Publication 1995 Issue No. 1 - January Abstract - On the Conversion Between Binary Code and Binary-Reflected Gray Code on Binary Cubes
On the Conversion Between Binary Code and Binary-Reflected Gray Code on Binary Cubes
January 1995 (vol. 44 no. 1)
pp. 47-53
 ASCII Text x S. Lennart Johnsson, Ching-Tien Ho, "On the Conversion Between Binary Code and Binary-Reflected Gray Code on Binary Cubes," IEEE Transactions on Computers, vol. 44, no. 1, pp. 47-53, January, 1995.
 BibTex x @article{ 10.1109/12.368010,author = {S. Lennart Johnsson and Ching-Tien Ho},title = {On the Conversion Between Binary Code and Binary-Reflected Gray Code on Binary Cubes},journal ={IEEE Transactions on Computers},volume = {44},number = {1},issn = {0018-9340},year = {1995},pages = {47-53},doi = {http://doi.ieeecomputersociety.org/10.1109/12.368010},publisher = {IEEE Computer Society},address = {Los Alamitos, CA, USA},}
 RefWorks Procite/RefMan/Endnote x TY - JOURJO - IEEE Transactions on ComputersTI - On the Conversion Between Binary Code and Binary-Reflected Gray Code on Binary CubesIS - 1SN - 0018-9340SP47EP53EPD - 47-53A1 - S. Lennart Johnsson, A1 - Ching-Tien Ho, PY - 1995KW - Gray-to-binary conversionKW - binary code encodingKW - Gray code encodingKW - hypercubesKW - permutationKW - routing algorithmKW - communication algorithmKW - all-port communication.VL - 44JA - IEEE Transactions on ComputersER -

We present a new algorithm for conversion between binary code and binary-reflected Gray code that requires approximately $\scriptstyle\left\{2K \over 3\right\}$ element transfers in sequence for K elements per node, compared to K element transfers for previously known algorithms. For a binary cube of $n = 2$ dimensions the new algorithm degenerates to yield a complexity of $\left\{K \over 2\right\} + 1$ element transfers, which is optimal. The new algorithm is optimal to within a multiplicative factor of $\scriptstyle\left\{4\over 3\right\}$ with respect to the best known lower bound for any routing strategy. We show that the minimum number of element transfers for minimum path length routing is $\left\{K\right\}$ with concurrent communication on all channels of every node of a binary cube.

[1] J. C. Cooley and J. W. Tukey,“An algorithm for the machine computation of complex fourier series,”Math. Computat., vol. 19, pp. 291–301, 1965.
[2] High Performance Fortran Forum. High performance fortran; language specification, version 1.0.,Scientific Programming, vol. 2, no. 1–2, pp. 1–170, 1993.
[3] I. Havel and J. Mravek,“B-valuations of graphs,”Czech. Math. J., vol. 22, pp. 338–351, 1972.
[4] W.D. Hillis, The Connection Machine.Cambridge, Mass.: The MIT Press, 1985.
[5] C.-T. Ho and S.L. Johnsson, “Embedding Meshes in Boolean Cubes by Graph Decomposition,” J. Parallel and Distributed Computing, vol. 8, pp. 325-339, 1990.
[6] S.L. Johnsson, "Communication Efficient Basic Linear Algebra Computations on Hypercube Architectures," J. Parallel and Distributed Computing, vol. 4, pp. 133-172, 1987.
[7] S. Lennart Johnsson and C.-T. Ho,“The complexity of reshaping arrays on Boolean cubes,”inFifth Distrib. Memory Computing Conf., IEEE Comput. Soc., Apr. 1990, pp. 370–377.
[8] E. M. Reingold, J. Nievergelt, and N. Deo,Combinatorial Algorithms. Englewood Cliffs, NJ: Prentice-Hall, 1977.
[9] Thinking Machines Corp., CM-200 Technical Summary, 1991.

Index Terms:
Gray-to-binary conversion, binary code encoding, Gray code encoding, hypercubes, permutation, routing algorithm, communication algorithm, all-port communication.
Citation:
S. Lennart Johnsson, Ching-Tien Ho, "On the Conversion Between Binary Code and Binary-Reflected Gray Code on Binary Cubes," IEEE Transactions on Computers, vol. 44, no. 1, pp. 47-53, Jan. 1995, doi:10.1109/12.368010