In this work we settle the complexity of three sketching problems. (i) We show that sketching vertex neighborhood sizes in graphs requires $\Omega(n^2)$ bits, standing in sharp contrast to the $\tilde{O}(n)$ complexity of sketching edge cuts. (ii) We obtain tight lower and upper bounds of $\tilde{\Theta}(n^2)$ for sketching coverage functions with additive and multiplicative errors. (iii) We prove an $\Omega(n^2)$ lower bound for sketching Random Utility Models under the $\ell_\infty$-norm, improving upon the previous $\Omega(n \log n)$ bound and matching a known upper bound to within logarithmic factors.
These bounds are obtained through a connection with the problem of sketching the intersection profile of a distribution $D$ on $2^{[n]}$. Specifically, we seek a succinct data structure that, for any query set $S \subseteq [n]$, approximates the quantity $\Pr_{T \sim D}[T \cap S \neq \varnothing]$ to within a small constant additive error.
One can obtain lower bounds for this latter problem directly from known results about the itemset frequency estimation problem in databases for which tight bounds are known.
As an additional contribution, we also provide an alternative proof for the intersection profile sketching lower bound, in the setting in which the accuracy parameter is constant. This proof relies solely on elementary probability avoiding the heavier machinery used in previous proofs.
The paper now derives the lower bound on intersection profiles from a 2016 result about itemset frequency sketches. As a result, the focus of the paper has shifted more towards the applications to other domains.
In this work, we study the space complexity of sketching the intersection profile of a distribution $D$ on $2^{[n]}$. Specifically, we seek a succinct data structure that, for any query set $S \subseteq [n]$, approximates the quantity $\Pr_{T \sim D}[T \cap S \neq \emptyset]$ to within a small constant additive error. Via a probabilistic packing argument, we show that the worst-case bit complexity of this problem is $\Omega(n^2)$, which we also prove to be tight.
We use this lower bound to settle the complexity of three sketching problems. (i) We show that sketching vertex neighborhood sizes in graphs requires $\Omega(n^2)$ bits, standing in sharp contrast to the $\tilde{O}(n)$ complexity of sketching edge cuts. (ii) We obtain tight lower and upper bounds of $\tilde{\Theta}(n^2)$ for sketching coverage functions with additive and multiplicative errors. (iii) We prove an $\Omega(n^2)$ lower bound for sketching Random Utility Models under the $\ell_\infty$-norm, improving upon the previous $\Omega(n \log n)$ bound and matching the upper bound to within logarithmic factors.
