Reconfiguring embedded task graphs in faulty hypercubes by automorphisms

The authors develop graph-theoretical techniques using automorphisms to solve the problem of reconfiguring embedded task graphs in faulty hypercubes. Four cases are considered: single node or link failure, small number (<5) of random node failures, multiple failures that are adjacent and large number (⩾5) of random node failures. For single node (link) failure, any arbitrary task graphs embedded in a hypercube can be remapped to another instance of that graph by simple bit-flip operations. It takes at most O(n) such operations in an n-dimensional hypercube. For a small number of faults (<5) and multiple adjacent node failures, necessary and sufficient conditions are derived under which the remapping can be achieved with these techniques. It is also shown that the general problem of reconfiguring multiple failures is as hard as the graph isomorphism problem. However, an algorithm is developed to determine if the reconfiguration can be achieved with the graph-theoretical techniques, when the necessary conditions are satisfied