[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84674-en":3,"doc-seo-84674-105":29,"detail-sidebar-cat-0-en-105":83},{"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},84674,4810365810221,"Aurora","https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d",8,"Research & Report","Approximate Single Source Dual Fault Tolerant Distance Oracle","Given an undirected weighted graph G with n vertices and m edges (edge weights in [1, W]) and a designated source vertex s, the work builds a single-source dual-fault-tolerant distance oracle. For a destination t and a set F of at most two faulty edges, the oracle returns a (1 + O(ε))-approximation of the shortest-path distance from s to t avoiding all edges in F. The construction uses O(n√n) space and supports O(1) query time, improving over known Ω(n2) space requirements for multi-fault (1+ε)-approximate oracles.","Approximate Single Source Dual Fault Tolerant Distance Oracle  \nKoustav Das  \nIIT Gandhinagar India  \n[koustav. das@iitgn. ac. in](koustav. das@iitgn. ac. in)  \nManoj Gupta IIT Gandhinagar India  \n[gmanoj@iitgn. ac. in](gmanoj@iitgn. ac. in)  \narXiv :2607 .02999v 1 [ cs .DS] 3 Jul 2026  \nAbstract  \nWe are given an undirected weighted graph G with n vertices and m edges, edge weights in [1, W], anda designated source vertex s. We design a single source dual fault tolerant distance oracle for G. Given a destination vertex t and a set F of at most two faulty edges, the oracle returns a ˜( 1 + O (ε))-approximatio˜n of  \nthe weight of the shortest path from the source s to t avoiding F. Our oracle uses O(n √n ) space1 and has O (1) query time.  \nPrior to our result, single source sin˜gle fault tolerant oracles were known to return a (1 +ε) approximation of  \nthe weight of the shortest path using O (n) space and O (1) query time. However, extending these approaches to multiple faults remained an open problem. Indeed, all (1 +ε)-approximate distance oracles that handle multiple faults require Ω (n2 ) space. We break this bound by presenting the first dual fault tolerant distance oracle with o (n2 ) space.  \n1 Introduction  \nIn many real-world scenarios, we are interested in computing the shortest distances from a specific location to all other reachable locations in a network. Such networks can be modelled as graphs, where vertices represent places and edges capture connectivity between them. This naturally leads to the classical single source shortest paths (SSSP) problem, where the objective is to preprocess a graph so that distances from a designated source sto any other vertex can be efficiently retrieved in response to queries.  \nQUERY(s, t) : Find the weight of the shortest path from s to t.  \nFor unweighted graphs, the single source shortest paths (SSSP) can be efficiently computed using a breadthfirst search (BFS). The distances from the source to all other vertices can be stored in O(n) space, allowing O (1) query time per vertex, where n and m denote the number of vertices and edges, respectively. For weighted graphs, classical SSSP algorithms such as Dijkstra’s algorithm allow us to preprocess the graph in O(m+nlog n) time so that distances from a designated source can be queried in constant time. Over the years, a rich body of work has advanced the classical SSSP problem [FW93, FW90, Tho00, Ram96, Ram97, Hag00] . Later, Duan, Mao, Shu, and Yin [DMSY23 ] developed a randomized algorithm for the SSSP problem with a time complexity of O (mp log n log log n) . Recently, Duan, Mao, Mao, Shu, Yin [DMM+ 25] achieved further progress by presenting a deterministic O (m log2/3 n)-time algorithm for directed graphs with non-negative edge weights.  \nHowever, in practice, some connections may fail or become temporarily unavailable. We model these disruptions as faulty edges, i.e., edges removed from the graph. The task is to determine the length of the shortest path from a specified source vertex s to any other vertex t ∈ V while avoiding faulty edges. Our objective is to answer the following query efficiently:  \nQUERY(t, F): Find the weight of the shortest path from s to t avoiding all the edges contained in F  \n1poly (log1+εnW) factors are hidden in the ˜O notation.  \nAn algorithm may preprocess the graph G and create a data structure to quickly answer the above query. Such an algorithm (and the data structure) is called a distance oracle in literature. Since we are dealing with a single source s and must output shortest paths avoiding faults, our oracle is called Single Source fault tolerant distance oracle.  \nLet us first see a simple single source fault tolerant distance oracle. One could naively precompute and store exact shortest paths from the source s to all other vertices under all possible faulty edge scenarios. However, this approach is computationally prohibitive, as both the preprocessing time and space taken by the data","cbCaip2TCJMh6oTm","https://ap.wps.com/l/cbCaip2TCJMh6oTm","pdf",471326,1,30,"English","en",105,"# Abstract\n# Introduction\n## Single Source Shortest Paths and Fault Modeling\n## Distance Oracles and Fault Tolerant Variants\n## Naive Approaches and Motivation\n## Problem Definition: Queries with Fault Sets\n## Approximation Guarantees and Related Work","[{\"question\":\"Why is the result significant compared with prior multi-fault distance oracles?\",\"answer\":\"Prior (1 + ε)-approximate distance oracles that handle multiple faults require Ω(n2) space; this work presents a dual fault tolerant oracle that breaks that bound with subquadratic space.\"}]",1784197600,76,{"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":78,"head_meta":80,"extra_data":82,"updated_unix":27},"approximate-single-source-dual-fault-tolerant-distance-oracle","",{"@graph":35,"@context":77},[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/approximate-single-source-dual-fault-tolerant-distance-oracle/84674/",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],{"name":72,"@type":73,"acceptedAnswer":74},"Why is the result significant compared with prior multi-fault distance oracles?","Question",{"text":75,"@type":76},"Prior (1 + ε)-approximate distance oracles that handle multiple faults require Ω(n2) space; this work presents a dual fault tolerant oracle that breaks that bound with subquadratic space.","Answer","https://schema.org",{"og:url":51,"og:type":79,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":81,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":84},[85,89,93,97,102,107,112,114,119,122,126],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":86,"show_sort_weight":87,"slug":88},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":90,"show_sort_weight":91,"slug":92},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":94,"show_sort_weight":95,"slug":96},"Exam",70,"exam",{"id":98,"doc_module":4,"doc_module_name":45,"category_name":99,"show_sort_weight":100,"slug":101},5,"Comic",60,"comic",{"id":103,"doc_module":4,"doc_module_name":45,"category_name":104,"show_sort_weight":105,"slug":106},6,"Technology",50,"technology",{"id":108,"doc_module":4,"doc_module_name":45,"category_name":109,"show_sort_weight":110,"slug":111},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":21,"slug":113},"research-report",{"id":115,"doc_module":4,"doc_module_name":45,"category_name":116,"show_sort_weight":117,"slug":118},9,"Religion & Spirituality",20,"religion-spirituality",{"id":117,"doc_module":4,"doc_module_name":45,"category_name":120,"show_sort_weight":117,"slug":121},"World Cup","world-cup",{"id":123,"doc_module":4,"doc_module_name":45,"category_name":124,"show_sort_weight":123,"slug":125},10,"Lifestyle","lifestyle",{"id":127,"doc_module":4,"doc_module_name":45,"category_name":128,"show_sort_weight":98,"slug":129},19,"General","general"]