Reliable broadcasting in hypercubes with random link and node failures

We consider the problem of broadcasting in an n-node hypercube whose links and nodes fail independently with given probabilities p < 1 and q < 1, respectively. Information held in a fault-free node, called the source, has to reach all other fault-free nodes. Messages may be directly transmitted to adjacent nodes only, and every node may communicate with at most one neighbour in a unit of time. A message can be transmitted only if both communicating neighbours and the link joining them are fault-free. For parameters p and q satisfying (1 -p) (1-q) ≥ 0.99 (e.g. p = q = 0.5%), we give an algorithm working in time O (log n) and broadcasting source information to all fault-free nodes with probability exceeding 1- cn^-ε for some positive constant ε , c depending on p and q but not depending on n.