Long paths in hypercubes with a quadratic number of faults
A path between distinct vertices u and v of the n -dimensional hypercube Qn Q n avoiding a given set of f  faulty vertices is called long if its length is at least 2n-2f-2 2 n - 2 f - 2 . We present a function ?(n)=?(n2) ? ( n ) = ? ( n 2 ) such that if f??(n) f ? ? ( n ) then there is a long fault-free path between every pair of distinct vertices of the largest fault-free block of Qn Q n . Moreover, the bound provided by ?(n) ? ( n ) is asymptotically optimal. Furthermore, we show that assuming f??(n) f ? ? ( n ) , the existence of a long fault-free path between an arbitrary pair of vertices may be verified in polynomial time with respect to n and, if the path exists, its construction performed in linear time with respect to its length.
long path hypercube faulty vertex linear-time algorithm