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On the Conversion Between Binary Code and Binary-Reflected Gray Code on Binary Cubes
January 1995 (vol. 44 no. 1)
pp. 47-53

We present a new algorithm for conversion between binary code and binary-reflected Gray code that requires approximately \scriptstyle{2K \over 3} 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 {K \over 2} + 1 element transfers, which is optimal. The new algorithm is optimal to within a multiplicative factor of \scriptstyle{4\over 3} 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 {K} with concurrent communication on all channels of every node of a binary cube.

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Index Terms:
Gray-to-binary conversion, binary code encoding, Gray code encoding, hypercubes, permutation, routing algorithm, communication algorithm, all-port communication.
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
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