[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-86004-en":3,"doc-seo-86004-105":30,"detail-sidebar-cat-0-en-105":92},{"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},86004,1099514067415,"Rowan","https://ap-avatar.wpscdn.com/avatar/100002539d78ffe74a7?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779092875211072502",8,"Research & Report","Fully Dynamic Edge Connectivity in Õ(n^(1213)) Time","Fully dynamic edge connectivity focuses on maintaining the edge connectivity λG of an n-vertex graph under edge insertions and deletions. The work presents a randomized algorithm for dynamic simple graphs with worst-case update and query time O(n^12/13), covering all values of λG. This achieves the first sublinear-in-n bounds for every connectivity regime. It further derives deterministic variants: exact edge connectivity with n^1+o(1) worst-case time (or O(n) amortized), and a deterministic algorithm for dynamic unweighted multigraphs with O(n^3/2) worst-case update and query time.","arXiv :2607 . 10689v 1 [ cs .DS] 12 Jul 2026  \nFully Dynamic Edge Connectivity in ˜O(n12/13) Time  \nYotam Kenneth-Mordoch∗ Robert Krauthgamer†  \nJuly 14, 2026  \nAbstract  \nIn the (fully) dynamic edge connectivity problem, the goal is to maintain the edge connectivity λG of an n-vertex graph G that undergoes edge insertions and deletions. Our main result is a randomized algorithm for mai˜ntaining edge connectivity in dynamic simple graphs using worst  \ncase update and query time O(n12/13), for all values of λG . This is the first algorithm that haso (n) update and query time, as all existing algorithms achieve this only when λ G is below n1/11 or above n 1/2 (up to polylogarithmic factors) . We then use the tools developed for this purpose to design two additional algorithms. The first one is a determin˜istic algorithm for the exact same  \ntask, that uses n 1+o(1) worst-case update and query time or O (n) amortized update and query time; this gives a polynomial improvement over existing deterministic algorithms. The second one is˜a deterministic algorithm for the same task but in dynamic unweighted multigraphs, that  \nuses O(n3/2) worst-case update and query time.  \n∗ Weizmann Institute of Science, Rehovot, Israel ([yotam.kenneth@weizmann.ac.il](yotam.kenneth@weizmann.ac.il)).  \n†Weizmann Institute of Science, Rehovot, Israel ([robert.krauthgamer@weizmann.ac.il](robert.krauthgamer@weizmann.ac.il)) . The Harry Weinrebe Professorial Chair of Computer Science. Work partially supported by the Israel Science Foundation grant 1336/23, by the Israeli Council for Higher Education (CHE) via the Weizmann Data Science Research Center, and by a research grant from the Estate of Harry Schutzman.  \n1 Introduction  \nThe edge connectivity of an (unweighted) graph G, denoted λG , is the minimum number of edges that need to be removed from G to disconnect it. This fundamental graph parameter has been studied across a wide range of computational models, including the classical sequential setting [NI92, KS96 , SW97 , Kar00 , GMW20 , MN20], parallel computation [GG18], the distributed Congest model [GK13, DHNS19 , DEMN21], graph streams [MN20, AD21], and the cut-query and quantum query models [RSW18, AEG+22 , AL21] . We focus on maintaining the edge connectivity in the fully dynamic setting, where the input graph G = (V, E) on a fixed set of n vertices that undergoes edge insertions and deletions. In this setting, the goal is to minimize the time it takes to update the data structure after an edge insertion or deletion (called update time) and the time it takes to query the edge connectivity (called query time) . Throughout, the worst-case update time is the maximum time taken by the algorithm to process any single update in any update sequence. When we mention only the update time it is implied that the query time is the same. Similarly, when the type of update time is not specified, we refer to the worst-case and not, say, amortized.  \nMaintaining edge connectivity in the fully dynamic setting has a rich history spanning over 40 years [Fre85, GI91] . Early work focused on the regime where the edge connectivity λ G is a small constant, giving efficient algorithms for λG ∈ {1, 2 , 3 , 4} [Fre85, GI91 , EGIN97 , HdLT01 , KKM13] . Recent work on the bounded connectivity regime has given efficient algorithms for values of λ G up to no(1) using no(1) worst-case update time [JST24, EHL25 , EHL26] .  \nThe first algorithm for polynomial values of λG was given in [Tho07]; it uses ˜O( √nλ7max) worstca˜ se update time, where λmax is the maximum edge connectivity throughout the execution and O˜ (·) hides polylogarithmic factors in n. A similar technique was later used in [dVC25] to obtain O ( √nλ5m.5ax) worst-case update time, which is sublinear in the number of vertices whe˜n λ max \u003C  \nn 1/11/polylog(n) . Unfortunately, when λmax ≥ n3/11, this bound is no better than O (n2 ), the trivial solution of recomputing the edge connect˜ivity from scratch after each ","cbCailVHHAZrCG8N","https://ap.wps.com/l/cbCailVHHAZrCG8N","pdf",473189,5,1,27,"English","en",105,"# Introduction\n## Results","[{\"question\":\"What problem does the document address in fully dynamic graphs?\",\"answer\":\"It addresses maintaining the edge connectivity λG of a graph while edges are inserted and deleted dynamically.\"},{\"question\":\"What is the main performance guarantee of the proposed randomized algorithm?\",\"answer\":\"For dynamic simple graphs, it provides worst-case update and query time O(n^12/13) for all values of λG.\"},{\"question\":\"What deterministic algorithms are presented as extensions?\",\"answer\":\"One deterministic algorithm maintains exact edge connectivity with n^1+o(1) worst-case update/query time (or O(n) amortized), and another handles dynamic unweighted multigraphs with O(n^3/2) worst-case update/query time.\"}]",1784207719,68,{"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":87,"head_meta":89,"extra_data":91,"updated_unix":28},"fully-dynamic-edge-connectivity-in-on1213-time","",{"@graph":36,"@context":86},[37,54,69],{"@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":53},"https://docshare.wps.com/document/fully-dynamic-edge-connectivity-in-on1213-time/86004/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":24,"description":14,"dateModified":62,"datePublished":63,"encodingFormat":61,"isAccessibleForFree":64,"interactionStatistic":65},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-07-27","2026-07-16",true,{"@type":66,"interactionType":67,"userInteractionCount":20},"InteractionCounter",{"@type":68},"ViewAction",{"@type":70,"mainEntity":71},"FAQPage",[72,78,82],{"name":73,"@type":74,"acceptedAnswer":75},"What problem does the document address in fully dynamic graphs?","Question",{"text":76,"@type":77},"It addresses maintaining the edge connectivity λG of a graph while edges are inserted and deleted dynamically.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"What is the main performance guarantee of the proposed randomized algorithm?",{"text":81,"@type":77},"For dynamic simple graphs, it provides worst-case update and query time O(n^12/13) for all values of λG.",{"name":83,"@type":74,"acceptedAnswer":84},"What deterministic algorithms are presented as extensions?",{"text":85,"@type":77},"One deterministic algorithm maintains exact edge connectivity with n^1+o(1) worst-case update/query time (or O(n) amortized), and another handles dynamic unweighted multigraphs with O(n^3/2) worst-case update/query time.","https://schema.org",{"og:url":52,"og:type":88,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":90,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":93},[94,98,102,106,110,115,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":95,"show_sort_weight":96,"slug":97},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":99,"show_sort_weight":100,"slug":101},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":103,"show_sort_weight":104,"slug":105},"Exam",70,"exam",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},"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":20,"slug":138},19,"General","general"]