[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85221-en":3,"doc-seo-85221-105":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},85221,13056703019662,"Evangeline","https://ap-avatar.wpscdn.com/avatar/be000253a8e92610077?_k=1778726343310543188",8,"Research & Report","A Faster Closest-Point Algorithm for the A*ₙ Lattices","The dual root lattice A∗n is a key structure in quantization, coding, and estimation, representable as the projection of Zn+1 onto the sum-zero hyperplane. The paper addresses the closest-point problem: for a query y, compute x∈A∗n minimizing ‖y−x‖2. Building on the fastest prior linear-time method (MCSQ) that uses bucket sort, it introduces a faster linear-time approach using only prefix aggregates—count and residual sum—avoiding in-bucket sorting and traversal, with scaled arithmetic enabling exact integer-only computation for rational inputs.","arXiv :2607 . 10479v 1 [ cs .IT] 11 Jul 2026  \nA Faster Closest-Point Algorithm  \nfor the A∗n Lattices  \nYuriy A. Reznik  \nMassachusetts Institute of Technology  \n[yreznik@mit.edu](yreznik@mit.edu)  \nAbstract  \nThe dual root lattice A∗n is an important lattice in quantization, coding, and estimation. It can be represented as the projection of the integer lattice Zn+1 onto the n-dimensional hyperplane whose coordinates sum to zero. This representation makes A∗n particularly natural for quantizing simplex-constrained data, such as histograms and probability distributions.  \nThis paper studies the closest-point problem for A∗n: given a query vector y, find the lattice point x ∈ A∗n minimizing ∥y − x∥2 . The fastest previously known method is the linear-time algorithm of McKilliam, Clarkson, Smith, and Quinn (MCSQ), which employs bucket sort as a core operation.  \nWe present a faster linear-time algorithm. The key observation is that the closest-point objective depends on the rounding residuals only through two prefix aggregates: a count and a residual sum. Hence the elements inside each bucket never need to be sorted, stored, or traversed. This replaces the linked-list traversal and pointer chasing of MCSQ with a single bucketing passover two flat arrays with counting-sort-style accumulates. A scaled objective further makes most of the computation exact integer arithmetic, and when the input coordinates are rationals with a common denominator, for example, histograms or empirical distibutions, the entire algorithm becomes exact and integer-only.  \nExperiments on an Intel Core i9-13900H show speedups of about 1 .8 to 3 .0 over MCSQ for n = 2 , . . . , 100, with larger gains at higher dimensions. The proposed algorithm is also noticeably faster than Conway and Sloan methods for other root lattices, including An , D∗n, and E8 . An open-source implementation is available in the fanstar project [12] .  \n1 Introduction  \n1.1 The closest-point problem  \nA lattice is a discrete set of points obtained by taking all integer linear combinations of a set of basis vectors: if B is a generator matrix, then L = {Bw : w ∈ Zn } . Given a query vector y, the closest-point problem, also called the nearest-lattice-point problem, is to find  \nx∗ = arg min∥y − x∥2 .  \nx∈L  \nIn vector quantization, x∗ is the minimum-distortion reproduction point for y. In communication over a Gaussian channel, x∗ is the maximum-likelihood decoded lattice point [1] .  \nThis paper concerns the closest-point problem for the dual root lattice A∗n . We use the standard representation in which A∗n is embedded in Rn+1 but lies on the n-dimensional hyperplane H ={x ∈ Rn+1 : 1Tx = 0}, where 1 is the all-ones vector. Throughout the paper we write N = n + 1  \nfor the ambient dimension. Let Q = I − 11T /N be the orthogonal projector onto H. In the normalization used here,  \nA∗n = QZN = 􀀚 k − 1TNk 1 : k ∈ ZN􀀛 .  \nThus every point of A∗n can be represented by an integer vector k ∈ ZN, followed by projection: x = Qk. This representation is not unique, since Q(k + c1) = Qk for every integer c, but it is very convenient algorithmically.  \nThe input to the algorithms in this paper is a real vector y ∈ RN; the output is a lattice point x = Qk ∈ A∗n for some integer representative k. Since all lattice points lie in H, an arbitrary input y0 = Qy0 + t1 may first be projected: for every x ∈ A∗n , ∥y0 − x∥2 = ∥Qy0 − x∥2 + t2 N, and the second term is independent of x. Therefore, unless stated otherwise, we assume that the query vector has already been projected and satisfies 1Ty = 0 .  \n1.2 Why A∗n is useful  \nThe lattice A∗n has several attractive geometric and computational properties. In low dimensions it gives exceptionally efficient sphere coverings; in fact, it gives the thinnest known covering in all dimensions up to eight [3] . It includes familiar low-dimensional lattices as special cases: A ∗2 is equivalent to the hexagonal lattice, and A∗3 is equivalent to the body-centered cubic l","cbCaidSnQtijfwqY","https://ap.wps.com/l/cbCaidSnQtijfwqY","pdf",291296,3,1,11,"English","en",105,"# Abstract\n# Introduction\n## The closest-point problem\n## Why A∗n is useful\n## Prior closest-point algorithms for A∗n\n## Contribution of this paper","[{\"question\":\"What problem does the paper focus on for A∗n lattices?\",\"answer\":\"It studies the closest-point (nearest-lattice-point) problem for A∗n, finding x∈A∗n that minimizes the Euclidean distance ‖y−x‖2 for a query vector y.\"},{\"question\":\"What makes the new algorithm faster than the previous MCSQ method?\",\"answer\":\"The objective depends only on two prefix aggregates—count and residual sum—so elements inside each bucket do not need to be sorted, stored, or traversed. This reduces memory access and pointer-chasing overhead compared with MCSQ’s linked-list bucket traversal.\"},{\"question\":\"When does the algorithm become exact integer-only?\",\"answer\":\"When input coordinates are rationals sharing a common denominator, such as in histograms or empirical distributions, the scaled objective enables computation to be exact using only integer arithmetic.\"}]",1784201816,28,{"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},"a-faster-closest-point-algorithm-for-the-a-lattices","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,50],{"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":20},"https://docshare.wps.com/document/research-report/",{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/a-faster-closest-point-algorithm-for-the-a-lattices/85221/",4,{"url":51,"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","2026-07-23","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 focus on for A∗n lattices?","Question",{"text":75,"@type":76},"It studies the closest-point (nearest-lattice-point) problem for A∗n, finding x∈A∗n that minimizes the Euclidean distance ‖y−x‖2 for a query vector y.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What makes the new algorithm faster than the previous MCSQ method?",{"text":80,"@type":76},"The objective depends only on two prefix aggregates—count and residual sum—so elements inside each bucket do not need to be sorted, stored, or traversed. This reduces memory access and pointer-chasing overhead compared with MCSQ’s linked-list bucket traversal.",{"name":82,"@type":73,"acceptedAnswer":83},"When does the algorithm become exact integer-only?",{"text":84,"@type":76},"When input coordinates are rationals sharing a common denominator, such as in histograms or empirical distributions, the scaled objective enables computation to be exact using only integer arithmetic.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"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":52,"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"]