Sketching Intersection Profiles: A Simple Proof and Three Applications

2026-06-11

AI Summary

This work investigates the sketch complexity of vertex neighborhood size, coverage functions, and random utility models on graphs, establishing their information-theoretic lower bounds. By introducing a sketching framework based on distributional intersection profiles, the study connects these problems to frequent itemset estimation in databases and presents a streamlined proof technique relying solely on elementary probability theory, thereby avoiding more intricate conventional tools. The main contributions include strengthening the lower bound for random utility models from Ω(n log n) to Ω̃(n²), and demonstrating that all three problems require Ω̃(n²) bits of space. Notably, the coverage function achieves a tight bound of Θ̃(n²), while the lower bound for random utility models matches the best-known upper bound up to logarithmic factors.

Abstract

In this work we settle the complexity of three sketching problems. (i) We show that sketching vertex neighborhood sizes in graphs requires $Ω(n^2)$ bits, standing in sharp contrast to the $\tilde{O}(n)$ complexity of sketching edge cuts. (ii) We obtain tight lower and upper bounds of $\tildeΘ(n^2)$ for sketching coverage functions with additive and multiplicative errors. (iii) We prove an $Ω(n^2)$ lower bound for sketching Random Utility Models under the $\ell_\infty$-norm, improving upon the previous $Ω(n \log n)$ 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.

Problem

• sketching
• intersection profile
• coverage functions
• Random Utility Models
• neighborhood sizes

Innovation

• sketching
• intersection profile
• lower bounds
• coverage functions
• Random Utility Models