[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84250-en":3,"doc-seo-84250-105":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},84250,13056703019662,"Evangeline","https://ap-avatar.wpscdn.com/avatar/be000253a8e92610077?_k=1778726343310543188",8,"Research & Report","Faster Randomized and Deterministic k-Clustering on Graphs","Paper studies (k,z)-clustering and k-center on graphs, where (k,z)-clustering unifies k-median (z=1) and k-means (z=2). It presents three main algorithmic results: a first deterministic O(m)-time (2+ε)-approximation for k-center that resolves a cited open problem and also extends to k-center with outliers; a randomized O(m)-time O(1)-approximation for (k,z)-clustering; and a deterministic O(poly(c))-approximation running in O(m^(1+1/c)) using Thorup–Zwick distance oracle techniques.","arXiv :2607 .076 15v 1 [ cs .DS] 8 Jul 2026  \nFaster Randomized and Deterministic k-Clustering on Graphs Sebastian Forster∗1, Yasamin Nazari†2,3, Rajath Rao K.N.‡2, and Antonis Skarlatos §4  \n1 Department of Computer Science, University of Salzburg, Austria  \n2Vrije Universiteit Amsterdam, The Netherlands  \n3 CWI Amsterdam, The Netherlands  \n4 Department of Computer Science, University of Warwick, Coventry, UK  \nJuly 9, 2026  \nAbstract  \nIn this paper, we study the (k, z)-clustering and k-center problems on graphs, where (k, z)-clustering generalizes the k-median (z = 1) and k-means (z = 2) problems. In this work, we obtain the following main results:  \n• Our first contribution is the first ˜deterministic algorithm for k-center on graphs that  \nachieves a (2+ε)-approximation in O (m) time. This affirmatively resolves an open problem raised by Abboud, Cohen-Addad, Lee, and Manurangsi [1] . Our techniques also extend to the k-center with outliers problem, where up to t points may be discarded.  \n• Our second contribution is a rando˜mized algorithm for (k, z)-clustering on graphs that achieves an O(1)-approximation in O (m) time˜, which in particular covers k-median (z = 1)  \nand k-means (z = 2) . Prior to this work, an O (m)-time randomized algorithm was known for k-med˜ian by Thorup [52] . Moreover, a recent work of Jiang, Jin, Lou, and Lu [41]  \nachieves O (m1+o(1) ) time for general z via local search.  \n• Finally, we design a deterministi˜c algorithm for (k, z)-clustering on graphs that achieves  \na O (poly(c))-approximation in O (m1+1/c ) time, for a positive parameter c. To obtain this result, we use techniques from the Thorup–Zwick distance oracle [55] . This technical connection may be of independent interest, considering the wide application of [55] in various computational settings.  \nWe emphasize that most of our algorithms are incremental in the sense that for any given parameter k, they return a sequence of centers such that any prefix of length ℓ ≤ k yields an ℓ-clustering solution with a constant-factor approximation.  \n∗ [forster@cs.sbg.ac.at](forster@cs.sbg.ac.at)[ ](forster@cs.sbg.ac.at)†[y.nazari@vu.nl](y.nazari@vu.nl)  \n‡[r.raokamblanagendra@vu.nl](r.raokamblanagendra@vu.nl)  \n§ [antonis.skarlatos@warwick.ac.uk](antonis.skarlatos@warwick.ac.uk)  \nContents  \n1 Introduction 3  \n1.1 Our Results and the State of the Art ........................... 4  \n2 Technical Overview 7  \n3 Deterministic Greedy k-Center and Faster k-Center with Outlier 8  \n4 Greedy (k, z)-Clustering Algorithm 11  \n5 Randomized (k, z)-Clustering Algorithm on Graphs 12  \n5.1 Analysis of the Randomized (k, z)-Clustering Algorithm ................ 14  \n5.1.1 Approximation Ratio ................................ 17  \n6 Deterministic (k, z)-Clustering Algorithm on Graphs 18  \n6.1 Thorup–Zwick distance oracle ............................... 19  \n6.2 Approximate Balls via Truncated Bunches and Clusters................. 20  \nA Approximation of Value(·, ·) 27  \nB k-Center with Outliers 27  \nB.1 2-Approximation for Constant k ............................. 28  \nB.2 Randomized Bicriteria Approximation for General k .................. 30  \nC Cohen’s Algorithm 32  \n1 Introduction  \nClustering is a fundamental task in data mining and unsupervised learning with numerous applications across various domains [37 , 32 , 8 , 2] . Intuitively speaking, the goal of a clustering algorithm is to group the input data together, in structures called “clusters”, such that similar data points are contained in the same cluster.  \nA very popular, and arguably also very intuitive, clustering paradigm is partitional clustering in which the clusters are represented by a set of centers of a prespecified size k and each point in the data set is assigned to the cluster of its closest center. The quality of a solution is measured based on an objective on the distances of points to their closest center.  \nFormally, this type of k-clustering can be formulated as an optimization proble","cbCaisuA2x3ODgMo","https://ap.wps.com/l/cbCaisuA2x3ODgMo","pdf",830823,6,1,32,"English","en",105,"# Introduction\n## Our Results and the State of the Art\n# Technical Overview\n# Deterministic Greedy k-Center and Faster k-Center with Outlier\n# Greedy (k, z)-Clustering Algorithm\n# Randomized (k, z)-Clustering Algorithm on Graphs\n# Deterministic (k, z)-Clustering Algorithm on Graphs","[{\"question\":\"What does (k,z)-clustering generalize in this paper?\",\"answer\":\"It generalizes k-median when z=1 and k-means when z=2 by minimizing the sum of powered distances to the nearest chosen center.\"},{\"question\":\"What is the main deterministic result for the k-center problem?\",\"answer\":\"The paper gives a deterministic algorithm achieving a (2+ε)-approximation for k-center on graphs in O(m) time, resolving an open problem cited in the abstract.\"},{\"question\":\"How do the randomized and deterministic approaches differ for (k,z)-clustering?\",\"answer\":\"The randomized algorithm provides an O(1)-approximation for (k,z)-clustering in O(m) time, while the deterministic method achieves an O(poly(c))-approximation in O(m^(1+1/c)) time using Thorup–Zwick distance oracle techniques.\"}]",1784194366,81,{"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},"faster-randomized-and-deterministic-k-clustering-on-graphs","",{"@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/faster-randomized-and-deterministic-k-clustering-on-graphs/84250/",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 does (k,z)-clustering generalize in this paper?","Question",{"text":76,"@type":77},"It generalizes k-median when z=1 and k-means when z=2 by minimizing the sum of powered distances to the nearest chosen center.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"What is the main deterministic result for the k-center problem?",{"text":81,"@type":77},"The paper gives a deterministic algorithm achieving a (2+ε)-approximation for k-center on graphs in O(m) time, resolving an open problem cited in the abstract.",{"name":83,"@type":74,"acceptedAnswer":84},"How do the randomized and deterministic approaches differ for (k,z)-clustering?",{"text":85,"@type":77},"The randomized algorithm provides an O(1)-approximation for (k,z)-clustering in O(m) time, while the deterministic method achieves an O(poly(c))-approximation in O(m^(1+1/c)) time using Thorup–Zwick distance oracle 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