This Article 
   
 Share 
   
 Bibliographic References 
   
 Add to: 
 
Digg
Furl
Spurl
Blink
Simpy
Google
Del.icio.us
Y!MyWeb
 
 Search 
   
Placement of the Processors of a Hypercube
June 1991 (vol. 40 no. 6)
pp. 714-722

The authors formalize the problem of minimizing the length of the longest interprocessor wire as the problem of embedding the processors of a hypercube onto a rectangular mesh, so as to minimize the length of longest wire. Where neighboring nodes of the mesh are taken as being at unit distance from one another, and where wires are constrained to be laid out as horizontal and vertical wires, the length of the wire joining nodes u and v of the mesh equals the graph-theoretic distance between u and v. The problem of minimizing delays due to interprocessor communication is then modeled as the problem of embedding the vertices of a hypercube onto the nodes of a mesh, so as to minimize dilation. Two embeddings which achieve dilations that (for large n) are within 26% of the lower bound for square meshes and within 12% for meshes with aspect ratio 2 are presented.

[1] R. Aleliunas and A. L. Rosenberg, "On embedding rectangular grids in square grids,"IEEE Trans. Comput., vol. C-31, pp. 907-913, 1982.
[2] S. Bhatt, F. Chung, T. Leighton, and A. Rosenberg, "Optimal simulations of tree machines," inProc. 27th Annu. IEEE Symp. Foundations Comput. Sci., 1986, pp. 272-282.
[3] G. S. Bloom and S. W. Golomb, "Applications of numbered undirected graphs,"Proc. IEEE, vol. 65, pp. 562-569, 1977.
[4] M. Y. Chan, "Dilation-2 embeddings of grids into hypercubes," inProc. 1988 Int. Conf. Parallel Processing, vol. 3, 1988, pp. 295-298.
[5] M. Y. Chan and F. Y. L. Chin, "On embedding rectangular grids in hypercubes,"IEEE Trans. Comput., vol. 37, pp. 1285-1288, 1988.
[6] P. Z. Chinn, J. Chvatalova, A. K. Dewdney, and N. E. Gibbs, "The bandwidth problem for graphs and matrices--A survey,"J. Graph Theory, vol. 6, pp. 223-254, 1982.
[7] W. J. Dally, "Wire efficient VLSI multiprocessor communication networks," inProc. Stanford Conf. Advanced Res. VLSI, 1987, pp. 391-415.
[8] W. Feller,An Introduction to Probability Theory and Its Applications, Vol. 1. New York: Wiley, 1950.
[9] M. J. Flynn, "Very high-speed computer systems,"Proc. IEEE, vol. 54, pp. 1901-1909, 1966.
[10] M. R. Garey, R. L. Graham, D. S. Johnson, and D. E. Knuth, "Complexity results for bandwidth minimization,"SIAM J. Appl. Math., vol. 34, pp. 477-495, 1978.
[11] L. H. Harper, "Optimal assignments of numbers to vertices,"J. Soc. Industrial Appl. Math, vol. 12, pp. 131-135, 1964.
[12] L. H. Harper, "Optimal numberings and isoperimetric problems on graphs,"J. Combinatorial Theory, vol. 1, pp. 385-393, 1966.
[13] L. H. Harper, "Chassis layout and isoperimetric problems,"Jet Propulsion Laboratory Space Projects Summary 37-66, vol. 2, pp. 37-42, 1970.
[14] C.-T. Ho and S. L. Johnsson, "On the embedding of meshes in Boolean cubes," inProc. 1987 Int. Conf. Parallel Processing, 1987, pp. 188-191.
[15] D. Kratsch, "Finding the minimum bandwidth of an interval graph,"Inform. Computat., vol. 74, pp. 140-158, 1987.
[16] T.-H. Lai and W. White, "Mapping pyramid algorithms into hypercubes,"J. Parallel Comput., vol. 8, 1990.
[17] Y. E. Ma and L. Tao, "Embeddings among toruses and meshes," inProc. 1987 Int. Conf. Parallel Processing, 1987, pp. 178-187.
[18] G. Mitchison and R. Durbin, "Optimal numberings of an n×n array,"SIAM J. Algebraic Discrete Methods, vol. 7, pp. 571-582, 1986.
[19] B. Monien, "The bandwidth minimization problem for caterpillars with hair length 3 is NP-complete,"SIAM J. Algebraic Discrete Methods, vol. 7, pp. 505-512, 1986.
[20] A. M. Odlyzko and H. S. Wilf, "Bandwidths and profiles of trees,"J. Combinatorial Theory, Series B, vol. 42, pp. 348-370, 1987.
[21] C. H. Papadimitriou, "The NP-completeness of the bandwidth minimization problem,"Computing, vol. 16, pp. 263-270, 1976.
[22] M. S. Paterson, W. L. Ruzzo, and L. Snyder, "Bounds on minimax edge length for complete binary trees," inProc. 13th Annu. ACM Symp. Theory Comput., 1981, pp. 293-299.
[23] A. G. Ranade and S. L. Johnsson, "The communication efficiency of meshes, boolean cubes and cube connected cycles for wafer scale integration," inProc. 1987 Int. Conf. Parallel Processing, 1987, pp. 479-482.
[24] C. L. Seitz, "Self-timed VLSI systems," inProc. Caltech Conf. Very Large Scale Integration, 1979, pp. 345-355.
[25] C. L. Seitz, "The Cosmic Cube,"Commun. ACM, pp. 22-33, Jan. 1985.
[26] V. I. Smirnov,A Course in Higher Mathematics, Vol. 3, Part 2. Oxford, England: Pergamon, 1964, translated by D. E. Brown.
[27] A. P. Sprague, "Problems in VLSI layout design," Ph.D. dissertation, Ohio State Univ., 1988.
[28] A. Y. Wu, "Embedding of tree networks into hypercubes,"J. Parallel Distributed Comput., vol. 2, pp. 239-249, 1985.

Index Terms:
processors embedding; minimizing; longest interprocessor wire; hypercube; rectangular mesh; neighboring nodes; graph-theoretic distance; delays; delays; graph theory; hypercube networks; minimisation of switching nets.
Citation:
T.-H. Lai, A.P. Sprague, "Placement of the Processors of a Hypercube," IEEE Transactions on Computers, vol. 40, no. 6, pp. 714-722, June 1991, doi:10.1109/12.90250
Usage of this product signifies your acceptance of the Terms of Use.