{"id":"https://openalex.org/W2087183702","doi":"https://doi.org/10.1137/s009753970138390x","title":"Phase Change of Limit Laws in the Quicksort Recurrence under Varying Toll Functions","display_name":"Phase Change of Limit Laws in the Quicksort Recurrence under Varying Toll Functions","publication_year":2002,"publication_date":"2002-01-01","ids":{"openalex":"https://openalex.org/W2087183702","doi":"https://doi.org/10.1137/s009753970138390x","mag":"2087183702"},"language":"en","primary_location":{"id":"doi:10.1137/s009753970138390x","is_oa":false,"landing_page_url":"https://doi.org/10.1137/s009753970138390x","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/A5003984705","display_name":"Hsien\u2010Kuei Hwang","orcid":"https://orcid.org/0000-0002-9410-6476"},"institutions":[{"id":"https://openalex.org/I84653119","display_name":"Academia Sinica","ror":"https://ror.org/05bxb3784","country_code":"TW","type":"facility","lineage":["https://openalex.org/I84653119"]}],"countries":["TW"],"is_corresponding":true,"raw_author_name":"Hsien-Kuei Hwang","raw_affiliation_strings":["Academia Sinica )"],"affiliations":[{"raw_affiliation_string":"Academia Sinica )","institution_ids":["https://openalex.org/I84653119"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5056428246","display_name":"Ralph Neininger","orcid":"https://orcid.org/0000-0003-3975-1293"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Ralph Neininger","raw_affiliation_strings":[],"affiliations":[]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5003984705"],"corresponding_institution_ids":["https://openalex.org/I84653119"],"apc_list":null,"apc_paid":null,"fwci":13.8823,"has_fulltext":false,"cited_by_count":88,"citation_normalized_percentile":{"value":0.99642127,"is_in_top_1_percent":true,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":90,"max":99},"biblio":{"volume":"31","issue":"6","first_page":"1687","last_page":"1722"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11152","display_name":"Stochastic processes and statistical mechanics","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11152","display_name":"Stochastic processes and statistical mechanics","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11901","display_name":"Bayesian Methods and Mixture Models","score":0.9993000030517578,"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/T11716","display_name":"Random Matrices and Applications","score":0.9977999925613403,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/quicksort","display_name":"Quicksort","score":0.9784303903579712},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.686657190322876},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6477969288825989},{"id":"https://openalex.org/keywords/limit-of-a-function","display_name":"Limit of a function","score":0.5972961783409119},{"id":"https://openalex.org/keywords/tree-traversal","display_name":"Tree traversal","score":0.5941159725189209},{"id":"https://openalex.org/keywords/limit","display_name":"Limit (mathematics)","score":0.547599196434021},{"id":"https://openalex.org/keywords/partition","display_name":"Partition (number theory)","score":0.5039638876914978},{"id":"https://openalex.org/keywords/permutation","display_name":"Permutation (music)","score":0.4637596607208252},{"id":"https://openalex.org/keywords/sorting","display_name":"Sorting","score":0.4603772759437561},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.4566371738910675},{"id":"https://openalex.org/keywords/binary-logarithm","display_name":"Binary logarithm","score":0.45486021041870117},{"id":"https://openalex.org/keywords/tree","display_name":"Tree (set theory)","score":0.44219517707824707},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.4109903573989868},{"id":"https://openalex.org/keywords/sorting-algorithm","display_name":"Sorting algorithm","score":0.3304068446159363},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.24501550197601318},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.09227541089057922},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.07304245233535767}],"concepts":[{"id":"https://openalex.org/C2778937211","wikidata":"https://www.wikidata.org/wiki/Q486598","display_name":"Quicksort","level":4,"score":0.9784303903579712},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.686657190322876},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6477969288825989},{"id":"https://openalex.org/C19448537","wikidata":"https://www.wikidata.org/wiki/Q33540","display_name":"Limit of a function","level":3,"score":0.5972961783409119},{"id":"https://openalex.org/C140745168","wikidata":"https://www.wikidata.org/wiki/Q1210082","display_name":"Tree traversal","level":2,"score":0.5941159725189209},{"id":"https://openalex.org/C151201525","wikidata":"https://www.wikidata.org/wiki/Q177239","display_name":"Limit (mathematics)","level":2,"score":0.547599196434021},{"id":"https://openalex.org/C42812","wikidata":"https://www.wikidata.org/wiki/Q1082910","display_name":"Partition (number theory)","level":2,"score":0.5039638876914978},{"id":"https://openalex.org/C21308566","wikidata":"https://www.wikidata.org/wiki/Q7169365","display_name":"Permutation (music)","level":2,"score":0.4637596607208252},{"id":"https://openalex.org/C111696304","wikidata":"https://www.wikidata.org/wiki/Q2303697","display_name":"Sorting","level":2,"score":0.4603772759437561},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.4566371738910675},{"id":"https://openalex.org/C63553672","wikidata":"https://www.wikidata.org/wiki/Q581168","display_name":"Binary logarithm","level":2,"score":0.45486021041870117},{"id":"https://openalex.org/C113174947","wikidata":"https://www.wikidata.org/wiki/Q2859736","display_name":"Tree (set theory)","level":2,"score":0.44219517707824707},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4109903573989868},{"id":"https://openalex.org/C108094655","wikidata":"https://www.wikidata.org/wiki/Q181593","display_name":"Sorting algorithm","level":3,"score":0.3304068446159363},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.24501550197601318},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.09227541089057922},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.07304245233535767},{"id":"https://openalex.org/C78458016","wikidata":"https://www.wikidata.org/wiki/Q840400","display_name":"Evolutionary biology","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},{"id":"https://openalex.org/C24890656","wikidata":"https://www.wikidata.org/wiki/Q82811","display_name":"Acoustics","level":1,"score":0.0}],"mesh":[],"locations_count":5,"locations":[{"id":"doi:10.1137/s009753970138390x","is_oa":false,"landing_page_url":"https://doi.org/10.1137/s009753970138390x","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:CiteSeerX.psu:10.1.1.14.2407","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.14.2407","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://algo.stat.sinica.edu.tw/counter.asp?paper=pdf/q.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.25.5940","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.25.5940","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://neyman.mathematik.uni-freiburg.de/homepages/neininger/qfinal.ps","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.57.6588","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.57.6588","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://algo.stat.sinica.edu.tw/counter.asp?paper=ps-gz/q.ps.gz","raw_type":"text"},{"id":"pmh:oai:freidok.uni-freiburg.de:44381","is_oa":false,"landing_page_url":"https://freidok.uni-freiburg.de/data/44381","pdf_url":null,"source":{"id":"https://openalex.org/S4306401057","display_name":"FreiDok plus (Universit\u00e4tsbibliothek Freiburg)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I161046081","host_organization_name":"University of Freiburg","host_organization_lineage":["https://openalex.org/I161046081"],"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. - 31, 6 (2002) , 1687-1722, ISSN: 0097-5397","raw_type":"article"}],"best_oa_location":null,"sustainable_development_goals":[{"score":0.49000000953674316,"id":"https://metadata.un.org/sdg/16","display_name":"Peace, Justice and strong institutions"}],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":70,"referenced_works":["https://openalex.org/W7828374","https://openalex.org/W68608586","https://openalex.org/W192578068","https://openalex.org/W583356579","https://openalex.org/W1490652101","https://openalex.org/W1497373391","https://openalex.org/W1511251174","https://openalex.org/W1529110373","https://openalex.org/W1546072503","https://openalex.org/W1546740483","https://openalex.org/W1563213005","https://openalex.org/W1585070348","https://openalex.org/W1585200013","https://openalex.org/W1597637931","https://openalex.org/W1968569626","https://openalex.org/W1968621471","https://openalex.org/W1968751976","https://openalex.org/W1970628287","https://openalex.org/W1979228272","https://openalex.org/W1982839286","https://openalex.org/W1985870994","https://openalex.org/W1996376703","https://openalex.org/W1996508950","https://openalex.org/W2000066175","https://openalex.org/W2005935699","https://openalex.org/W2018919817","https://openalex.org/W2019765856","https://openalex.org/W2022718961","https://openalex.org/W2026287107","https://openalex.org/W2027952317","https://openalex.org/W2035490304","https://openalex.org/W2038681797","https://openalex.org/W2041147307","https://openalex.org/W2045594573","https://openalex.org/W2047659988","https://openalex.org/W2053101698","https://openalex.org/W2056099894","https://openalex.org/W2056643960","https://openalex.org/W2056975861","https://openalex.org/W2057352619","https://openalex.org/W2059983062","https://openalex.org/W2063369100","https://openalex.org/W2065095710","https://openalex.org/W2066363650","https://openalex.org/W2066969204","https://openalex.org/W2067707835","https://openalex.org/W2084532840","https://openalex.org/W2086486876","https://openalex.org/W2086893133","https://openalex.org/W2090041152","https://openalex.org/W2091386360","https://openalex.org/W2092816446","https://openalex.org/W2095581856","https://openalex.org/W2097457356","https://openalex.org/W2104846491","https://openalex.org/W2105422170","https://openalex.org/W2113913510","https://openalex.org/W2115337421","https://openalex.org/W2124919561","https://openalex.org/W2149468555","https://openalex.org/W2201329049","https://openalex.org/W2254463431","https://openalex.org/W2318794083","https://openalex.org/W2498839538","https://openalex.org/W2752885492","https://openalex.org/W2798461040","https://openalex.org/W2890747390","https://openalex.org/W4235295823","https://openalex.org/W4252889257","https://openalex.org/W4298050965"],"related_works":["https://openalex.org/W2125028276","https://openalex.org/W2154737298","https://openalex.org/W2037221665","https://openalex.org/W4289305650","https://openalex.org/W2898932275","https://openalex.org/W71033917","https://openalex.org/W65866683","https://openalex.org/W2139781203","https://openalex.org/W2087183702","https://openalex.org/W4224111978"],"abstract_inverted_index":{"We":[0,82],"characterize":[1],"all":[2],"limit":[3,75],"laws":[4,76],"of":[5,91],"the":[6,18,48,56,89],"quicksort-type":[7],"random":[8],"variables":[9],"defined":[10],"recursively":[11],"by":[12],"${\\cal":[13,44],"L}(X_n)=":[14],"{\\cal":[15],"L}(X_{I_n}+X^*_{n-1-I_n}+T_n)$":[16],"when":[17,78],"\"toll":[19,49],"function\"":[20,50],"Tn":[21,51,79],"varies":[22],"and":[23,43],"satisfies":[24],"general":[25],"conditions,":[26],"where":[27],"(Xn),":[28],"(Xn*),":[29],"(In,":[30],"Tn)":[31],"are":[32],"independent,":[33],"In":[34],"is":[35,62,70],"uniformly":[36],"distributed":[37],"over":[38],"{0,":[39],".":[40,41],".,n-1},":[42],"L}(X_n)={\\cal":[45],"L}(X_n^\\ast)$.":[46],"When":[47],"(cost":[52],"needed":[53],"to":[54,95,107],"partition":[55],"original":[57],"problem":[58],"into":[59],"smaller":[60],"subproblems)":[61],"small":[63],"(roughly":[64],"$\\limsup_{n\\rightarrow\\infty}\\log":[65],"E(T_n)/\\log":[66],"n\\le":[67],"1/2$),":[68],"Xn":[69],"asymptotically":[71],"normally":[72],"distributed;":[73],"nonnormal":[74],"emerge":[77],"becomes":[80],"larger.":[81],"give":[83],"many":[84],"new":[85],"examples":[86],"ranging":[87],"from":[88,102],"number":[90],"exchanges":[92],"in":[93],"quicksort":[94],"sorting":[96],"on":[97],"a":[98],"broadcast":[99],"communication":[100],"model,":[101],"an":[103],"in-situ":[104],"permutation":[105],"algorithm":[106],"tree":[108],"traversal":[109],"algorithms,":[110],"etc.":[111]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2022,"cited_by_count":2},{"year":2021,"cited_by_count":4},{"year":2019,"cited_by_count":5},{"year":2018,"cited_by_count":1},{"year":2017,"cited_by_count":1},{"year":2016,"cited_by_count":4},{"year":2015,"cited_by_count":3},{"year":2014,"cited_by_count":8},{"year":2013,"cited_by_count":3},{"year":2012,"cited_by_count":1}],"updated_date":"2026-04-04T16:13:02.066488","created_date":"2025-10-10T00:00:00"}
