Long paths in hypercubes with a quadratic number of faults

Abstract

A path between distinct vertices u and v of the n-dimensional hypercube Q n avoiding a given set of f faulty vertices is called long if its length is at least 2 n - 2 f - 2 . We present a function ϕ ( n ) = Θ ( n 2 ) such that if f ⩽ ϕ ( n ) then there is a long fault-free path between every pair of distinct vertices of the largest fault-free block of Q n . Moreover, the bound provided by ϕ ( n ) is asymptotically optimal. Furthermore, we show that assuming f ⩽ ϕ ( 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.