectures,"J. Parallel Distributed Comput., pp. 133-172, 1987.
[9] S. L. Johnsson and C. T. Ho, "Optimum broadcasting and personalized communication in hypercubes,"IEEE Trans. Comput., vol. 38, pp. 1249-1268, 1989.
[10] D. W. Krumme, K. N. Venkataraman. and G. Cybenko. "The token exchange problem," Tufts Univ., Tech. Rep. 88-2, 1988.
[11] O. A. McBryan and E. F. Van de Velde, "Hypercube algorithms and implementations,"SIAM J. Scientif. Statist. Comput., vol. 8, no. 2, pp. s227-s287, Mar. 1987.
[12] C. Ozveren, "Communication aspects of parallel processing," Lab. for Information and Decision Systems Rep. LIDS-P-1721, M.I.T., Cambridge, MA, 1987.
[13] S. M. Ross,Stochastic Processes. New York: Wiley, 1983.
[14] Y. Saad and M. H. Schultz, "Data communication in hypercubes,"J. Parallel Distributed Comput., vol. 6, pp. 115-135, 1989.
[15] Y. Saad and M. H. Schultz, "Data communication in parallel architectures," Yale Univ., Report, Mar. 1986.
[16] Y. Saad and M. H. Schultz, "Topological properties of hypercubes."IEEE Trans. Comput., vol. 37, pp. 867-872, 1988.
[17] G. Stamoulis and J. Tsitsiklis, "Efficient routing schemes for multiple broadcasts in hypercubes," Lab. for Information and Decision Systems, Rep. LIDS-P-1948, Feb. 1990.
[18] O. F. Stout and B. Wagar, "Passing messages in link-bound hypercubes," inProc. 1986 Hypercube Conf., SIAM, Philadelphia, PA, 1987, pp. 251-257.
[19] D. M. Topkis. "Concurrent broadcast for information dissemination,"IEEE Trans. Software Eng., vol. 13, pp. 207-231, 1983.
[20] E. A. Varvarigos, "Optimal communication algorithms for multiprocessor computers," M.S. thesis, Dept. of Elec. Eng. and Comput. Sci., M.I.T.; also Center for Intelligent Control Systems Rep., CICS-TH-192, Jan. 1990.
[21] E. A. Varvarigos and D. P. Bertsekas, "Optimal communication algorithms for isotropic tasks in hypercubes and wraparound meshes," Lab. for Information and Decision Systems. Rep. LIDS-P-1972, May 1990.

Index Terms:
Index Termsmultinode broadcast; hypercubes; rings; randomly distributed length of packets;probabilistic rule; hypercube networks; token networks
E.A. Varvarigos, D.T. Bertsekas, "Multinode Broadcast in Hypercubes and Rings with Randomly Distributed Length of Packets," IEEE Transactions on Parallel and Distributed Systems, vol. 4, no. 2, pp. 144-154, Feb. 1993, doi:10.1109/71.207590
Usage of this product signifies your acceptance of the Terms of Use.