[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82564-en":3,"doc-seo-82564-105":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},82564,962075006959,"Anda","https://ap-avatar.wpscdn.com/avatar/e0002397efbe92a78e?_k=1776741047341049297",8,"Research & Report","Space-Optimal Sensitivity Oracles for Single-Source Mincuts","Research focuses on Single-Source Mincut Sensitivity Oracles for undirected multigraphs with a fixed source s. Given an edge insertion or failure query, the oracle reports the affected vertices whose mincut-to-s value changes. Existing insertion oracles achieve extremely compact O(n) space, while failure queries are typically handled with O(n^2) space methods. This work introduces an optimal O(n) space oracle with O(n) time output for failure queries, plus subquadratic-space variants with near-optimal query performance.","arXiv :2607 .00894v 1 [ cs .DS] 1 Jul 2026  \nSpace-Optimal Sensitivity Oracles for Single-Source Mincuts  \nKoustav Bhanja * Merav Parter † Asaf Petruschka ‡  \nAbstract  \nLet G be an undirected multi-graph on n vertices, with a designated source vertex s. We study Single-Source Mincut Sensitivity Oracles: compact data structures that, when queried with an edge e, report those affected vertices whose mincut value to s changes upon the insertion or failure of e.  \nInsertion queries were treated by Baswana, Gupta, and Knollmann [Algorithmica ’22], who showed an extremely compact oracle with only O (n) space. In this work, we consider edge failure queries, which are of even greater interest, but far more challenging. The current-best approaches give O (n2 ) space: either using n − 1 fixed-pair oracles of O (n) space each, based on the Picard-Queyranne representation [MPS ’80], or using the O(n2 ) space all-pairs oracle by Baswana and Pandey [SODA’22] .  \n• Our key result is an optimal O (n) space single-source mincut sensitivity oracle for edge failure queries. It reports the set of affected vertices in O (n) time, thus matching the state-of-the-art bounds for the insertion case.  \n• Additionally, we provide oracles with near-optimal query times at the cost of increasing the space to O(n1.5 ) . They can determine if any given vertex is affected by an insertion/failure of an edge in O(log n) time, or reports all affected vertices in amortized O(log3 n) time per vertex. Such oracles of subquadratic space were previously unknown, even for insertion.  \nOur main technical contribution is in establishing novel and intricate connections between two seemingly distant objects, representing two different families of mincuts. The first is the DAG representation of farthest mincuts to the source, which was the central tool introduced by Baswana, Gupta, and Knollmann. The second is the Connectivity Carcass for Steiner mincuts of Dinitz and Vainshtein [STOC’94], which generalizes well-known cactus representations of global mincuts. Our work demonstrates the relatively unexplored potential of the carcass beyond its “obvious” Steiner mincuts scope.  \n* Weizmann Institute of Science, Israel. Email: [koustav.bhanja@weizmann.ac.il](koustav.bhanja@weizmann.ac.il. Supported by the Koshland Prize)[. Supported by the Koshland Prize](koustav.bhanja@weizmann.ac.il. Supported by the Koshland Prize)[ ](koustav.bhanja@weizmann.ac.il. Supported by the Koshland Prize)fellowship, and Merav Parter’s European Research Council (ERC) grant under the European Union’s Horizon 2020 research and innovation programme, grant agreement No. 949083.  \n†Weizmann Institute of Science, Israel. Email: [merav.parter@weizmann.ac.il](merav.parter@weizmann.ac.il. Supported partially by the European)[. Supported partially by the European](merav.parter@weizmann.ac.il. Supported partially by the European)[ ](merav.parter@weizmann.ac.il. Supported partially by the European)Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme, grant agreement No. 949083.  \n‡Weizmann Institute of Science, Israel. Email: [asaf.petruschka@weizmann.ac.il](asaf.petruschka@weizmann.ac.il. Supported by an Azrieli Founda)[. Supported by an Azrieli Founda](asaf.petruschka@weizmann.ac.il. Supported by an Azrieli Founda)tion fellowship, and by Merav Parter’s European Research Council (ERC) grant under the European Union’s Horizon 2020 research and innovation programme, grant agreement No. 949083.  \nContents  \n1 Introduction 1  \n2 Preliminaries 3  \n2.1 Essentials of the Connectivity Carcass .................................. 4  \n3 Technical Overview 6  \n3.1 From Insertion to Failure ......................................... 6  \n3.2 Key Insight: The Bridge to the Carcass .................................. 7  \n3.3 Utilizing Skeletons for O(m) Space and O(n) Query Time ....................... 8  \n3.4 Utilizing Projections Towards O(n) Space ................................ 9 ","cbCaiubO0YdFptxH","https://ap.wps.com/l/cbCaiubO0YdFptxH","pdf",2646615,1,50,"English","en",105,"# Introduction\n# Preliminaries\n## Essentials of the Connectivity Carcass\n# Technical Overview\n## From Insertion to Failure\n## Key Insight: The Bridge to the Carcass\n## Utilizing Skeletons for O(m) Space and O(n) Query Time\n## Utilizing Projections Towards O(n) Space\n## Organization\n# Carcass Tools\n# The Representatives Framework\n# Anchors and Query Translation\n# Mincuts Splitting the LCA\n## Construction\n## Query Algorithm\n## Correctness\n# Mincuts Not Splitting the LCA\n## Construction\n## Query Algorithm\n## Correctness\n# Data Structures for Fast Queries: Output-Sensitive and One-Destination\n## Ancestry and LCA in Farthest Mincut DAG\n## Extensions of The Representatives Framework","[{\"question\":\"What problem do Single-Source Mincut Sensitivity Oracles address?\",\"answer\":\"They provide compact data structures for undirected multigraphs with a fixed source s, answering edge insertion or failure queries by reporting vertices whose mincut value to s changes.\"},{\"question\":\"How do edge failure queries differ from edge insertion queries in this work?\",\"answer\":\"Failure queries are highlighted as more challenging, with best-known approaches using O(n^2) space, while insertion queries can already be answered with extremely compact O(n) space.\"},{\"question\":\"What is the main space and query-time result for edge failure queries?\",\"answer\":\"The work presents an optimal O(n) space oracle for edge failure queries that reports all affected vertices in O(n) time, matching state-of-the-art bounds from the insertion case.\"}]",1784181553,126,{"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":86,"head_meta":88,"extra_data":90,"updated_unix":27},"space-optimal-sensitivity-oracles-for-single-source-mincuts","",{"@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/space-optimal-sensitivity-oracles-for-single-source-mincuts/82564/",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 do Single-Source Mincut Sensitivity Oracles address?","Question",{"text":75,"@type":76},"They provide compact data structures for undirected multigraphs with a fixed source s, answering edge insertion or failure queries by reporting vertices whose mincut value to s changes.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How do edge failure queries differ from edge insertion queries in this work?",{"text":80,"@type":76},"Failure queries are highlighted as more challenging, with best-known approaches using O(n^2) space, while insertion queries can already be answered with extremely compact O(n) space.",{"name":82,"@type":73,"acceptedAnswer":83},"What is the main space and query-time result for edge failure queries?",{"text":84,"@type":76},"The work presents an optimal O(n) space oracle for edge failure queries that reports all affected vertices in O(n) time, matching state-of-the-art bounds from the insertion 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