[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83622-en":3,"doc-seo-83622-105":29,"detail-sidebar-cat-0-en-105":90},{"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":4,"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},83622,16904993612988,"Olivia Brown","https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd",8,"Research & Report","Scalable and Distributed Silhouette Approximation","Silhouette analysis serves as a widely used internal quality measure for evaluating the quality of a k-clustering, both at the element level (local silhouette) and at the clustering level (global silhouette). Exact computation under general metric spaces requires Θ(n^2) distance evaluations, making it impractical for modern large-scale datasets. Existing approximate approaches lack provable, controllable guarantees. This work presents rigorous sampling-based algorithms estimating local and global silhouette with additive error O(ε) with probability at least 1−δ, and provides a scalable MapReduce/MPC distributed design.","Scalable and Distributed Silhouette Approximation  \narXiv :2607 .0 1993v 1 [ cs .DS] 2 Jul 2026  \nIlie Sarpe  \n[ilsarpe@kth.se](ilsarpe@kth.se)  \nKTH Royal Institute of Technology  \nFederico Altieri  \n[altierif87@gmail.com](altierif87@gmail.com)  \nUniversity of Padova  \nAndrea Pietracaprina [capri@dei.unipd.it](capri@dei.unipd.it)[ ](capri@dei.unipd.it)University of Padova  \nGeppino Pucci  \n[geppo@dei.unipd.it](geppo@dei.unipd.it)[ ](geppo@dei.unipd.it)University of Padova  \nFabio Vandin  \n[fabio.vandin@unipd.it](fabio.vandin@unipd.it)  \nUniversity of Padova  \nAbstract  \nThe silhouette is one of the most widely used measures to assess the quality of a 􀀺 -clustering of a dataset of 􀀽 elements. Its popularity stems from the fact that its evaluation requires no information beyond the clustering assignment. In addition, the silhouette is extremely easy to interpret, with a plethora of applications in various domains. The silhouette provides a score to measure the quality of a clustering as a whole or for each individual element. However, the exact computation of the: (􀀸) silhouette of each element of a dataset; and (􀀸􀀸) the global silhouette of the entire clustering; require Θ(􀀽2 ) distance calculations, under general metrics. The quadratic complexity Θ(􀀽2 ) is extremely prohibitive, especially on massive modern datasets. Surprisingly, existing approximate methods using O(􀀽2 ) distance calculations are heuristics and do not offer provable and controllable guarantees on the quality of their results.  \nIn this work, we introduce the first rigorous and efficient algorithms to estimate: (􀀸) the (local) silhouette of each element of a dataset; and (􀀸􀀸) the (global) silhouette; of any metric 􀀺-clustering. Our methods are based on sampling, performing O(􀀽􀀺􀁙−2 ln (􀀽􀀺/􀁘)) distance computations, and providing estimates with additive error O(􀁙) with probability at least 1 − 􀁘 . That is, the user-defined parameters 􀁙 and 􀁘 in (0, 1) control the trade-off between accuracy and efficiency. Furthermore, we introduce a scalable and distributed design of our methods for the MapReduce and Massively Parallel Computing (MPC) frameworks. Our distributed algorithms require a constant number of rounds and sublinear local memory, under practical parameter setting. Finally, we perform extensive experiments to compare our methods with state-of-the-art approaches. The results show that our new techniques yield the best trade-off between accuracy and efficiency for both local and global silhouette estimation. In addition, our methods scale efficiently to massive datasets for which an exact computation of the silhouette is not practical.  \n1 Introduction  \nClustering is of fundamental importance for data analysis, with ubiquitous applications in various areas including pattern recognition [Xu and Tian, 2015], bioinformatics and biomedicine [Ikotunet al., 2023], and data management [Aggarwal and Reddy, 2014, Karypis and Kumar, 1998] . Broadly, clustering a dataset requires grouping elements with high similarity, and separating dissimilar elements [Kleinberg, 2002] . The clustering of a dataset is often viewed as an optimization problem, where the objective varies according to the desired properties of the optimal clustering. A large number of algorithms have been designed to cluster a dataset according to various objectives [Hennig et al., 2015] . In particular, recent research for clustering methods has its focus on the design of rigorous and scalable approaches to cluster massive datasets [Bargerand Feldman, 2016, Ceccarello et al., 2017, 2019, Malkomes et al., 2015, Mazzetto et al., 2019, Pranjal and Balcan, 2013] .  \nThe vast number of clustering objectives and methods, introduce the need for rigorous methods to identify “good quality” clusterings of a dataset [Liu et al., 2010, Schubert, 2023] . Clustering evaluation (or simply validation) assesses the quality of a given clustering relying on an evaluation measure. An evaluation measure is either external ","cbCaipcxkmo6L4dd","https://ap.wps.com/l/cbCaipcxkmo6L4dd","pdf",1898757,1,50,"English","en",105,"# Introduction\n## Clustering and validation\n## Silhouette as an internal measure\n## Local and global silhouette definitions","[{\"question\":\"Why is exact silhouette computation too expensive on large datasets?\",\"answer\":\"Exact local and global silhouette computations require Θ(n^2) distance calculations under general metrics, which becomes prohibitive for massive modern datasets.\"},{\"question\":\"What problem do the proposed methods address?\",\"answer\":\"They introduce efficient, provably correct algorithms to estimate both the local silhouette of each element and the global silhouette of a k-clustering for any metric space.\"},{\"question\":\"How do the methods balance accuracy and efficiency in practice?\",\"answer\":\"The sampling approach uses user-controlled parameters (ε, δ) in (0,1), where ε sets the additive error target and δ controls the failure probability, enabling a tunable trade-off.\"}]",1784189340,126,{"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":85,"head_meta":87,"extra_data":89,"updated_unix":27},"scalable-and-distributed-silhouette-approximation","",{"@graph":35,"@context":84},[36,53,67],{"@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/scalable-and-distributed-silhouette-approximation/83622/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":61,"encodingFormat":60,"isAccessibleForFree":62,"interactionStatistic":63},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-16",true,{"@type":64,"interactionType":65,"userInteractionCount":4},"InteractionCounter",{"@type":66},"ViewAction",{"@type":68,"mainEntity":69},"FAQPage",[70,76,80],{"name":71,"@type":72,"acceptedAnswer":73},"Why is exact silhouette computation too expensive on large datasets?","Question",{"text":74,"@type":75},"Exact local and global silhouette computations require Θ(n^2) distance calculations under general metrics, which becomes prohibitive for massive modern datasets.","Answer",{"name":77,"@type":72,"acceptedAnswer":78},"What problem do the proposed methods address?",{"text":79,"@type":75},"They introduce efficient, provably correct algorithms to estimate both the local silhouette of each element and the global silhouette of a k-clustering for any metric space.",{"name":81,"@type":72,"acceptedAnswer":82},"How do the methods balance accuracy and efficiency in practice?",{"text":83,"@type":75},"The sampling approach uses user-controlled parameters (ε, δ) in (0,1), where ε sets the additive error target and δ controls the failure probability, enabling a tunable trade-off.","https://schema.org",{"og:url":51,"og:type":86,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":88,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":91},[92,96,100,104,109,113,118,121,126,129,133],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":93,"show_sort_weight":94,"slug":95},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":97,"show_sort_weight":98,"slug":99},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":101,"show_sort_weight":102,"slug":103},"Exam",70,"exam",{"id":105,"doc_module":4,"doc_module_name":45,"category_name":106,"show_sort_weight":107,"slug":108},5,"Comic",60,"comic",{"id":110,"doc_module":4,"doc_module_name":45,"category_name":111,"show_sort_weight":21,"slug":112},6,"Technology","technology",{"id":114,"doc_module":4,"doc_module_name":45,"category_name":115,"show_sort_weight":116,"slug":117},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":119,"slug":120},30,"research-report",{"id":122,"doc_module":4,"doc_module_name":45,"category_name":123,"show_sort_weight":124,"slug":125},9,"Religion & Spirituality",20,"religion-spirituality",{"id":124,"doc_module":4,"doc_module_name":45,"category_name":127,"show_sort_weight":124,"slug":128},"World Cup","world-cup",{"id":130,"doc_module":4,"doc_module_name":45,"category_name":131,"show_sort_weight":130,"slug":132},10,"Lifestyle","lifestyle",{"id":134,"doc_module":4,"doc_module_name":45,"category_name":135,"show_sort_weight":105,"slug":136},19,"General","general"]