When trying to process a data stream in small space, how important is the order in which the data arrive? Are there problems that are unsolvable when the ordering is worst case, but that can be solved (with high probability) when the order is chosen uniformly at random? If we consider the stream as if ordered by an adversary, what happens if we restrict the power of the adversary? We study these questions in the context of quantile estimation, one of the most well studied problems in the data-stream model. Our results include an O(polylogn)-space, O(log log n)-pass algorithm for exact selection in a randomly ordered stream of n elements. This resolves an open question of Munro and Paterson [Theoret. Comput. Sci., 23 (1980), pp. 315-323]. We then demonstrate an exponential separation between the random-order and adversarial-order models: using O(polylog n) space, exact selection requires O(log n/log log n) passes in the adversarial-order model. This lower bound, in contrast to previous results, applies to fully general randomized algorithms and is established via a new bound on the communication complexity of a natural pointer-chasing style problem. We also prove the first fully general lower bounds in the random-order model:. finding an element with rank n/2 +/- n(delta) in the single-pass random-order model with probability at least 9/10 requires Omega(root n(1-3 delta)/log n) space.
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