[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-86008-en":3,"doc-seo-86008-105":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},86008,1099514067415,"Rowan","https://ap-avatar.wpscdn.com/avatar/100002539d78ffe74a7?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779092875211072502",8,"Research & Report","Rectilinear Matching to the Integer Grid in Nearly-Linear Time","Rectilinear matching to the integer grid assigns n planar points to distinct integer lattice points to minimize total ℓ1 movement, while the infinite target grid requires identifying a finite relevant subset without losing optimality. The work proves a geometric compression theorem that constructs a candidate set C of asymptotically optimal size O(n) in O(n log^2 n) time, enabling ℓp-optimal assignments for all p ∈ [1,∞]. For ℓ1, the approach combines C with a sparse network representation and a nearly-linear time minimum-cost flow algorithm.","arXiv :2607 . 10703v1 [ cs .CG] 12 Jul 2026  \nRectilinear Matching to the Integer Grid in Nearly-Linear Time  \nYu Gao  \nAbstract  \nRectilinear matching to the integer grid asks to assign each of n points in R2 to a distinct  \npoint of Z2 , minimizing total ℓ 1 movement. The main difficulty is that the target set is infinite: one must first identify a finite set of relevant grid points without losing optimality.  \nWe prove a geometric compression theorem for this infinite-target problem. In O (n log2 n) time, we construct a set C of asymptotically optimal size O(n) such that, simultaneously for every p ∈ [1 , ∞ ], some optimal ℓp assignment uses only points of C. The construction is independent of the subsequent optimization algorithm and of the coordinate spread.  \nFor the rectilinear case, we combine this candidate set with a linear-size sparse network representation of ℓ1 distances. In the word-RAM model with O(1)-word dyadic coordinates and O(log n) fractional bits, a nearly-linear time minimum-cost flow algorithm then gives a randasopetmpraizolsedacho gexivacCest aomanlbre(nitditlwhogite( hx1i/exstεi)pn)-egtcitefimdniet(re1uneoεingme) attpimrpicreoOmx(nchat)ii.Tngonhiafoslgrimorevpritheryovmfiesxeththdeeinssttaeanmgederacpid(n2ate)  \n1 Introduction  \nWe study a common core legalization task in many geometric optimization pipelines: the input isan indexed sequence P = (h1 , ... , hn) . Each hi is a point in R2 ; repeated locations are allowed. The task is to find an injective map  \nµ : [n] ,→ Z2  \nminimizing  \nn  \nX ∥hi − µ(i)∥ 1 .  \ni=1  \nThe injectivity constraint ensures that no two objects are assigned to the same grid point, and the ℓ 1 objective is the rectilinear movement cost. The target set is the entire integer grid, not a finite set listed in the input. Thus an exact algorithm must identify a relevant set of grid points and optimize the assignment to those points. Our main contribution is an explicit construction of such a set with O (n) grid points.  \nWe are not aware of a prior worst-case subquadratic exact algorithm for the unrestricted infinite-grid problem defined above. The standard reduction that retains n nearest grid points per input, combined with the finite geometric partial-matching algorithm of Agarwal, Chang, and XgrEviiadeo pry[AoiliCntstXsed,19bp]ro,eigaknivitesngisatnnioear(n2rbthie)tr-trafimrroielymexShiau,ctpaapondlgosethriaethnomopthe.tirFomnl eaas1csihignnpinpmutuestntopomccuintatpychhaiets, rehmtioaostinutne1ntonhfeistahrlieesstmt. At least one listed point is therefore unused, and reassigning hi to that point does not increase the cost. Restricting the target set to the union of these nearest grid points thus preserves the optimal value. This candidate set may, however, have size N = Θ(n2 ) . Applying the exact geometric partial-matching algorithm [ACX19] to the resulting finite instance takes (n2 +N) = (n2 ) time.  \nEven when approximate solutions are allowed, the quadratic-size candidate set remains the main bottleneck. A nearly-linear time approximation algorithm for finite geometric matching or transportation still takes Ω(n2 ) time after this reduction because the reduced input can have size Ω(n2 ) in the worst case.  \ntsWo thimuleetaginnivteeoeguaerslryagcnrioddnotmaiiFnziser,dstfo,erxweaevctserahylgowℓportnithohmatrm,ratushnninge tinlearg egeinxpt gelrixpcidietlpctyoiecndot(nruat)ctitilemblaseet fsooenrte reofopctOtiilmi(nnael)argasmrsiidgatchponmieingntsnt Second, for the ℓ 1 norm, we optimize over these points without building the complete bipartite graph, using a sparse minimum-cost flow representation. Since an explicit assignment contains n target grid points, any algorithm that outputs the assignment requires Ω(n) time. Thus our exwbipepeaactlsrteoitdoe-a(natinc)rahiunnnfaginstalg tapgorimpritheoxmiismsopattotitimonhealailnugpporutitothppomoinlyutlognds aarerndithththme ℓeilcpinfanoeactrr-mosrisbze.yFcoaarpnepdvlyideryingatefixexseeitd integer p ≥sting geometr1ic, In real-world","cbCaighExUXFSkDg","https://ap.wps.com/l/cbCaighExUXFSkDg","pdf",396662,4,1,29,"English","en",105,"# Introduction\n## Our Results","[{\"question\":\"What problem does rectilinear matching to the integer grid solve?\",\"answer\":\"It matches n points in R2 injectively to distinct points in Z2 while minimizing the sum of rectilinear (ℓ1) movement costs.\"},{\"question\":\"Why is the infinite integer grid a challenge for exact algorithms?\",\"answer\":\"Because the algorithm must replace an unbounded target set by a finite candidate set that is guaranteed not to exclude any optimal solution.\"},{\"question\":\"What is the main contribution of the paper?\",\"answer\":\"A geometric compression theorem that constructs a candidate set C of size O(n) in O(n log^2 n) time, such that for every p ∈ [1,∞] some optimal ℓp assignment uses only points from C.\"}]",1784207754,73,{"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},"rectilinear-matching-to-the-integer-grid-in-nearly-linear-time","",{"@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/rectilinear-matching-to-the-integer-grid-in-nearly-linear-time/86008/",{"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","2026-07-26","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 rectilinear matching to the integer grid solve?","Question",{"text":75,"@type":76},"It matches n points in R2 injectively to distinct points in Z2 while minimizing the sum of rectilinear (ℓ1) movement costs.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"Why is the infinite integer grid a challenge for exact algorithms?",{"text":80,"@type":76},"Because the algorithm must replace an unbounded target set by a finite candidate set that is guaranteed not to exclude any optimal solution.",{"name":82,"@type":73,"acceptedAnswer":83},"What is the main contribution of the paper?",{"text":84,"@type":76},"A geometric compression theorem that constructs a candidate set C of size O(n) in O(n log^2 n) time, such that for every p ∈ [1,∞] some optimal ℓp assignment uses only points from 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