{"ID":2856665,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2510.10431","arxiv_id":"2510.10431","title":"Explicit Min-wise Hash Families with Optimal Size","abstract":"We study explicit constructions of min-wise hash families and their extension to $k$-min-wise hash families. Informally, a min-wise hash family guarantees that for any fixed subset $X\\subseteq[N]$, every element in $X$ has an equal chance to have the smallest value among all elements in $X$; a $k$-min-wise hash family guarantees this for every subset of size $k$ in $X$. Min-wise hash is widely used in many areas of computer science such as sketching, web page detection, and $\\ell_0$ sampling. The classical works by Indyk and Pătraşcu and Thorup have shown $Θ(\\log(1/δ))$-wise independent families give min-wise hash of multiplicative (relative) error $δ$, resulting in a construction with $Θ(\\log(1/δ)\\log N)$ random bits. Based on a reduction from pseudorandom generators for combinatorial rectangles by Saks, Srinivasan, Zhou and Zuckerman, Gopalan and Yehudayoff improved the number of bits to $O(\\log N\\log\\log N)$ for polynomially small errors $δ$. However, no construction with $O(\\log N)$ bits (polynomial size family) and sub-constant error was known before. In this work, we continue and extend the study of constructing ($k$-)min-wise hash families from pseudorandomness for combinatorial rectangles and read-once branching programs. Our main result gives the first explicit min-wise hash families that use an optimal (up to constant) number of random bits and achieve a sub-constant (in fact, almost polynomially small) error, specifically, an explicit family of $k$-min-wise hash with $O(k\\log N)$ bits and $2^{-O(\\log N/\\log\\log N)}$ error. This improves all previous results for any $k=\\log^{O(1)}N$ under $O(k \\log N)$ bits. Our main techniques involve several new ideas to adapt the classical Nisan-Zuckerman pseudorandom generator to fool min-wise hashing with a multiplicative error.","short_abstract":"We study explicit constructions of min-wise hash families and their extension to $k$-min-wise hash families. Informally, a min-wise hash family guarantees that for any fixed subset $X\\subseteq[N]$, every element in $X$ has an equal chance to have the smallest value among all elements in $X$; a $k$-min-wise hash family...","url_abs":"https://arxiv.org/abs/2510.10431","url_pdf":"https://arxiv.org/pdf/2510.10431v3","authors":"[\"Xue Chen\",\"Shengtang Huang\",\"Xin Li\"]","published":"2025-10-12T03:47:48Z","proceeding":"cs.DS","tasks":"[\"cs.DS\",\"cs.DM\"]","methods":"[]","has_code":false}