[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-151696-en":3,"doc-seo-151696-105":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":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":27,"seo_description":14,"update_tm":28,"read_time":29},151696,5909877438554,"Maeve","https://ap-avatar.wpscdn.com/avatar/5600025385ad2bf12a7?_k=1778553567797529272",8,"Research & Report","Fully-Dynamic Graph Sparsifiers Against an Adaptive Adversary - Article No. 20 - ICALP 2022","Designing efficient dynamic graph algorithms against an adaptive adversary is a major goal in dynamic graph algorithms, with recent progress for matching and decremental shortest paths. This paper introduces efficient adaptive algorithms for maintaining multiple sparsifier types under an adaptive adversary: polylog(n)-spanners, O(k)-approximate cut sparsifiers, and polylog(n)-approximate spectral sparsifiers. The results work against a stronger adversary that accesses random bits, and can be converted to worst-case update times with sub-polynomial factors. The techniques use an expander reduction and proactive resampling to control adversarial dependencies.","Fully-Dynamic Graph Sparsifiers Against an Adaptive Adversary  \nAaron Bernstein  \nRutgers University, Piscataway, NJ, USA  \nJan van den Brand  \nSimons Institute, Berkeley, CA, USA  \nUniversity of California Berkeley, CA, USA  \nMaximilian Probst Gutenberg  \nETH Zürich, Switzerland  \nDanupon Nanongkai  \nUniversity of Copenhagen, Denmark  \nKTH Royal Institute of Technology, Stockholm, Sweden  \nThatchaphol Saranurak  \nUniversity of Michigan, Ann Arbor, MI, USA  \nAaron Sidford  \nStanford University, CA, USA  \nHe Sun  \nUniversity of Edinburgh, UK  \n~~ Abstract ~~  \nDesigning efficient dynamic graph algorithms against an adaptive adversary is a major goal in the field of dynamic graph algorithms and has witnessed many exciting recent developments in, e.g., dynamic matching (Wajc STOC’20) and decremental shortest paths (Chuzhoy and Khanna STOC’19) . Compared to other graph primitives (e.g. spanning trees and matchings), designing such algorithms for graph spanners and (more broadly) graph sparsifiers poses a unique challenge since there is no fast deterministic algorithm known for static computation and the lack of a way to adjust the output slowly (known as “small recourse/replacements”) .  \nThis paper presents the first non-trivial efficient adaptive algorithms for maintaining many  \nsparsifiers against an adaptive ad˜versary. Specifically, we present algorithms that maintain 1. a polylog(n)-spanner of size O (n) in polylog˜ (n) amo˜rtized update time,  \n2. an O (k)-approximate cut sparsifier of size O (n) in O(n1/k ) amortized update time, and  \n3. a polylog(n)-approximate spectral sparsifier in polylog(n) amortized update time.  \nOur bounds are the first non-trivial ones even when only the recourse is concerned. Our results hold even against a stronger adversary, who can access the random bits previously used by the algorithms and the amortized update time of all algorithms can be made worst-case by paying sub-polynomial factors. Our spanner result resolves an open question by Ahmed et al. (2019) and our results and techniques imply additional improvements over existing results, including (i) answering open questions about decremental single-source shortest paths by Chuzhoy and Khanna (STOC’19) and Gutenberg and Wulff-Nilsen (SODA’20), implying a nearly-quadratic time algorithm for approximating minimum-cost unit-capacity flow and (ii) de-amortizing a result of Abraham et al. (FOCS’16) for dynamic spectral sparsifiers.  \nOur results are based on two novel techniques. The first technique is a generic black-box reduction that allows us to assume that the graph is initially an expander with almost uniform-degree and, more importantly, stays as an almost uniform-degree expander while undergoing only edge deletions. The second technique is called proactive resampling: here we constantly re-sample parts of the input graph so that, independent of an adversary’s computational power, a desired structure of the underlying graph can be always maintained. Despite its simplicity, the analysis of this sampling scheme is far from trivial, because the adversary can potentially create dependencies between the random choices used by the algorithm. We believe these two techniques could be useful for developing other adaptive algorithms.  \n© Aaron Bernstein, Jan van den Brand, Maximilian Probst Gutenberg,  \nDanupon Nanongkai, Thatchaphol Saranurak, Aaron Sidford, and He Sun;  \nlicensed under Creative Commons License CC-BY 4.0  \n49th International Colloquium on Automata, Languages, and Programming (ICALP 2022) .  \nEditors: Mikołaj Bojańczyk, Emanuela Merelli, and David P. Woodruff; Article No. 20; pp. 20:1–20:20  \nLeibniz International Proceedings in Informatics  \n Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany  \nS  \nC  \nT  \nA  \nE  \n20:2 Fully-Dynamic Graph Sparsifiers Against an Adaptive Adversary  \n2012 ACM Subject Classification Theory of computation → Dynamic graph algorithms Keywords and phrases dynamic graph algo","cbCaisrPw2ljmuNp","https://ap.wps.com/l/cbCaisrPw2ljmuNp","pdf",761037,1,20,"English","en",105,"# Introduction\n## Dynamic graph algorithms and adversary models\n## Problem focus: maintaining spanners and sparsifiers adaptively","[{\"question\":\"What problem does the paper address in dynamic graph algorithms?\",\"answer\":\"It studies efficient maintenance of graph spanners and sparsifiers when the adversary is adaptive, meaning updates can depend on prior algorithm behavior.\"},{\"question\":\"What three main sparsifier/spanner constructions does the paper provide?\",\"answer\":\"It presents algorithms for maintaining (1) polylog(n)-spanners, (2) O(k)-approximate cut sparsifiers, and (3) polylog(n)-approximate spectral sparsifiers.\"},{\"question\":\"Which two core techniques underpin the adaptive algorithms?\",\"answer\":\"The paper uses a generic black-box reduction to an almost uniform-degree expander under edge deletions, and proactive resampling that repeatedly resamples parts of the graph to preserve desired structure.\"}]","Fully-Dynamic Graph Sparsifiers Against an Adaptive Adversary - Article No. 20 - ICALP 2022 | PDF",1787847520,50,{"code":4,"msg":31,"data":32},"ok",{"site_id":24,"language":23,"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-graph-sparsifiers-against-an-adaptive-adversary-article-no-20-icalp-2022","",{"@graph":36,"@context":86},[37,54,69],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":20},"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-graph-sparsifiers-against-an-adaptive-adversary-article-no-20-icalp-2022/151696/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":23,"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","2026-09-04","2026-08-27",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 paper address in dynamic graph algorithms?","Question",{"text":76,"@type":77},"It studies efficient maintenance of graph spanners and sparsifiers when the adversary is adaptive, meaning updates can depend on prior algorithm behavior.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"What three main sparsifier/spanner constructions does the paper provide?",{"text":81,"@type":77},"It presents algorithms for maintaining (1) polylog(n)-spanners, (2) O(k)-approximate cut sparsifiers, and (3) polylog(n)-approximate spectral sparsifiers.",{"name":83,"@type":74,"acceptedAnswer":84},"Which two core techniques underpin the adaptive algorithms?",{"text":85,"@type":77},"The paper uses a generic black-box reduction to an almost uniform-degree expander under edge deletions, and proactive resampling that repeatedly resamples parts of the graph to preserve desired structure.","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":24},{"code":4,"msg":5,"data":93},[94,98,102,106,111,115,120,123,127,130,134],{"id":20,"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":107,"doc_module":4,"doc_module_name":46,"category_name":108,"show_sort_weight":109,"slug":110},5,"Comic",60,"comic",{"id":112,"doc_module":4,"doc_module_name":46,"category_name":113,"show_sort_weight":29,"slug":114},6,"Technology","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":21,"slug":126},9,"Religion & Spirituality","religion-spirituality",{"id":21,"doc_module":4,"doc_module_name":46,"category_name":128,"show_sort_weight":21,"slug":129},"World Cup","world-cup",{"id":131,"doc_module":4,"doc_module_name":46,"category_name":132,"show_sort_weight":131,"slug":133},10,"Lifestyle","lifestyle",{"id":135,"doc_module":4,"doc_module_name":46,"category_name":136,"show_sort_weight":107,"slug":137},19,"General","general"]