{"id":"https://openalex.org/W2019579910","doi":"https://doi.org/10.1137/0222064","title":"Probabilistic Analysis of Disjoint Set Union Algorithms","display_name":"Probabilistic Analysis of Disjoint Set Union Algorithms","publication_year":1993,"publication_date":"1993-10-01","ids":{"openalex":"https://openalex.org/W2019579910","doi":"https://doi.org/10.1137/0222064","mag":"2019579910"},"language":"en","primary_location":{"id":"doi:10.1137/0222064","is_oa":false,"landing_page_url":"https://doi.org/10.1137/0222064","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"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5109915650","display_name":"B\u00e9la Bollob\u00e1s","orcid":"https://orcid.org/0000-0002-9819-7213"},"institutions":[{"id":"https://openalex.org/I121820613","display_name":"Louisiana State University","ror":"https://ror.org/05ect4e57","country_code":"US","type":"education","lineage":["https://openalex.org/I121820613"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"B\u00e9la Bollob\u00e1s","raw_affiliation_strings":["Univ. of Cambridge, Cambridge, UK and Louisiana State Univ., Baton Rouge"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Univ. of Cambridge, Cambridge, UK and Louisiana State Univ., Baton Rouge","institution_ids":["https://openalex.org/I121820613"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5078144452","display_name":"Istv\u00e1n Simon","orcid":"https://orcid.org/0000-0003-1207-4018"},"institutions":[{"id":"https://openalex.org/I127339247","display_name":"California State University System","ror":"https://ror.org/020qm1538","country_code":"US","type":"education","lineage":["https://openalex.org/I127339247"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Istvan Simon","raw_affiliation_strings":["California State University\u2014Hayward,"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"California State University\u2014Hayward,","institution_ids":["https://openalex.org/I127339247"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":8,"citation_normalized_percentile":{"value":0.14816895,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":96},"biblio":{"volume":"22","issue":"5","first_page":"1053","last_page":"1074"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10720","display_name":"Complexity and Algorithms in Graphs","score":0.9998000264167786,"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"}},"topics":[{"id":"https://openalex.org/T10720","display_name":"Complexity and Algorithms in Graphs","score":0.9998000264167786,"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/T12288","display_name":"Optimization and Search Problems","score":0.9994999766349792,"subfield":{"id":"https://openalex.org/subfields/1705","display_name":"Computer Networks and Communications"},"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/T10374","display_name":"Advanced Graph Theory Research","score":0.9909999966621399,"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":0.7614995241165161},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.6769353151321411},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6351841688156128},{"id":"https://openalex.org/keywords/binary-logarithm","display_name":"Binary logarithm","score":0.5675661563873291},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.5168972015380859},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.3317338228225708}],"concepts":[{"id":"https://openalex.org/C45340560","wikidata":"https://www.wikidata.org/wiki/Q215382","display_name":"Disjoint sets","level":2,"score":0.7614995241165161},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6769353151321411},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6351841688156128},{"id":"https://openalex.org/C63553672","wikidata":"https://www.wikidata.org/wiki/Q581168","display_name":"Binary logarithm","level":2,"score":0.5675661563873291},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.5168972015380859},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3317338228225708}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1137/0222064","is_oa":false,"landing_page_url":"https://doi.org/10.1137/0222064","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"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320322077","display_name":"Korea Institute for International Economic Policy","ror":"https://ror.org/04vtz2g46"}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":18,"referenced_works":["https://openalex.org/W1655990431","https://openalex.org/W1969148345","https://openalex.org/W1986022261","https://openalex.org/W1986261983","https://openalex.org/W1988401041","https://openalex.org/W1989351810","https://openalex.org/W2009232680","https://openalex.org/W2012335680","https://openalex.org/W2044146725","https://openalex.org/W2051748083","https://openalex.org/W2055754268","https://openalex.org/W2061496873","https://openalex.org/W2146162554","https://openalex.org/W2242150843","https://openalex.org/W4210622426","https://openalex.org/W4210731599","https://openalex.org/W4241669766","https://openalex.org/W4312512934"],"related_works":["https://openalex.org/W1971174658","https://openalex.org/W2099195351","https://openalex.org/W2051487156","https://openalex.org/W2073681303","https://openalex.org/W4256429076","https://openalex.org/W2012063272","https://openalex.org/W2348092930","https://openalex.org/W2062850566","https://openalex.org/W4308716484","https://openalex.org/W1553466895"],"abstract_inverted_index":{"A":[0],"number":[1],"of":[2,11,38,69,97,109,127,154,212],"open":[3],"questions":[4],"are":[5,131,140],"settled":[6],"about":[7],"the":[8,35,39,106,110,151,209],"expected":[9,36,107,125,152,210],"costs":[10,126],"two":[12],"disjoint":[13],"set":[14],"Union":[15],"and":[16,22,71,92,129,199],"Find":[17],"algorithms":[18],"raised":[19],"by":[20],"Knuth":[21],"Sch\u00f6nhage":[23],"[Theoret.":[24],"Comput.":[25,77],"Sci.,":[26],"6":[27],"(1978),":[28],"pp.":[29,81],"281\u2013315].":[30],"This":[31],"paper":[32],"shows":[33],"that":[34,105,150],"time":[37,86,108],"Weighted":[40],"Quick-Find":[41,112],"(QFW)":[42],"algorithm":[43,114],"to":[44,88],"perform":[45,89],"$(n":[46,136],"-":[47,137,176,202,216],"1)$":[48,138],"randomly":[49,158,225],"chosen":[50,159,226],"unions":[51,91,139,160,227],"is":[52,102,115,148,161,181,228],"$cn":[53],"+":[54,120,163,183,187],"o({n":[55],"/":[56,118,170,214,218,222],"{\\log":[57],"n}})$,":[58],"where":[59,172,189],"$c":[60],"=":[61,156,167],"2.0847":[62],"\\ldots":[63],"$":[64,224],".":[65],"Through":[66],"an":[67],"observation":[68],"Tarjan":[70],"Van":[72],"Leeuwen":[73],"in":[74],"[J.":[75],"Assoc.":[76],"Mach.,":[78],"22":[79],"(1975),":[80],"215\u2013225]":[82],"this":[83,179],"implies":[84],"linear":[85],"bounds":[87],"$O(n)$":[90],"finds":[93],"for":[94,145],"a":[95],"class":[96],"other":[98,143],"union-find":[99],"algorithms.":[100],"It":[101],"also":[103],"proved":[104],"Unweighted":[111],"(QF)":[113],"${{n^2":[116],"}":[117],"{8":[119],"O(n(\\log":[121],"n)^2":[122],")}}$.":[123],"The":[124],"QFW":[128,146],"QF":[130],"analyzed":[132],"when":[133],"fewer":[134],"than":[135],"performed.":[141],"Among":[142],"results,":[144],"it":[147],"shown":[149],"cost":[153,180,211],"$m":[155,166],"o(n)$":[157],"$m(1":[162,182],"o(1))$.":[164],"If":[165],"{{\\alpha":[168],"n}":[169],"2}$,":[171],"$\\alpha":[173,196],"\\leqslant":[174,205],"e^{":[175],"2}":[177,203],"$,":[178],"\\epsilon":[184],"(\\alpha":[185,191],")":[186,192,204],"o(1))$,":[188],"$\\epsilon":[190,200],"\\to":[193,197],"0$":[194,198],"as":[195],"(e^{":[201],".026$.":[206],"For":[207],"QF,":[208],"${n":[213],"{2":[215],"n^{{2":[217],"3}}":[219,223],"}}(\\log":[220],"n)^{{2":[221],"$O(n\\log":[229],"n)$.":[230]},"counts_by_year":[{"year":2021,"cited_by_count":2},{"year":2013,"cited_by_count":1},{"year":2012,"cited_by_count":2}],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2025-10-10T00:00:00"}
