A time-randomness tradeoff for oblivious routing

Three parameters characterize the performance of a probabilistic algorithm: <italic>T</italic>, the runtime of the algorithm; <italic>Q</italic>, the probability that the algorithm fails to complete the computation in the first <italic>T</italic> steps and <italic>R</italic>, the amount of randomness used by the algorithm, measured by the entropy of its random source. We present a tight tradeoff between these three parameters for the problem of oblivious packet routing on <italic>N</italic>-vertex bounded-degree networks. We prove a (1 - <italic>Q</italic>) log <italic>N</italic>/<italic>T</italic> - log <italic>Q</italic> - <italic>&Ogr;</italic>(1) lower bound for the entropy of a random source of any oblivious packet routing algorithm that routes an arbitrary permutation in <italic>T</italic> steps with probability 1 - <italic>Q</italic>. We show that this lower bound is almost optimal by proving the existence, for every <italic>e</italic><supscrpt>3</supscrpt> log <italic>N</italic> ≤ <italic>T</italic> ≤ <italic>N</italic><supscrpt>1/2</supscrpt>, of an oblivious algorithm that terminates in <italic>T</italic> steps with probability 1 - <italic>Q</italic> and uses (1-<italic>Q</italic>+<italic>&ogr;</italic>(1))log<italic>N</italic>/<italic>T</italic>-log<italic>Q</italic> independent random bits. We complement this result with an explicit construction of a family of oblivious algorithms that use less than a factor of log <italic>N</italic> more random bits than the optimal algorithm achieving the same run-time.