Frequency Estimation with One-Sided Error

Frequency estimation is one of the most fundamental problems in streaming algorithms. Given a stream $S$ of elements from some universe $U=\{1 \ldots n\}$, the goal is to compute, in a single pass, a short sketch of $S$ so that for any element $i \in U$, one can estimate the number $x_i$ of times $i$ occurs in $S$ based on the sketch alone. Two state of the art solutions to this problems are the Count-Min and Count-Sketch algorithms. The frequency estimator $\tilde{x}$ produced by Count-Min, using $O(1/\varepsilon \cdot \log n)$ dimensions, guarantees that $\|\tilde{x}-x\|_{\infty} \le \varepsilon \|x\|_1$ with high probability, and $\tilde{x} \ge x$ holds deterministically. Also, Count-Min works under the assumption that $x \ge 0$. On the other hand, Count-Sketch, using $O(1/\varepsilon^2 \cdot \log n)$ dimensions, guarantees that $\|\tilde{x}-x\|_{\infty} \le \varepsilon \|x\|_2$ with high probability. A natural question is whether it is possible to design the best of both worlds sketching method, with error guarantees depending on the $\ell_2$ norm and space comparable to Count-Sketch, but (like Count-Min) also has the no-underestimation property. Our main set of results shows that the answer to the above question is negative. We show this in two incomparable computational models: linear sketching and streaming algorithms. We also study the complementary problem, where the sketch is required to not over-estimate, i.e., $\tilde{x} \le x$ should hold always.