{"id":"https://openalex.org/W1500601723","doi":"https://doi.org/10.1007/978-3-540-30195-0_10","title":"Identifying Clusters from Positive Data","display_name":"Identifying Clusters from Positive Data","publication_year":2004,"publication_date":"2004-01-01","ids":{"openalex":"https://openalex.org/W1500601723","doi":"https://doi.org/10.1007/978-3-540-30195-0_10","mag":"1500601723"},"language":"en","primary_location":{"id":"doi:10.1007/978-3-540-30195-0_10","is_oa":false,"landing_page_url":"https://doi.org/10.1007/978-3-540-30195-0_10","pdf_url":null,"source":{"id":"https://openalex.org/S106296714","display_name":"Lecture notes in computer science","issn_l":"0302-9743","issn":["0302-9743","1611-3349"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"book series"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Lecture Notes in Computer Science","raw_type":"book-chapter"},"type":"conference-paper","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://eprints.qut.edu.au/13145/1/13145.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5111850796","display_name":"John Case","orcid":null},"institutions":[{"id":"https://openalex.org/I86501945","display_name":"University of Delaware","ror":"https://ror.org/01sbq1a82","country_code":"US","type":"education","lineage":["https://openalex.org/I86501945"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"John Case","raw_affiliation_strings":["Computer and Information Sciences Department, University of Delaware, Newark, DE, 19716-2586, USA","Computer and Information Sciences Department, University of Delaware, Newark, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Computer and Information Sciences Department, University of Delaware, Newark, DE, 19716-2586, USA","institution_ids":["https://openalex.org/I86501945"]},{"raw_affiliation_string":"Computer and Information Sciences Department, University of Delaware, Newark, USA","institution_ids":["https://openalex.org/I86501945"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5101627589","display_name":"Sanjay Jain","orcid":"https://orcid.org/0000-0001-6798-8330"},"institutions":[{"id":"https://openalex.org/I165932596","display_name":"National University of Singapore","ror":"https://ror.org/01tgyzw49","country_code":"SG","type":"education","lineage":["https://openalex.org/I165932596"]}],"countries":["SG"],"is_corresponding":false,"raw_author_name":"Sanjay Jain","raw_affiliation_strings":["School of Computing, National University of Singapore, 117543, Singapore","School of Computing, National University of Singapore, Singapore"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Computing, National University of Singapore, 117543, Singapore","institution_ids":["https://openalex.org/I165932596"]},{"raw_affiliation_string":"School of Computing, National University of Singapore, Singapore","institution_ids":["https://openalex.org/I165932596"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5027881741","display_name":"\u00c9ric Martin","orcid":"https://orcid.org/0000-0001-8832-7478"},"institutions":[{"id":"https://openalex.org/I31746571","display_name":"UNSW Sydney","ror":"https://ror.org/03r8z3t63","country_code":"AU","type":"education","lineage":["https://openalex.org/I31746571"]}],"countries":["AU"],"is_corresponding":false,"raw_author_name":"Eric Martin","raw_affiliation_strings":["School of Computer Science and Engeniering, UNSW, Sydney, NSW 2052, Australia","School of Computer Science and Engeniering, UNSW, Sydney, Australia"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Computer Science and Engeniering, UNSW, Sydney, NSW 2052, Australia","institution_ids":["https://openalex.org/I31746571"]},{"raw_affiliation_string":"School of Computer Science and Engeniering, UNSW, Sydney, Australia","institution_ids":["https://openalex.org/I31746571"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5008299613","display_name":"Arun Sharma","orcid":"https://orcid.org/0000-0002-5794-733X"},"institutions":[{"id":"https://openalex.org/I160993911","display_name":"Queensland University of Technology","ror":"https://ror.org/03pnv4752","country_code":"AU","type":"education","lineage":["https://openalex.org/I160993911"]}],"countries":["AU"],"is_corresponding":false,"raw_author_name":"Arun Sharma","raw_affiliation_strings":["Division of Research and Commercialization, Queensland University of Technology, GPO Box 2434, Brisbane, QLD 4001, Australia","Division of Research and Commercialization, Queensland University of Technology, Brisbane, Australia"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Division of Research and Commercialization, Queensland University of Technology, GPO Box 2434, Brisbane, QLD 4001, Australia","institution_ids":["https://openalex.org/I160993911"]},{"raw_affiliation_string":"Division of Research and Commercialization, Queensland University of Technology, Brisbane, Australia","institution_ids":["https://openalex.org/I160993911"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5085103227","display_name":"Frank Stephan","orcid":"https://orcid.org/0000-0001-9152-1706"},"institutions":[{"id":"https://openalex.org/I165932596","display_name":"National University of Singapore","ror":"https://ror.org/01tgyzw49","country_code":"SG","type":"education","lineage":["https://openalex.org/I165932596"]}],"countries":["SG"],"is_corresponding":false,"raw_author_name":"Frank Stephan","raw_affiliation_strings":["School of Computing, National University of Singapore, 117543, Singapore","School of Computing, National University of Singapore, Singapore"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Computing, National University of Singapore, 117543, Singapore","institution_ids":["https://openalex.org/I165932596"]},{"raw_affiliation_string":"School of Computing, National University of Singapore, Singapore","institution_ids":["https://openalex.org/I165932596"]}]}],"institutions":[],"countries_distinct_count":3,"institutions_distinct_count":4,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":{"value":5000,"currency":"EUR","value_usd":5392},"apc_paid":null,"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.04579017,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"36","issue":"1","first_page":"103","last_page":"114"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12072","display_name":"Machine Learning and Algorithms","score":0.9994000196456909,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T12072","display_name":"Machine Learning and Algorithms","score":0.9994000196456909,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11063","display_name":"Rough Sets and Fuzzy Logic","score":0.9970999956130981,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12002","display_name":"Computability, Logic, AI Algorithms","score":0.9965999722480774,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/disjoint-sets","display_name":"Disjoint sets","score":0.8148674964904785},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6971724033355713},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.5861772894859314},{"id":"https://openalex.org/keywords/cluster-analysis","display_name":"Cluster analysis","score":0.579393208026886},{"id":"https://openalex.org/keywords/numbering","display_name":"Numbering","score":0.5720652341842651},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.5228367447853088},{"id":"https://openalex.org/keywords/class","display_name":"Class (philosophy)","score":0.5150930285453796},{"id":"https://openalex.org/keywords/context","display_name":"Context (archaeology)","score":0.43982794880867004},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.22490417957305908},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.15880128741264343},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.08152437210083008}],"concepts":[{"id":"https://openalex.org/C45340560","wikidata":"https://www.wikidata.org/wiki/Q215382","display_name":"Disjoint sets","level":2,"score":0.8148674964904785},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6971724033355713},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5861772894859314},{"id":"https://openalex.org/C73555534","wikidata":"https://www.wikidata.org/wiki/Q622825","display_name":"Cluster analysis","level":2,"score":0.579393208026886},{"id":"https://openalex.org/C191916993","wikidata":"https://www.wikidata.org/wiki/Q362019","display_name":"Numbering","level":2,"score":0.5720652341842651},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.5228367447853088},{"id":"https://openalex.org/C2777212361","wikidata":"https://www.wikidata.org/wiki/Q5127848","display_name":"Class (philosophy)","level":2,"score":0.5150930285453796},{"id":"https://openalex.org/C2779343474","wikidata":"https://www.wikidata.org/wiki/Q3109175","display_name":"Context (archaeology)","level":2,"score":0.43982794880867004},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.22490417957305908},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.15880128741264343},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.08152437210083008},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.0},{"id":"https://openalex.org/C151730666","wikidata":"https://www.wikidata.org/wiki/Q7205","display_name":"Paleontology","level":1,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0}],"mesh":[],"locations_count":8,"locations":[{"id":"doi:10.1007/978-3-540-30195-0_10","is_oa":false,"landing_page_url":"https://doi.org/10.1007/978-3-540-30195-0_10","pdf_url":null,"source":{"id":"https://openalex.org/S106296714","display_name":"Lecture notes in computer science","issn_l":"0302-9743","issn":["0302-9743","1611-3349"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"book series"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Lecture Notes in Computer Science","raw_type":"book-chapter"},{"id":"doi:10.1137/050629112","is_oa":false,"landing_page_url":"https://doi.org/10.1137/050629112","pdf_url":null,"source":{"id":"https://openalex.org/S153560523","display_name":"SIAM Journal on Computing","issn_l":"0097-5397","issn":["0097-5397","1095-7111"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Computing","raw_type":"journal-article"},{"id":"pmh:oai:eprints.qut.edu.au:13145","is_oa":true,"landing_page_url":null,"pdf_url":"https://eprints.qut.edu.au/13145/1/13145.pdf","source":{"id":"https://openalex.org/S4306402607","display_name":"QUT ePrints (Queensland University of Technology)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I160993911","host_organization_name":"Queensland University of Technology","host_organization_lineage":["https://openalex.org/I160993911"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"SIAM Journal on Computing","raw_type":"Contribution to Journal"},{"id":"pmh:oai:scholarbank.nus.edu.sg:10635/41132","is_oa":false,"landing_page_url":"http://scholarbank.nus.edu.sg/handle/10635/41132","pdf_url":null,"source":{"id":"https://openalex.org/S7407052290","display_name":"National University of Singapore","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Scopus","raw_type":"Conference Paper"},{"id":"pmh:oai:scholarbank.nus.edu.sg:10635/43097","is_oa":false,"landing_page_url":"http://scholarbank.nus.edu.sg/handle/10635/43097","pdf_url":null,"source":{"id":"https://openalex.org/S7407052290","display_name":"National University of Singapore","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Scopus","raw_type":"Article"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.97.1785","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.97.1785","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.comp.nus.edu.sg/~sanjay/paps/cluster.pdf","raw_type":"text"},{"id":"mag:1604536697","is_oa":false,"landing_page_url":"https://eprints.qut.edu.au/13145/","pdf_url":null,"source":null,"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":null},{"id":"mag:1500601723","is_oa":false,"landing_page_url":"https://dblp.uni-trier.de/db/journals/eccc/eccc11.html#ECCC-TR04-058","pdf_url":null,"source":{"id":"https://openalex.org/S11189987","display_name":"Electronic colloquium on computational complexity","issn_l":"1433-8092","issn":["1433-8092"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":null,"raw_source_name":"Electronic colloquium on computational complexity","raw_type":null}],"best_oa_location":{"id":"pmh:oai:eprints.qut.edu.au:13145","is_oa":true,"landing_page_url":null,"pdf_url":"https://eprints.qut.edu.au/13145/1/13145.pdf","source":{"id":"https://openalex.org/S4306402607","display_name":"QUT ePrints (Queensland University of Technology)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I160993911","host_organization_name":"Queensland University of Technology","host_organization_lineage":["https://openalex.org/I160993911"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"SIAM Journal on Computing","raw_type":"Contribution to Journal"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G5657896296","display_name":"Self-Reference, Complexity, and Learning","funder_award_id":"0208616","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320306114","display_name":"U.S. Department of Agriculture","ror":"https://ror.org/01na82s61"},{"id":"https://openalex.org/F4320310369","display_name":"National ICT Australia","ror":"https://ror.org/03q397159"},{"id":"https://openalex.org/F4320315885","display_name":"Australian Government","ror":"https://ror.org/0314h5y94"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W1500601723.pdf","grobid_xml":"https://content.openalex.org/works/W1500601723.grobid-xml"},"referenced_works_count":21,"referenced_works":["https://openalex.org/W166170480","https://openalex.org/W234856483","https://openalex.org/W623765668","https://openalex.org/W1969005071","https://openalex.org/W1970570926","https://openalex.org/W1971784203","https://openalex.org/W1985133821","https://openalex.org/W2019001600","https://openalex.org/W2020133060","https://openalex.org/W2064496697","https://openalex.org/W2067541950","https://openalex.org/W2073020732","https://openalex.org/W2111004121","https://openalex.org/W2118587067","https://openalex.org/W2184820246","https://openalex.org/W2606195968","https://openalex.org/W2799061466","https://openalex.org/W4213286322","https://openalex.org/W4230725619","https://openalex.org/W4241020758","https://openalex.org/W7037119483"],"related_works":["https://openalex.org/W1604536697","https://openalex.org/W3027144919","https://openalex.org/W2052337727","https://openalex.org/W42223060","https://openalex.org/W2142539551","https://openalex.org/W2084815663","https://openalex.org/W2083563352","https://openalex.org/W2140501550","https://openalex.org/W2933218595","https://openalex.org/W2003510477","https://openalex.org/W1985276029","https://openalex.org/W2078696977","https://openalex.org/W2121058991","https://openalex.org/W2619090478","https://openalex.org/W2024489128","https://openalex.org/W229605563","https://openalex.org/W2756143566","https://openalex.org/W1977622349","https://openalex.org/W3101855747","https://openalex.org/W1995176020"],"abstract_inverted_index":{"The":[0,61,144],"present":[1],"work":[2],"studies":[3],"clustering":[4,45,77,234,236,337,341,360,384],"from":[5],"an":[6,72,123,356,392],"abstract":[7],"point":[8],"of":[9,18,34,40,53,56,97,99,178,183,196,205,224,237,239,249,252,267,275,286,292,294],"view":[10],"and":[11,50,150,186,288,379],"investigates":[12],"its":[13],"properties":[14,149,155],"in":[15,81,221,256],"the":[16,82,130,170,180,184,191,194,206,222,230,233,243,247,265,273,284,289,300],"framework":[17],"inductive":[19],"inference.":[20],"Any":[21],"class":[22,62,115,185,198,207,248,274,386],"S":[23,63],"considered":[24],"is":[25,47,64,71,116,121,152,208,281,297,309,353,388],"given":[26,298],"by":[27],"a":[28,48,93,127,187,197,211,257],"hypothesis":[29,212],"space,":[30],"i.e.,":[31],"numbering,":[32],"$A_0,A_1,\\ldots$":[33],"nonempty":[35,51],"recursively":[36],"enumerable":[37],"(r.e.)":[38],"subsets":[39],"${\\mathbb{N}}$":[41],"or":[42,159],"${\\mathbb{Q}^k}$.":[43],"A":[44,114],"task":[46,78],"finite":[49,94],"set":[52,95],"r.e.":[54],"indices":[55,98],"pairwise":[57,100],"disjoint":[58,101],"such":[59,103,122],"sets.":[60],"said":[65],"to":[66,92,134,299,324],"be":[67,262],"clusterable":[68,282,390],"if":[69,119,283],"there":[70,120],"algorithm":[73,124],"which,":[74],"for":[75,87,162,214,232,312],"every":[76],"I,":[79],"converges":[80],"limit":[83],"on":[84,202,345],"any":[85,385],"text":[86,231],"$\\bigcup_{i":[88],"\\in":[89,106,111,136,141],"I}":[90,112,142],"A_i$":[91],"J":[96,128],"clusters":[102,268,287,296],"that":[104,355],"$\\bigcup_{j":[105,135],"J}":[107,137],"A_j":[108,138],"=":[109],"\\bigcup_{i":[110,140],"A_i$.":[113,143],"called":[117],"semiclusterable":[118],"which":[125,203,348,387],"finds":[126],"with":[129,264,278,316],"last":[131],"condition":[132],"relaxed":[133],"\\supseteq":[139],"relationship":[145],"between":[146],"natural":[147],"topological":[148,181,326],"clusterability":[151,195],"investigated.":[153],"Topological":[154],"can":[156,261],"provide":[157],"sufficient":[158,311],"necessary":[160],"conditions":[161,175],"clusterability,":[163],"but":[164,343],"they":[165],"cannot":[166],"characterize":[167],"it.":[168],"On":[169,190],"one":[171],"hand,":[172,193],"many":[173,238,254],"interesting":[174],"make":[176],"use":[177],"both":[179],"structure":[182],"well-chosen":[188],"numbering.":[189],"other":[192],"does":[199],"not":[200,310,389],"depend":[201],"numbering":[204],"used":[209],"as":[210,269,302],"space":[213,260],"the\\break":[215],"clusterer.":[216],"These":[217],"ideas":[218],"are":[219,321,329,349],"demonstrated":[220],"context":[223],"naturally":[225],"geometrically":[226],"defined":[227],"classes.":[228],"Besides":[229],"task,":[235],"these":[240,295],"classes":[241,313,320,347,364],"requires":[242],"following":[244],"additional":[245,270,303,307],"information:":[246],"convex":[250],"hulls":[251],"finitely":[253],"points":[255],"rational":[258],"vector":[259],"clustered":[263],"number":[266,285,291],"information.":[271,304],"Interestingly,":[272],"polygons":[276],"(together":[277],"their":[279,325],"interiors)":[280],"overall":[290],"vertices":[293],"clusterer":[301],"Intriguingly,":[305],"this":[306],"information":[308],"including":[314],"figures":[315],"holes.":[317],"While":[318],"some":[319,346],"unclusterable":[322],"due":[323],"structure,":[327],"others":[328],"only":[330],"computationally":[331,339,362],"intractable.":[332],"An":[333],"oracle":[334,357,376,382],"might":[335],"permit":[336],"all":[338,361],"intractable":[340],"tasks":[342],"fail":[344],"topologically":[350],"difficult.":[351],"It":[352],"shown":[354],"E":[358],"permits":[359,383],"difficult":[363],"iff":[365],"$E":[366],"\\geq_T":[367,371],"K":[368,378],"\\wedge":[369],"E'":[370],"K\"$.":[372],"Furthermore,":[373],"no":[374,380],"1-generic":[375],"below":[377],"2-generic":[381],"without":[391],"oracle.":[393]},"counts_by_year":[],"updated_date":"2026-08-05T07:39:15.569665","created_date":"2025-10-10T00:00:00"}