[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84208-en":3,"doc-seo-84208-105":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},84208,962075114765,"Quinn","https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd",8,"Research & Report","Non-minimal k-Perfect Hashing: Tight Lower Bounds and an Application to Fast Static Hash Tables","Minimal perfect hashing maps a static set of n keys into n bins without collisions, and it extends naturally to minimal k-perfect hashing (bins of capacity k) and non-minimal perfect hashing with load factor α\u003C1 by adding spare capacity. Non-minimal k-PHFs are studied here for cache-resident use in accelerating static hash tables via branchless lookups. The work derives tight space lower bounds for all α∈(0,1] and k≥1, showing strong space reductions when α\u003C1 and k≥2. It also develops a tuned PtrHash-based k-PHF with empirical speedups up to 1.5×.","arXiv :2607 .07257v 1 [ cs .DS] 8 Jul 2026  \nNon-minimal k-perfect hashing: Tight lower  \nbounds and an application to fast static hash tables  \nRagnar Groot Koerkamp \\#  Karlsruhe Institute of Technology, Germany Stefan Hermann \\#   \nKarlsruhe Institute of Technology, Germany Peter Sanders \\#   \nKarlsruhe Institute of Technology, Germany Stefan Walzer \\#   \nKarlsruhe Institute of Technology, Germany  \n~~ Abstract ~~  \nA minimal perfect hash function (minimal PHF) is a data structure mapping a static set of n keys to n bins without collisions. Two natural generalizations are minimal k-PHFs where n keys are mapped to n/k bins of capacity k each, and (non-minimal) PHFs with load factor α \u003C 1 where the number of bins is increased by a factor of 1/α, resulting in spare capacity.  \nWhile there has been a recent surge of interest in perfect hashing generally, non-minimal k-PHFshave not been systematically studied despite a natural use case of speeding up static hash tables: The idea is that a small cache-resident k-PHF maps each key x to a cache-line-sized bin of capacity k where x resides. Ideally, this yields a branchless lookup operation with a single cache miss working at high load factors for positive and negative queries alike.  \nOur main theoretical contribution is to determine tight space lower bounds for k-PHFs for all pairs of α ∈ (0 , 1] and k ≥ 1. It turns out that combining α \u003C 1 and k ≥ 2 drastically reduces the space of k-PHFs, e.g. for (k, α) = (16 , 0.8) the space lower bound is 0.027 bits per key while for (k, α) = (16 , 1.0) and (k, α) = (1 , 0.8) the lower bounds are higher by factors of ≈ 8 and ≈ 32 , respectively. On the practical side, we develop a k-PHF based on PtrHash and tune it for use in static hash tables. Empirically, our implementation produces k-PHFs of size roughly 50% above the lower bound. A static hash set based on this k-PHF is consistently at least as fast as other hash sets for negative and mixed queries. On two of the three tested architectures it achieves up to 1 .5 × speedup for large n ≥ 30M where a 1-PHF does not fit in cache.  \n2012 ACM Subject Classification Theory of computation → Bloom filters and hashing; Theory of computation → Data structures design and analysis; Information Systems → Point lookups  \nKeywords and phrases Compressed Data Structures, k-Perfect Hashing, Hash Table, Space Lower Bound  \nRelated Version Extended version: TODOARXIVLINK [26]  \nSupplementary Material Software: [https://github.com/RagnarGrootKoerkamp/static-hash-sets](https://github.com/RagnarGrootKoerkamp/static-hash-sets)  \n 1  Introduction  \nGiven a universe of u keys and an input set of n keys, a k-perfect hash function (PHF) maps the input keys to b bins such that there are at most k keys in each bin. The challenge is to represent the k-PHF as a data structure without storing the input set itself. We refer to α := ~~n~~bk as the load of a k-PHF and call a k-PHF minimal when α = 1 . Table 1, as well as the survey [35], give an overview of construction techniques and lower bounds. In this paper, we initiate the study of the most difficult case of non-minimal k-PHF with k ≥ 2. We derive tight lower bounds on the space of a non-minimal k-PHF for all pairs of α and k (Section 2), extend PtrHash [25] into k-PtrHash, a fast and practical k-PHF with about 50%  \n2 Non-minimal k-perfect hashing  \n\n| k = 1 , α = 1 | Bucket Placement [19 , 2 , 47 , 28 , 25 , 4], Recursive Splitting [16 , 38 , 5], Fingerprinting [3, 10 , 40 , 34], Hypergraph / Retrieval [7, 14 , 43 , 6 , 20 , 52 , 36 , 37 , 27] Lower Bound: n · log2 (e) [44] |\n| --- | --- |\n| k = 1 , α \u003C 1 | Bucket Placement [19, 2, 47, 28, 25, 4], Hypergraph / Retrieval [11, 36]\u003Cbr>Lower Bound: n · 􀀀log2 (e) +􀀀 1α − 1􀀁log2 (1 − α)􀀁 [2] |\n| k ≥ 2 , α = 1 | Bucket Placement & Recursive Splitting & Thresholding [29]\u003Cbr>Lower Bound: n · 􀀀log2 (e) − log2 (kk /k!)/k 􀀁 [42, 2, 31] |\n| k ≥ 2 , α \u003C 1 | Bucket Placement [2] and [Section 3]\u003Cbr>Lower Bound (new): n · 􀀀log2 􀀀 ~","cbCainFu8jRSeye8","https://ap.wps.com/l/cbCainFu8jRSeye8","pdf",1059911,4,1,27,"English","en",105,"# Introduction\n# Non-minimal k-perfect hashing\n## Theory: Tight lower bounds","[{\"question\":\"What problem does non-minimal k-perfect hashing address in static hash tables?\",\"answer\":\"It provides a compact representation of mappings where each cache-line-sized bin holds at most k keys, enabling fast lookups with low branching and cache-friendly memory access.\"},{\"question\":\"How are load factor α and parameter k defined for a k-perfect hash function?\",\"answer\":\"A k-PHF maps n keys into b bins so that each bin contains at most k keys; the load is α := n/b, and the function is minimal when α=1.\"},{\"question\":\"What is the paper’s main theoretical contribution?\",\"answer\":\"It determines tight space lower bounds for k-PHFs for every pair α∈(0,1] and k≥1, including cases where combining α\\u003c1 with k≥2 substantially reduces required space.\"}]",1784193963,68,{"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},"non-minimal-k-perfect-hashing-tight-lower-bounds-and-an-application-to-fast-static-hash-tables","",{"@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/non-minimal-k-perfect-hashing-tight-lower-bounds-and-an-application-to-fast-static-hash-tables/84208/",{"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 non-minimal k-perfect hashing address in static hash tables?","Question",{"text":75,"@type":76},"It provides a compact representation of mappings where each cache-line-sized bin holds at most k keys, enabling fast lookups with low branching and cache-friendly memory access.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How are load factor α and parameter k defined for a k-perfect hash function?",{"text":80,"@type":76},"A k-PHF maps n keys into b bins so that each bin contains at most k keys; the load is α := n/b, and the function is minimal when α=1.",{"name":82,"@type":73,"acceptedAnswer":83},"What is the paper’s main theoretical contribution?",{"text":84,"@type":76},"It determines tight space lower bounds for k-PHFs for every pair α∈(0,1] and k≥1, including cases where combining α\u003C1 with k≥2 substantially reduces required space.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]