[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85007-en":3,"doc-seo-85007-105":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},85007,13056703019404,"Miles","https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0",8,"Research & Report","Stochastic Online Euclidean TSP","Euclidean traveling salesman problem studies assigning revealed points in Euclidean space to time slots while minimizing total travel distance under a fixed distance function. In online Euclidean TSP, points arrive sequentially and once a point is assigned to a slot it cannot be reassigned, with competitive ratios known for adversarial and stochastic settings. The work introduces a deterministic algorithm for stochastic online Euclidean TSP, proving expected competitive bounds for both high-dimensional cases and the one-dimensional case, and evaluates the approach experimentally.","Stochastic Online Euclidea~~n~~ TSP  \n    June 15, 2 026  \nAbstract  \nIn the Euclidean travelling salesman problem (Euclidean TSP), a salesman must visit 􀀁 points in Euclidean space, while minimizing the travel distance, according to the Euclidean distance function. In online Euclidean TSP, introduced by Abrahamsen, Bercea, Beretta, Klausen and Kozma [ESA 2024], the points are revealed one at a time, and a time slot must be assigned before the next is revealed. Once a point is assigned to a time slot, it can never be reassigned to another time slot. There are 􀀁 time slots. Euclidean online TSP is a high-dimensional generalization of online sorting, introduced by Aamand, Abrahamsen, Beretta and Kleist [SODA 2023] . Bertram [ESA 2025] showed an algorithm that achieves a competitive ratio of 􀀂( √􀀁 ) in the worst case. In stochastic online Euclidean TSP, the points are sampled uniformly and independently in the unit 􀀆-cube. Kalavas, Platanos and Tolias [STACS 2026] presented an algorithm achieving a competitive ratio of 􀀂(log2 􀀁 ) with high probability for stochastic online Euclidean TSP.  \nWe present a simple algorithm that for 􀀆 ≥ 2 achieves an expected competitive ratio of 􀀂 (1), and for 􀀆 = 1 achieves an expected competitive ratio of 􀀂(log 􀀁 ) , matching the lower bound by Hu [SODA 2026] for 􀀆 = 1. The algorithm is deterministic, and the expectation is due to the stochastic input. We also show that in the variant where there are more time slots than points, i.e. , ⌈ (1 + 􀀓)􀀁⌉ time slots and 􀀆 = 1 , our algorithm achieves an expected competitive ratio of 􀀂 ( 1 + log 􀀓−1) . We also survey algorithms from the literature.  \nWe experimentally evaluate our algorithm, which reveals that in all variants the constant factor hidden asymptotically is small. We also evaluate the algorithms from the literature.  \nResumé  \nI det euklidiske travelling salesman problem (euklidisk TSP), skal en sælger besøge 􀀁 punkter idet euklidiske rum, mens rejsedistancen minimeres, beregnet med den euklidiske distancefunktion. I online euklidisk TSP, introduceret af Abrahamsen, Bercea, Beretta, Klausen og Kozma [ESA 2024], bliver punkterne afsløret et punkt ad gangen, og før det næste punkt bliver afsløret, skal punktet tildeles et tidspunkt. Efter at et punkt har fået tildelt et tidspunkt, kan detaldrig tildeles et andet tidspunkt i stedet. Der er 􀀁 tidspunkter. Euklidisk online TSP er en højdimensionel generalisering af online sorting, introduceret af Aamand, Abrahamsen, Beretta og Kleist [SODA 2023]. Bertram [ESA 2025] præsenterede en algoritme som opnår en kompetitiv ratio på 􀀂( √􀀁  ) i værste tilfælde. I stokastisk online euklididsk TSP bliver punkterne genererettilfældigt, uniformt og uafhængigt i 􀀆-enhedskuben. Kalavas, Platanos og Tolias [STACS 2026] præsenterede en algoritme som opnår en kompetitiv ratio på 􀀂(log2 􀀁 ) med høj sandsynlighed for stokastisk online euklidisk TSP.  \nVi præsenterer en simpel algoritme, som for 􀀆 ≥ 2 opnår en forventet kompetitiv ratio på 􀀂(1), og for 􀀆 = 1 opnår en forventet kompetitiv ratio på 􀀂 (log 􀀁 ) , hvilket passer med den nedre grænse af Hu [SODA 2026] for 􀀆 = 1. Algoritmen er deterministisk og forventningen kommer fra det stokastiske input. Vi viser også at i varianten hvor der er flere tidspunkter end punkter, for eksempel ⌈ (1 + 􀀓)􀀁⌉ tidspunkter og 􀀆 = 1 , opnår vores algoritme en forventet kompetitiv ratio på 􀀂( 1 + log 􀀓−1) . Vi undersøger også algoritmer fra litteraturen.  \nVi evaluerer vores algoritme eksperimentelt, hvilket viser, at I alle varianter er konstantfaktoren, som er skjult af asymptotikken, lille. Vi evaluerer også algoritmer fra litteraturen.  \nContents  \n1. Introduction .............................................................................. 4  \n1.1. Terminology ......................................................................... 5  \n1.2. Structure of the Thesis .............................................................. 5  \n2. Previous Work ...................................","cbCaicLv3E4Q1HTw","https://ap.wps.com/l/cbCaicLv3E4Q1HTw","pdf",1364803,1,47,"English","en",105,"# Introduction\n## Terminology\n## Structure of the Thesis\n# Previous Work\n## Adversarial Online Sorting\n## Adversarial Online TSP\n## Stochastic Online Sorting\n## Stochastic Online TSP\n## Stochastic Online Sorting with Larger Array\n# Our Algorithm\n# Analysis\n## Stochastic Online Sorting\n## Stochastic Online Sorting with Larger Array\n## Stochastic Online TSP\n# Experimental Evaluation","[{\"question\":\"What problem does stochastic online Euclidean TSP address?\",\"answer\":\"Stochastic online Euclidean TSP minimizes travel distance when points sampled uniformly and independently in a unit cube are revealed one by one and must be assigned to time slots immediately.\"},{\"question\":\"How does the proposed algorithm perform in different dimensions?\",\"answer\":\"For dimension d≥2 it achieves an expected competitive ratio proportional to φ(1), and for d=1 it achieves an expected competitive ratio proportional to φ(log n), matching the known lower bound for d=1.\"},{\"question\":\"What is the key constraint that makes the online variant challenging?\",\"answer\":\"After assigning a revealed point to a time slot, the point cannot be reassigned to a different slot, so decisions must be made irrevocably as the input is revealed.\"}]",1784200201,118,{"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},"stochastic-online-euclidean-tsp","",{"@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/stochastic-online-euclidean-tsp/85007/",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},"What problem does stochastic online Euclidean TSP address?","Question",{"text":74,"@type":75},"Stochastic online Euclidean TSP minimizes travel distance when points sampled uniformly and independently in a unit cube are revealed one by one and must be assigned to time slots immediately.","Answer",{"name":77,"@type":72,"acceptedAnswer":78},"How does the proposed algorithm perform in different dimensions?",{"text":79,"@type":75},"For dimension d≥2 it achieves an expected competitive ratio proportional to φ(1), and for d=1 it achieves an expected competitive ratio proportional to φ(log n), matching the known lower bound for d=1.",{"name":81,"@type":72,"acceptedAnswer":82},"What is the key constraint that makes the online variant challenging?",{"text":83,"@type":75},"After assigning a revealed point to a time slot, the point cannot be reassigned to a different slot, so decisions must be made irrevocably as the input is 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