A 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 by proving the existence, for every $e^{3} \log N \leqq T \leqq N^{{1 / 2}}$, of an oblivious algorithm that terminates in T steps with probability $1 - Q$ and uses $(1- Q + o(1)) \log ({N / T}) - \log Q$ independent random bits. This result is complemented with an explicit construction of a family of o...