Estimating the Range of a Function in an Online Setting

Consider an unknown function L(·) : {1, · · · , d} → {1, · · · , r} with range R = {L(i)|i = 1, · · · , d}. Given d, r, , δ > 0 we show how to compute an estimate p̃ such that with probability at least 1 − δ we have ||R|/r − p̃| ≤ p̃. This is an estimate with a fixed relative error, which is stronger than finding an estimate with a fixed absolute error. This calculation can be performed efficiently in one pass through the domain of L (allowing the the method to be used in online situations) using only O ( log r (log log r + log 1/δ) / 2 ) words of storage. The method is based on pairwise-independent pseudo-random variables.