This Article 
 Bibliographic References 
 Add to: 
Efficient Routing and Sorting Schemes for de Bruijn Networks
November 1997 (vol. 8 no. 11)
pp. 1157-1170

Abstract—We consider the problems of routing and sorting on a de Bruijn network. First, we show that any deterministic oblivious routing scheme for permutation routing on a d-ary de Bruijn network with N = dn nodes, in the worst case, will take $\Omega (\sqrt N)$ steps under the single-port model. This improves the existing lower bounds provided d is not a constant. We also show that the lower bound is indeed a tight one. Second, we present a deterministic nonoblivious permutation routing algorithm which runs in O(d·n2) time on a d-ary de Bruijn network with N = dn nodes. This algorithm is currently the fastest known nonoblivious deterministic routing algorithm for de Bruijn networks of arbitrary degree. Finally, we present an efficient general sorting algorithm for the de Bruijn networks of arbitrary degree. This algorithm is the best sorting algorithm known so far. It runs in O((log d) ·d·n2) time for directed de Bruijn network with dn nodes, degree d, and diameter n. As a corollary, we show that on a binary de Bruijn network of N nodes, our sorting scheme requires at most 2 log2N steps.

[1] S. Akers, D. Harel, and B. Krishnamurthy, "The Star Graph: An Attractive Alternative to the n-Cube," Proc. Int'l Conf. Parallel Processing, pp. 393-400, 1987.
[2] S.G. Akl, Parallel Sorting Algorithms.Orlando, Fla.: Academic Press Inc., 1985.
[3] R. Aleliunas, "Randomized Parallel Communication," ACM-SIGOPS Symp. Principles of Distributed Systems, pp. 60-72, 1982.
[4] K. Batcher, "Sorting Networks and Their Applications," Proc. AFIPS Spring Joint Computing Conf., pp. 307-314, 1968.
[5] A. Borodin and J.E. Hopcroft, “Routing, Merging and Sorting on Parallel Models of Computation,” Proc. STOC, pp. 338-344, 1982.
[6] M. Blum, R.W. Floyd, V. Pratt, R.L. Rivest, and R.E. Tarjan, "Time Bounds for Selection," J. Computer and System Sciences, vol. 7, no. 4, pp. 448-461, 1973.
[7] J.C. Bermond and C. Peyrat, "de Bruijn and Kautz Networks: A Competitor for the Hypercube?," Hypercube and Distributed Computers, F. André, and J.P. Verjus, eds., pp. 279-293. Elsevier Science Publishers, 1989.
[8] S.B. Choi and A.K. Somani, "Rearrangeable Circuit-Switched Architectures for Routing Permutations," J. Parallel and Distributed Computing, vol. 19, no. 2, pp. 125-130, Oct. 1993.
[9] O. Collins, S. Dolinar, R. McEliece, and F. Pollara, "A VLSI Decomposition of the De Bruijn Graph," J. ACM, vol. 39, pp. 931-948, Oct. 1992..
[10] T.H. Cormen,C.E. Leiserson, and R.L. Rivest,Introduction to Algorithms.Cambridge, Mass.: MIT Press/McGraw-Hill, 1990.
[11] R. Cypher and C.G. Plaxton, "Deterministic Sorting in Nearly Logarithmic Time on the Hypercube and Related Computers," J. Computer and System Sciences, vol. 47, pp. 501-548, Dec. 1993.
[12] N.G. de Bruijn, "A Combinatorial Problem," Proc. Akademe Van Wetenschappen, vol. 49, pp. 758-764, 1946.
[13] A.H. Esfahanian and S.L. Hakimi, “Fault-Tolerant Routing in de Bruijn Communication Networks,” IEEE Trans. Computers, vol. 34, no. 9, pp. 777-788, Sept. 1985.
[14] M.D. Grammatikakis, D.F. Hsu, and F.K. Hwang, "Adaptive and Oblivious Algorithms for D-Cube Permutation Routing," Algorithms and Computation,. LNCS, no. 762, pp. 167-175. Springer-Verlag, 1993.
[15] D.F. Hsu and D.S.L. Wei, "Permutation Routing and Sorting on Directed de Bruijn Networks," Proc. Int'l Conf. Parallel Processing, vol. 1, Architecture, pp. 96-100, P. Banerjee, ed. CRC Press, 1995.
[16] G. Waldman, J. Wootton, and G. Hobson, “Visual Detection with Search: An Empirical Model,” IEEE Trans. Systems, Man, and Cybernetics, vol. 21, pp. 596-606, 1991.
[17] D. Knuth, The Art of Computer Programming, vol. 3: Sorting and Searching. Addison-Wesley, 1973.
[18] F.T. Leighton,Introduction to Parallel Algorithms and Architectures: Arrays, Trees, Hypercubes.San Mateo, Calif.: Morgan Kaufmann, 1992.
[19] F.T. Leighton,B. Maggs, and S. Rao,"Universal Packet Routing Algorithms," Proc. 29th Ann. Symp. Foundations of Computer Science, pp. 256-269, 1988.
[20] Z. Liu and T.Y. Sung, "Routing and Transmitting Problems in de Bruijn Networks," IEEE Trans. Computers, vol. 45, no. 9, pp. 1,056-1,062, Sept. 1996.
[21] A. Menn and A.K. Somani, "An Efficient Sorting Algorithm for the Star Graph Interconnection Network," Proc. Int'l Conf. Parallel Processing, vol. 3, pp. 1-8, 1990.
[22] M. Palis, S. Rajasekaran, and D.S.L. Wei, “Packet Routing and PRAM Emulation on Star Graphs and Leveled Networks,” J. Parallel and Distributed Computing, vol. 20, no. 2, pp. 145-157, Feb. 1994.
[23] S. Rajasekaran and D.S.L. Wei, "Selection, Routing and Sorting on the Star Graph," J. Parallel and Distributed Computing, vol. 41, pp. 225-233, Apr. 1997.
[24] M.R. Samatham and D.K. Pradhan, "The de Bruijn Multiprocessor Network: A Versatile Parallel Processing and Sorting Network for VLSI," IEEE Trans. Computers, vol. 38, no. 4, pp. 567-581, Apr. 1989.
[25] L.G. Valiant and G.J. Brebner,"Universal Schemes for Parallel Communication," Proc. 13th Ann. ACM Symp. Theory of Computing, pp. 263-277, May 1981.
[26] D.S.L. Wei, "Quicksort and Permutation Routing on the Hypercube and de Bruijn Networks," Proc. Int'l Symp. Parallel Architectures, Algorithms, and Networks, pp. 238-245, S. Horiguchi et. al, eds. IEEE CS Press, 1994.

Index Terms:
Oblivious routing, nonoblivious routing, permutation routing, parallel sorting, de Bruijn network.
D. Frank Hsu, David S.L. Wei, "Efficient Routing and Sorting Schemes for de Bruijn Networks," IEEE Transactions on Parallel and Distributed Systems, vol. 8, no. 11, pp. 1157-1170, Nov. 1997, doi:10.1109/71.642950
Usage of this product signifies your acceptance of the Terms of Use.