Upper Tail Analysis of Bucket Sort and Random Tries

Bucket Sort is known to run in expected linear time when the input keys are distributed independently and uniformly at random in the interval [0, 1). The analysis holds even when a quadratic time algorithm is used to sort the keys in each bucket. We show how to obtain linear time guarantees on the running time of Bucket Sort that hold with very high probability. Specifically, we investigate the asymptotic behavior of the exponent in the upper tail probability of the running time of Bucket Sort. We consider large additive deviations from the expectation, of the form cn for large enough (constant) c, where n is the number of keys that are sorted.