{"id":"https://openalex.org/W2027482003","doi":"https://doi.org/10.1002/(sici)1098-2418(199908)15:1<25::aid-rsa2>3.0.co;2-r","title":"On the internal path length ofd-dimensional quad trees","display_name":"On the internal path length ofd-dimensional quad trees","publication_year":1999,"publication_date":"1999-08-01","ids":{"openalex":"https://openalex.org/W2027482003","doi":"https://doi.org/10.1002/(sici)1098-2418(199908)15:1<25::aid-rsa2>3.0.co;2-r","mag":"2027482003"},"language":"en","primary_location":{"id":"doi:10.1002/(sici)1098-2418(199908)15:1<25::aid-rsa2>3.0.co;2-r","is_oa":false,"landing_page_url":"https://doi.org/10.1002/(sici)1098-2418(199908)15:1<25::aid-rsa2>3.0.co;2-r","pdf_url":null,"source":{"id":"https://openalex.org/S59667848","display_name":"Random Structures and Algorithms","issn_l":"1042-9832","issn":["1042-9832","1098-2418"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320595","host_organization_name":"Wiley","host_organization_lineage":["https://openalex.org/P4310320595"],"host_organization_lineage_names":["Wiley"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Random Structures and Algorithms","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/A5056428246","display_name":"Ralph Neininger","orcid":"https://orcid.org/0000-0003-3975-1293"},"institutions":[{"id":"https://openalex.org/I161046081","display_name":"University of Freiburg","ror":"https://ror.org/0245cg223","country_code":"DE","type":"education","lineage":["https://openalex.org/I161046081"]}],"countries":["DE"],"is_corresponding":true,"raw_author_name":"Ralph Neininger","raw_affiliation_strings":["Institut fu\u0308r Mathematische Stochastik, Universita\u0308t Freiburg, Eckerstrasse 1, D-79104 Freiburg, Germany","Institut f\u00fcr Mathematische Stochastik, Universit\u00e4t Freiburg, Eckerstrasse 1, D\u201079104 Freiburg, Germany"],"affiliations":[{"raw_affiliation_string":"Institut fu\u0308r Mathematische Stochastik, Universita\u0308t Freiburg, Eckerstrasse 1, D-79104 Freiburg, Germany","institution_ids":["https://openalex.org/I161046081"]},{"raw_affiliation_string":"Institut f\u00fcr Mathematische Stochastik, Universit\u00e4t Freiburg, Eckerstrasse 1, D\u201079104 Freiburg, Germany","institution_ids":["https://openalex.org/I161046081"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5096949292","display_name":"Ludger R\ufffdschendorf","orcid":null},"institutions":[{"id":"https://openalex.org/I161046081","display_name":"University of Freiburg","ror":"https://ror.org/0245cg223","country_code":"DE","type":"education","lineage":["https://openalex.org/I161046081"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Ludger R\ufffdschendorf","raw_affiliation_strings":["Institut f\u00fcr Mathematische Stochastik, Universit\u00e4t Freiburg, Eckerstrasse 1, D\u201079104 Freiburg, Germany"],"affiliations":[{"raw_affiliation_string":"Institut f\u00fcr Mathematische Stochastik, Universit\u00e4t Freiburg, Eckerstrasse 1, D\u201079104 Freiburg, Germany","institution_ids":["https://openalex.org/I161046081"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5056428246"],"corresponding_institution_ids":["https://openalex.org/I161046081"],"apc_list":{"value":4330,"currency":"USD","value_usd":4330},"apc_paid":null,"fwci":4.6444,"has_fulltext":false,"cited_by_count":32,"citation_normalized_percentile":{"value":0.95063448,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":99},"biblio":{"volume":"15","issue":"1","first_page":"25","last_page":"41"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11152","display_name":"Stochastic processes and statistical mechanics","score":0.9998000264167786,"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":0.9998000264167786,"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/T11106","display_name":"Data Management and Algorithms","score":0.9995999932289124,"subfield":{"id":"https://openalex.org/subfields/1711","display_name":"Signal Processing"},"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/T11901","display_name":"Bayesian Methods and Mixture Models","score":0.9887999892234802,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7329750061035156},{"id":"https://openalex.org/keywords/path","display_name":"Path (computing)","score":0.48906952142715454},{"id":"https://openalex.org/keywords/path-length","display_name":"Path length","score":0.47386646270751953},{"id":"https://openalex.org/keywords/normalization","display_name":"Normalization (sociology)","score":0.45438846945762634},{"id":"https://openalex.org/keywords/tree","display_name":"Tree (set theory)","score":0.4219883680343628},{"id":"https://openalex.org/keywords/limiting","display_name":"Limiting","score":0.4211135506629944},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.37922123074531555},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.33881914615631104},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.14725252985954285}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7329750061035156},{"id":"https://openalex.org/C2777735758","wikidata":"https://www.wikidata.org/wiki/Q817765","display_name":"Path (computing)","level":2,"score":0.48906952142715454},{"id":"https://openalex.org/C129045301","wikidata":"https://www.wikidata.org/wiki/Q7144654","display_name":"Path length","level":2,"score":0.47386646270751953},{"id":"https://openalex.org/C136886441","wikidata":"https://www.wikidata.org/wiki/Q926129","display_name":"Normalization (sociology)","level":2,"score":0.45438846945762634},{"id":"https://openalex.org/C113174947","wikidata":"https://www.wikidata.org/wiki/Q2859736","display_name":"Tree (set theory)","level":2,"score":0.4219883680343628},{"id":"https://openalex.org/C188198153","wikidata":"https://www.wikidata.org/wiki/Q1613840","display_name":"Limiting","level":2,"score":0.4211135506629944},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.37922123074531555},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.33881914615631104},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.14725252985954285},{"id":"https://openalex.org/C19165224","wikidata":"https://www.wikidata.org/wiki/Q23404","display_name":"Anthropology","level":1,"score":0.0},{"id":"https://openalex.org/C31258907","wikidata":"https://www.wikidata.org/wiki/Q1301371","display_name":"Computer network","level":1,"score":0.0},{"id":"https://openalex.org/C127413603","wikidata":"https://www.wikidata.org/wiki/Q11023","display_name":"Engineering","level":0,"score":0.0},{"id":"https://openalex.org/C144024400","wikidata":"https://www.wikidata.org/wiki/Q21201","display_name":"Sociology","level":0,"score":0.0},{"id":"https://openalex.org/C78519656","wikidata":"https://www.wikidata.org/wiki/Q101333","display_name":"Mechanical engineering","level":1,"score":0.0},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1002/(sici)1098-2418(199908)15:1<25::aid-rsa2>3.0.co;2-r","is_oa":false,"landing_page_url":"https://doi.org/10.1002/(sici)1098-2418(199908)15:1<25::aid-rsa2>3.0.co;2-r","pdf_url":null,"source":{"id":"https://openalex.org/S59667848","display_name":"Random Structures and Algorithms","issn_l":"1042-9832","issn":["1042-9832","1098-2418"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320595","host_organization_name":"Wiley","host_organization_lineage":["https://openalex.org/P4310320595"],"host_organization_lineage_names":["Wiley"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Random Structures and Algorithms","raw_type":"journal-article"},{"id":"pmh:oai:freidok.uni-freiburg.de:21219","is_oa":false,"landing_page_url":"https://freidok.uni-freiburg.de/data/21219","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":"Random Structures and Algorithms. - 15, 1 (1999) , 25-41, ISSN: 1042-9832","raw_type":"article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":17,"referenced_works":["https://openalex.org/W1490652101","https://openalex.org/W1511251174","https://openalex.org/W1597637931","https://openalex.org/W1971711520","https://openalex.org/W1978606709","https://openalex.org/W1994166982","https://openalex.org/W1995847847","https://openalex.org/W2008048984","https://openalex.org/W2008196645","https://openalex.org/W2022718961","https://openalex.org/W2026409183","https://openalex.org/W2038681797","https://openalex.org/W2048858877","https://openalex.org/W2053101698","https://openalex.org/W2095581856","https://openalex.org/W2109399488","https://openalex.org/W4255565618"],"related_works":["https://openalex.org/W174598662","https://openalex.org/W2129550502","https://openalex.org/W2009808060","https://openalex.org/W2093961518","https://openalex.org/W2372801012","https://openalex.org/W2009843927","https://openalex.org/W2361133696","https://openalex.org/W2069032895","https://openalex.org/W2092513144","https://openalex.org/W1974037309"],"abstract_inverted_index":{"It":[0],"is":[1,21,71],"proved":[2],"that":[3],"the":[4,40,46,53,62,65,74,78,82,94,97,106,116],"internal":[5,66],"path":[6,67,98],"length":[7,99],"of":[8,27,35,39,45,64,81,96,108],"a":[9,24,28,101],"d-dimensional":[10],"quad":[11,109],"tree":[12,91],"after":[13],"normalization":[14],"converges":[15],"in":[16,105,113],"distribution.":[17],"The":[18,43,69],"limiting":[19,47],"distribution":[20,48],"characterized":[22],"as":[23,104],"fixed":[25],"point":[26],"random":[29],"affine":[30],"operator.":[31],"We":[32],"obtain":[33],"convergence":[34],"all":[36],"moments":[37,44,63],"and":[38,55],"Laplace":[41],"transforms.":[42],"can":[49],"be":[50],"evaluated":[51],"from":[52],"recursion":[54],"lead":[56],"to":[57,115],"first":[58],"order":[59],"asymptotics":[60],"for":[61,88],"lengths.":[68],"analysis":[70],"based":[72],"on":[73],"contraction":[75],"method.":[76],"In":[77],"final":[79],"part":[80],"paper":[83],"we":[84],"state":[85],"similar":[86,102],"results":[87],"general":[89],"split":[90],"models":[92],"if":[93],"expectation":[95],"has":[100],"expansion":[103],"case":[107],"trees.":[110,119],"This":[111],"applies":[112],"particular":[114],"m-ary":[117],"search":[118],"\u00a91999":[120],"John":[121],"Wiley":[122],"&":[123],"Sons,":[124],"Inc.":[125],"Random":[126],"Struct.":[127],"Alg.,":[128],"5:":[129],"25\u201341,":[130],"1999":[131]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":1},{"year":2018,"cited_by_count":2},{"year":2017,"cited_by_count":1},{"year":2016,"cited_by_count":3},{"year":2015,"cited_by_count":2},{"year":2012,"cited_by_count":1}],"updated_date":"2026-04-04T16:13:02.066488","created_date":"2025-10-10T00:00:00"}
