[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85866-en":3,"doc-seo-85866-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},85866,1099514068035,"Ezra","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",8,"Research & Report","Limited Independence Suffices for Large-k Min-wise Hashing","Min-wise hashing and its k-min-wise variant underpin similarity estimation, sampling, sketching, and streaming. A k-min-wise family ensures that for any fixed set X and any r ≤ k, every r-subset is approximately equally likely to be the bottom elements, within multiplicative error δ. Prior work required high independence and incurred an extra log log(1/δ) factor when k is large. This work reanalyzes s-wise independent polynomial hashing, proving k-min-wise behavior for s = O(k + log(1/δ)), yielding explicit seed length O(k log N) with polynomially small error for k = Θ(log N).","arXiv :2607 . 10255v 1 [ cs .DS] 11 Jul 2026  \nLimited Independence Suffices for Large-k Min-wise Hashing  \nHaoran Wang ∗  \nAbstract  \nMin-wise hashing and its k-min-wise variant are standard tools in similarity estimation, sampling, sketching, and streaming algorithms. A k-min-wise family requires that, for any fixed set X ⊆ [N] and each r ≤ k, every prescribed r-subset of X is approximately equally likely, up to multiplicative error δ, to be the r elements with the smallest hash values. The case k = 1 is ordinary min-wise hashing. Previous analyses show that t = O (log(1/δ) + k log log(1/δ))-wise independence suffices. For k = Θ(log N) and δ = N −c, the corresponding polynomial construction has seed length O (k log N log log N), losing a factor of log log N from the optimal O (k log N) . Recent work by Chen, Huang, and Li (CHL) constructs explicit k-min-wise families with an optimal seed length of O(k log N) for k = logO(1) N and almost-polynomial multiplicative error 2 −O(log N/ log log N) . This leaves open a question explicitly raised in their work: whether one can maintain the optimal O (k log N) seed length while achieving a polynomially small error, especially in the regime where k = Θ(log N) .  \nIn this work, we reanalyze the standard s-wise independent polynomial hash family fork-min-wise hashing. We prove that, for s = O (k + log(1/δ)), this family is k-min-wise with multiplicative error δ . Therefore, when k is at least a constant multiple of log(1/δ), the required independence degree is only O (k) . In particular, for k = Θ(log N) and δ = N −c, the polynomial hash family provides an explicit k-min-wise family with polynomially small error and seed length O (k log N), matching the optimal lower bound up to constant factors in this regime. The analysis focuses on the random threshold given by the largest hash value inside the prescribed bottom set. Instead of controlling the error at each individual threshold, we bound the error only after averaging over this random threshold. This gives a sharper total error bound, removes the extra log log(1/δ) factor in the large-k regime, and yields optimal O (k log N) seed length with polynomially small error when k = Θ(log N) .  \n∗ School of Mathematical Sciences, Peking University. Email: [whr.hrwang@gmail.com](whr.hrwang@gmail.com)  \n1 Introduction  \nMin-wise hashing, introduced by Broder, Charikar, Frieze, and Mitzenmacher [1], is a standard primitive in randomized algorithms, streaming, sketching, similarity estimation, near-duplicate detection, and sampling; see, for example, applications to similarity and rarity estimation [2, 6], near-duplicate detection [10], min-hash sketches [4], and ℓ0 sampling [5] . Let H be a family of functions from [N] to an ordered range [M] . The family H is approximately min-wise if, for every fixed set X ⊆ [N], each element of X is almost equally likely to attain the minimum hash value in X . The k-min-wise variant is the analogous property for bottom sets of size at most k: for every Y ⊆ X with |Y | ≤ k, the probability that Y forms the strict bottom |Y | elements of X is close to 􀀀 || XY || 􀀁 −1 .  \nWe use the standard multiplicative-error definition, where (1 ± δ)p denotes a value in [(1 −δ)p,(1+δ)p] . For [M] = {1,..., M}, a family H of functions h : [N] → [M] is k-min-wise with error δ if, for all X ⊆ [N] and all Y ⊆ X with |Y | ≤ k ,  \nhrH 􀀔my h (y) \u003C xin\\Y h (x)􀀕 = (1 ± δ) 􀀒 || XY|| 􀀓 −1 . (1.1)  \nThe trivial boundary cases are treated by the conventions stated in Section 2. For a fully random continuous ordering, the probability in (1.1) is exactly 􀀀 || XY|| 􀀁 −1 . However, for a finite range [M], potential ties across the boundary between Y and X \\ Y introduce a small additional error. Throughout the paper, we take M = Ω(N/δ), the standard regime in which fully random functions [N] → [M] already satisfy (1.1) within an O (δ) error [11, 7 , 3] .  \nThe multiplicative property of (1.1) is important. The fair probability itself can be v","cbCaieKRruFlwOt4","https://ap.wps.com/l/cbCaieKRruFlwOt4","pdf",397069,4,1,20,"English","en",105,"# Abstract\n# Introduction\n## Prior work and the gap","[{\"question\":\"What problem does the paper address in k-min-wise hashing?\",\"answer\":\"It studies how much independence is needed so that polynomial hash families remain k-min-wise with multiplicative error δ, especially in the large-k regime where k = Θ(log N).\"},{\"question\":\"What does the main theorem show about s-wise independence?\",\"answer\":\"For s = O(k + log(1/δ)), an s-wise independent polynomial hash family is k-min-wise with multiplicative error δ, implying only O(k) required independence when k dominates log(1/δ).\"},{\"question\":\"How does the paper achieve optimal seed length and error bounds?\",\"answer\":\"The analysis focuses on the random threshold given by the largest hash value within the bottom set, bounding error after averaging over this threshold, which removes the extra log log(1/δ) factor and yields seed length O(k log N) with polynomially small error for k = Θ(log N).\"}]",1784206785,50,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"limited-independence-suffices-for-large-k-min-wise-hashing","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":20},"https://docshare.wps.com/document/limited-independence-suffices-for-large-k-min-wise-hashing/85866/",{"url":52,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-24","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What problem does the paper address in k-min-wise hashing?","Question",{"text":75,"@type":76},"It studies how much independence is needed so that polynomial hash families remain k-min-wise with multiplicative error δ, especially in the large-k regime where k = Θ(log N).","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What does the main theorem show about s-wise independence?",{"text":80,"@type":76},"For s = O(k + log(1/δ)), an s-wise independent polynomial hash family is k-min-wise with multiplicative error δ, implying only O(k) required independence when k dominates log(1/δ).",{"name":82,"@type":73,"acceptedAnswer":83},"How does the paper achieve optimal seed length and error bounds?",{"text":84,"@type":76},"The analysis focuses on the random threshold given by the largest hash value within the bottom set, bounding error after averaging over this threshold, which removes the extra log log(1/δ) factor and yields seed length O(k log N) with polynomially small error for k = Θ(log 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