Estimating the range of a function in an online setting

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Abstract

Consider an unknown function L(·):{1,…,d}→{1,…,r} with range R={L(i)∣i=1,…,d}. Given d,r,ε,δ>0 we show how to compute an estimate p̃ such that with probability at least 1−δ we have ||R|/r−p̃|≤εp̃. 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(logr(log logr+log1/δ)/ε2) words of storage. The method is based on pairwise-independent pseudo-random variables.

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Work performed while at CombiChem, Inc.

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