[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82381-en":3,"doc-seo-82381-105":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},82381,1099514068365,"Aurelia","https://ap-avatar.wpscdn.com/avatar/10000253d8d9f28188e?_k=1776742907772140068",8,"Research & Report","Short Graph Sketches Suffice for Error-resilient Leader Verification in CONGEST","Locally Checkable Proofs (LCPs) verify global graph properties using locally checkable certificates distributed by a prover. This paper studies Locally Checkable Proofs-with-Errors (LCPE), where an adversary corrupts ε certificates, focusing on efficient implementations in the bandwidth-restricted CONGEST model. Efficient CONGEST verification is hindered by neighborhood information requirements, so the work introduces local graph sketches and imagined trees/certifications. The algorithm tolerates up to ε adversarial errors in O(ε2) rounds and is matched by a lower bound showing view distance ε is necessary.","arXiv :2607 .09522v1 [ cs .DC] 10 Jul 2026  \nShort Graph Sketches Suffice for Error-resilient Leader Verification in CONGEST  \nPawel Garncarek 1∗, Tomasz Jurdzi´nski 1†, Dariusz Kowalski2‡, Subhajit Pramanick 1§  \n1 Institute of Computer Science, University of Wroclaw, Poland  \n2 Department of Computer Science, Augusta University, USA  \nAbstract  \nLocally Checkable Proofs (LCPs) enable the verification of global graph properties from locally checkable certificates assigned by a prover. This framework has recently been extended to Locally Checkable Proofs-with-Errors (LCPE), in which an adversary may corrupt some of the certificates. Existing LCPE algorithms are designed for the LOCAL model, whose unbounded communication makes them unsuitable for direct implementation in the (logarithmically) bandwidth-restricted CONGEST model.  \nIn this paper, we initiate the study of efficient CONGEST implementations of LCPE through the unique-leader verification problem on trees. The main challenge is that tolerating ε certificate errors requires each node to reason about its (2ε+1)-hop neighborhood, whose exact topology cannot, in the worst case, be communicated efficiently in Congest – reconstructing it naively costs up to O(∆2ε+1 log n) bits. To overcome this bottleneck, we introduce local graph sketches, together with the notions of imagined trees and imagined certifications, which compactly encode—in only O(ε2 log n) bits at each node—precisely the information a node needs to decide. Using these techniques, we design an algorithm that tolerates up to ε adversarial certificate errors and computes the required sketches in O (ε2 ) communication rounds in the CONGEST model.  \nWe complement this with a matching impossibility result: even in the strictly more powerful LOCAL model, and even with certificates of unbounded size, no verification scheme with view distance at most ε can tolerate ε adversarial errors. Since LOCAL is strictly stronger than CONGEST, this lower bound carries over immediately, showing that a view distance beyond ε– and hence the wider neighborhood our algorithm summarizes – is unavoidable.  \nKeywords: Verification problem, Leader election, Locally Checkable Proof, Local Certification, Error-resilience, CONGEST Model.  \n∗[pgarn@cs.uni.wroc.pl](pgarn@cs.uni.wroc.pl)[ ](pgarn@cs.uni.wroc.pl)†[tju@cs.uni.wroc.pl](tju@cs.uni.wroc.pl)[ ](tju@cs.uni.wroc.pl)‡[darek.liv@gmail.com](darek.liv@gmail.com)  \n§ [suvo.iitg17@gmail.com](suvo.iitg17@gmail.com)  \n1 Introduction  \n1.1 Background and Motivation  \nIn general, nodes in a distributed system do not have a global view of the network. Instead, each node can access only a bounded neighborhood around itself (sometimes only its direct neighborhood), whereas many properties of interest are inherently global. Examples include determining whether the network is a tree, whether a graph is bipartite, whether a computed structure is a valid spanning tree, or whether the system remains consistent after transient faults. Such verification tasks arise in many settings, including self-stabilizing and fault-tolerant systems. For many global properties, however, purely local verification is impossible without additional information, while verifying them from scratch requires excessive communication. This motivates equipping each node with a small amount of auxiliary information that makes local verification possible.  \nThe notion of locally verifying graph properties with the help of auxiliary information assigned to the nodes is captured by the framework of proof labeling schemes (PLS), introduced by Korman, Kutten, and Peleg [26] . In this framework, a prover first assigns a certificate (a binary string) to each node. Subsequently, a distributed verification algorithm, called the verifier, runs at every node and uses its own certificate, together with the certificates of the neighboring nodes, to output a binary decision: either accept or reject. The aim of the verifier is to distinguish bet","cbCairCI3kL3AXtO","https://ap.wps.com/l/cbCairCI3kL3AXtO","pdf",558489,1,21,"English","en",105,"# Introduction\n## Background and Motivation\n# Erroneous Certificates","[{\"question\":\"What problem does the paper address in distributed graph verification?\",\"answer\":\"The paper addresses implementing error-resilient leader verification for LCPE in the CONGEST model, where communication bandwidth is restricted.\"},{\"question\":\"Why are existing LCPE algorithms difficult to use directly in CONGEST?\",\"answer\":\"They are designed for the LOCAL model with unbounded communication; in CONGEST, the exact topology of the required (2ε+1)-hop neighborhood cannot be communicated efficiently.\"},{\"question\":\"What is the main technique used to overcome the CONGEST bottleneck?\",\"answer\":\"The paper uses local graph sketches plus notions of imagined trees and imagined certifications to compactly encode the exact information each node needs, enabling O(ε2) round CONGEST verification.\"}]",1784180043,53,{"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},"short-graph-sketches-suffice-for-error-resilient-leader-verification-in-congest","",{"@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/short-graph-sketches-suffice-for-error-resilient-leader-verification-in-congest/82381/",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 problem does the paper address in distributed graph verification?","Question",{"text":75,"@type":76},"The paper addresses implementing error-resilient leader verification for LCPE in the CONGEST model, where communication bandwidth is restricted.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"Why are existing LCPE algorithms difficult to use directly in CONGEST?",{"text":80,"@type":76},"They are designed for the LOCAL model with unbounded communication; in CONGEST, the exact topology of the required (2ε+1)-hop neighborhood cannot be communicated efficiently.",{"name":82,"@type":73,"acceptedAnswer":83},"What is the main technique used to overcome the CONGEST bottleneck?",{"text":84,"@type":76},"The paper uses local graph sketches plus notions of imagined trees and imagined certifications to compactly encode the exact information each node needs, enabling O(ε2) round CONGEST verification.","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"]