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Keywords:

  • asymptotic relative efficiency;
  • cardinality;
  • data sketching;
  • data stream;
  • hash function;
  • maximum likelihood estimation;
  • space complexity;
  • stable distribution;
  • tail bounds

Abstract.  This article considers the problem of cardinality estimation in data stream applications. We present a statistical analysis of probabilistic counting algorithms, focusing on two techniques that use pseudo-random variates to form low-dimensional data sketches. We apply conventional statistical methods to compare probabilistic algorithms based on storing either selected order statistics, or random projections. We derive estimators of the cardinality in both cases, and show that the maximal-term estimator is recursively computable and has exponentially decreasing error bounds. Furthermore, we show that the estimators have comparable asymptotic efficiency, and explain this result by demonstrating an unexpected connection between the two approaches.