Information Theoretic Limits of Cardinality Estimation: Fisher Meets Shannon

Estimating the cardinality (number of distinct elements) of a large multiset is a classic problem in streaming and sketching, dating back to Flajolet and Martin's classic Probabilistic Counting (PCSA) algorithm from 1983. In this paper we study the intrinsic tradeoff between the space complexity of the sketch and its estimation error in the random oracle model. We define a new measure of efficiency for cardinality estimators called the Fisher-Shannon (Fish) number $\mathcal{H}/\mathcal{I}$. It captures the tension between the limiting Shannon entropy ($\mathcal{H}$) of the sketch and its normalized Fisher information ($\mathcal{I}$), which characterizes the variance of a statistically efficient, asymptotically unbiased estimator. Our results are as follows. We prove that all base-$q$ variants of Flajolet and Martin's PCSA sketch have Fish-number $H0/I0 \approx 1.98016$ and that every base-$q$ variant of (Hyper)LogLog has Fish-number worse than $H0/I0$, but that they tend to $H0/I0$ in the limit as $q\rightarrow \infty$. Here $H0,I0$ are precisely defined constants. We describe a sketch called Fishmonger that is based on a smoothed, entropy-compressed variant of PCSA with a different estimator function. It is proved that with high probability, Fishmonger processes a multiset of $[U]$ such that at all times, its space is $O(\log^2\log U) + (1+o(1))(H0/I0)b \approx 1.98b$ bits and its standard error is $1/\sqrt{b}$. We give circumstantial evidence that $H0/I0$ is the optimum Fish-number of mergeable sketches for Cardinality Estimation. We define a class of linearizable sketches and prove that no member of this class can beat $H0/I0$. The popular mergeable sketches are, in fact, also linearizable.