Hamming Codes, Hypercube Embeddings, and Fault Tolerance
William A. Aiello, Frank Thomson Leighton · SIAM Journal on Computing · 2007
In this paper, we devise a special 1-error-correcting code that enables us to embed the graph product of an $N/\log N$-node hypercube and a $\log N$-node complete graph into in an N-node hypercube with constant load, dilation, and congestion. We apply the result to construct improved embeddings of trees and other structures in a hypercube, and to design more efficient and robust algorithms for reconfiguring a hypercube around random or worst-case faults. The result has also been used subsequently by others to show that the N-node hypercube can emulate all N-node planar graphs with constant slowdown.