[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82416-en":3,"doc-seo-82416-105":29,"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":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},82416,13056703020460,"Valentina","https://ap-avatar.wpscdn.com/avatar/be000253dac470eee5d?_k=1778207105932848923",8,"Research & Report","Improved Approximation of Min Distances in Near Linear Time","Study of approximating the diameter of directed graphs under the min-distance measure dmin(u,v)=min(d(u,v),d(v,u)), where min-distance is non-metric and blocks many classical techniques. Builds on prior approximation–runtime tradeoffs, including subquadratic and near-linear constant approximations. Presents a randomized near-linear time 3-approximation to the min-diameter via a novel classification framework. Extends to multimode graphs and, for directed 2-mode graphs, achieves a 3-approximation near-linearly, improving over prior bounds.","arXiv :2607 .09588v 1 [ cs .DS] 10 Jul 2026  \nImproved Approximation of Min-Distances in Near-Linear Time  \nYael Kirkpatrick∗  \nMIT  \nAbstract  \nWe study the problem of approximating the diameter of directed graphs under the mindistance measure, defined as dmin (u, v) = min(d(u, v), d (v, u)) . Unlike standard shortest-path distance, min-distance is not a metric, which renders many classical techniques inapplicable. Prior work has therefore focused on approximating this parameter, culminating in anapproximatio˜n-runtime tradeoff by Dalirrooyfard et al. [ICALP’19] giving a 4k − 1 approx  \nimation in O (mn1/(k+1) ) time for any positive integer k and, more recently, the first nearlinear time constant approximation by Chechik and Zhang [FOCS’22], where they obtained a 4-approximation to the min-diameter.  \nIn this work we present a randomized near-linear time algorithm that achieves a 3-approximation to the min-diameter, outperforming all known approximation–runtime tradeoffs. Our approach introduces a novel type-classification framework that may be of independent interest.  \nWe further extend our techniques to the more general setting of multimode graphs, recently introduced as a generalization of min-distance by Kirkpatrick and Vassilevska W. [MFCS’25] . For directed 2-mode graphs, we obtain a 3-approximation to the diameter in near-linear time, dramatically improving over the previously best known n-approximation. Our results significantly narrow the gap between min-distance and multimode distance approximations, and open new directions for understanding graph parameters under non-metric distance measures.  \n1 Introduction  \nThe graph diameter, defined as the maximum shortest-path distance between pairs of vertices, isone of the most central and widely studied graph parameters. The diameter serves as an indicator of graph complexity, and consequently many practical applications involve estimating it [CGLM12 , MLH09 , TK11 , BCH+15] . As computing the exact diameter is hard under fine-grained assumptions [RVW13], much work has gone into constructing approximation algorithms [ACIM99 , RVW13 , BRS+21 , CLR+14 , CGR16], studying the hardness of approximating this value [DLVW25 , BRS+21 , Li21 , Bon21 , DW21] as well as studying it in specialized settings [CD94 , PR05 , PRT12 , AHR+19] .  \nIn undirected graphs, the natural distance between a pair of points u, v is the length of the shortest path between them, d (u, v) . In directed graphs, however, this notion is no longer symmetric and the shortest path (or one-way) distance, from u to v , d (u, v), may differ significantly from the distance in the opposite direction, d (v, u) . In the context of the graph diameter, this asymmetry can lead to often unsatisfying answers, as the diameter of a graph with more than one strongly connected component is infinite under this one-way notion of distance. This in part  \nhas motivated the study of various notions of distance in directed graph, including the roundtrip ∗[yaelkirk@mit.edu](yaelkirk@mit.edu), supported by NSF Grant No 2141064.  \ndistance, d (u, v) + d(v, u) [CW99], the max-distance dmax(u, v) := max(d(u, v), d (v, u)) [AVWW16] and the min-distance dmin(u, v) := min(d(u, v), d (v, u)) [AVWW16] .  \nAs with the standard definition of distance, computing the diameter under these various notions of distance requires Ω(m2−ε ) under the Strong Exponential Time Hypothesis (SETH) [AVWW16 , RVW13] so we resort to search for approximation algorithms instead. Among these notions, the roundtrip distance and the max-distance are metrics, and therefore many algorithms for the traditional diameter extend naturally to these settings as well [BRS+21 , CLR+14 , RVW13] . Themin-distance, in contrast, is not a metric since it does not satisfy the triangle inequality. This has inspired a long line of work on approximating graph parameters in the min-distance setting, as no existing algorithms extend to this setting directly.  \nIn this paper we study th","cbCaipXn4EkRFZeA","https://ap.wps.com/l/cbCaipXn4EkRFZeA","pdf",693899,1,23,"English","en",105,"# Introduction\n## Min-distance and related directed distance notions\n## Min-diameter approximation history and open problems\n## Multimode graphs and the paper’s contributions","[{\"question\":\"What is the min-distance measure and why is it challenging to use?\",\"answer\":\"Min-distance is defined as dmin(u,v)=min(d(u,v),d(v,u)). It is not a metric because it does not satisfy the triangle inequality, so many standard algorithmic tools do not directly apply.\"},{\"question\":\"What approximation result does the paper achieve for the min-diameter?\",\"answer\":\"It provides a randomized near-linear time algorithm that achieves a 3-approximation to the min-diameter, improving on previously known approximation–runtime tradeoffs.\"},{\"question\":\"How do the results extend beyond min-distance to multimode graphs?\",\"answer\":\"The paper extends its techniques to multimode graphs, a generalization of min-distance. For directed 2-mode graphs, it obtains a 3-approximation to the diameter in near-linear time, substantially improving over the best prior n-approximation.\"}]",1784180227,58,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"improved-approximation-of-min-distances-in-near-linear-time","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/improved-approximation-of-min-distances-in-near-linear-time/82416/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","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 is the min-distance measure and why is it challenging to use?","Question",{"text":75,"@type":76},"Min-distance is defined as dmin(u,v)=min(d(u,v),d(v,u)). It is not a metric because it does not satisfy the triangle inequality, so many standard algorithmic tools do not directly apply.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What approximation result does the paper achieve for the min-diameter?",{"text":80,"@type":76},"It provides a randomized near-linear time algorithm that achieves a 3-approximation to the min-diameter, improving on previously known approximation–runtime tradeoffs.",{"name":82,"@type":73,"acceptedAnswer":83},"How do the results extend beyond min-distance to multimode graphs?",{"text":84,"@type":76},"The paper extends its techniques to multimode graphs, a generalization of min-distance. For directed 2-mode graphs, it obtains a 3-approximation to the diameter in near-linear time, substantially improving over the best prior n-approximation.","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":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":45,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":45,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":45,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":45,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":45,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":45,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":45,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]