{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/towards-better-bounds-for-finding-quasi","title":"Towards Better Bounds for Finding Quasi-Identifiers","arxiv_id":"2211.13882","date":"2022-11-25","proceeding":null,"authors":["Ryan Hildebrant","Quoc-Tung Le","Duy-Hoang Ta","Hoa T. Vu"],"abstract":"We revisit the problem of finding small $\\epsilon$-separation keys introduced by Motwani and Xu (2008). In this problem, the input is $m$-dimensional tuples $x_1,x_2,\\ldots,x_n $. The goal is to find a small subset of coordinates that separates at least $(1-\\epsilon){n \\choose 2}$ pairs of tuples. They provided a fast algorithm that runs on $\\Theta(m/\\epsilon)$ tuples sampled uniformly at random. We show that the sample size can be improved to $\\Theta(m/\\sqrt{\\epsilon})$. Our algorithm also enjoys a faster running time. To obtain this result, we provide upper and lower bounds on the sample size to solve the following decision problem. Given a subset of coordinates $A$, reject if $A$ separates fewer than $(1-\\epsilon){n \\choose 2}$ pairs, and accept if $A$ separates all pairs. The algorithm must be correct with probability at least $1-\\delta$ for all $A$. We show that for algorithms based on sampling: - $\\Theta(m/\\sqrt{\\epsilon})$ samples are sufficient and necessary so that $\\delta \\leq e^{-m}$ and - $\\Omega(\\sqrt{\\frac{\\log m}{\\epsilon}})$ samples are necessary so that $\\delta$ is a constant. Our analysis is based on a constrained version of the balls-into-bins problem. We believe our analysis may be of independent interest. We also study a related problem that asks for the following sketching algorithm: with given parameters $\\alpha,k$ and $\\epsilon$, the algorithm takes a subset of coordinates $A$ of size at most $k$ and returns an estimate of the number of unseparated pairs in $A$ up to a $(1\\pm\\epsilon)$ factor if it is at least $\\alpha {n \\choose 2}$. We show that even for constant $\\alpha$ and success probability, such a sketching algorithm must use $\\Omega(mk \\log \\epsilon^{-1})$ bits of space; on the other hand, uniform sampling yields a sketch of size $\\Theta(\\frac{mk \\log m}{\\alpha \\epsilon^2})$ for this purpose.","url_abs":"https://arxiv.org/abs/2211.13882v3","url_pdf":"https://arxiv.org/pdf/2211.13882v3.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"towards-better-bounds-for-finding-quasi","repo_url":"https://github.com/ryanhilde/min_set_cover","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}