Time-randomness trade-off for oblivious routing

Three parameters characterize the performance of a probabilistic algorithm: T, the run-time of the algorithm; Q, the probability that the algorithm fails to complete the computation in the first T steps; and R, the amount of randomness used by the algorithm, measured by the entropy of its random source. A tight trade-off between these three parameters for the problem of oblivious packet routing on N-vertex bounded-degree networks is presented. A (1 - Q) log (N/T)-log Q-O(1) lower bound for the entropy of a random source of any oblivious packet routing algorithm that routes an arbitrary permutation in T steps with probability 1 - Q is proved. It is shown that this lower bound is almost optimal. This result is complemented with an explicit construction of a family of oblivious algorithms that use less than a factor of log N more random bits than the optimal algorithm achieving the same run-time.