Distinct Elements in Streams: An Algorithm for the (Text) Book

Given a data stream D = ⟨ a 1 , a 2 , . . . , a m ⟩ of m elements where each a i ∈ [ n ], the Distinct Elements problem is to estimate the number of distinct elements in D . Distinct Elements has been a subject of theoretical and empirical investigations over the past four decades resulting in space optimal algorithms for it. All the current state-of-the-art algorithms are, however, beyond the reach of an undergraduate textbook owing to their reliance on the usage of notions such as pairwise independence and universal hash functions. We present a simple, intuitive, sampling-based space-efficient algorithm whose description and the proof are accessible to undergraduates with the knowledge of basic probability theory.