{"id":"https://openalex.org/W2043368640","doi":"https://doi.org/10.1142/s0218195912500021","title":"SEPARABILITY OF POINT SETS BY k-LEVEL LINEAR CLASSIFICATION TREES","display_name":"SEPARABILITY OF POINT SETS BY k-LEVEL LINEAR CLASSIFICATION TREES","publication_year":2012,"publication_date":"2012-04-01","ids":{"openalex":"https://openalex.org/W2043368640","doi":"https://doi.org/10.1142/s0218195912500021","mag":"2043368640"},"language":"en","primary_location":{"id":"doi:10.1142/s0218195912500021","is_oa":false,"landing_page_url":"https://doi.org/10.1142/s0218195912500021","pdf_url":null,"source":{"id":"https://openalex.org/S25827807","display_name":"International Journal of Computational Geometry & Applications","issn_l":"0218-1959","issn":["0218-1959","1793-6357"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319815","host_organization_name":"World Scientific","host_organization_lineage":["https://openalex.org/P4310319815"],"host_organization_lineage_names":["World Scientific"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"International Journal of Computational Geometry &amp; Applications","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/A5037943415","display_name":"Esther M. Arkin","orcid":"https://orcid.org/0009-0005-8542-931X"},"institutions":[{"id":"https://openalex.org/I59553526","display_name":"Stony Brook University","ror":"https://ror.org/05qghxh33","country_code":"US","type":"education","lineage":["https://openalex.org/I59553526"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"ESTHER M. ARKIN","raw_affiliation_strings":["Department of Applied Mathematics and Statistics, Stony Brook University, Stony Brook, New York 11794, USA"],"affiliations":[{"raw_affiliation_string":"Department of Applied Mathematics and Statistics, Stony Brook University, Stony Brook, New York 11794, USA","institution_ids":["https://openalex.org/I59553526"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5075810612","display_name":"Delia Garijo","orcid":"https://orcid.org/0000-0002-0493-4754"},"institutions":[{"id":"https://openalex.org/I79238269","display_name":"Universidad de Sevilla","ror":"https://ror.org/03yxnpp24","country_code":"ES","type":"education","lineage":["https://openalex.org/I79238269"]}],"countries":["ES"],"is_corresponding":false,"raw_author_name":"DELIA GARIJO","raw_affiliation_strings":["Departamento de Matem\u00e1tica Aplicada I, Universidad de Sevilla, Avda. Reina Mercedes s/n, 41012 Sevilla, Spain"],"affiliations":[{"raw_affiliation_string":"Departamento de Matem\u00e1tica Aplicada I, Universidad de Sevilla, Avda. Reina Mercedes s/n, 41012 Sevilla, Spain","institution_ids":["https://openalex.org/I79238269"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5019566285","display_name":"Alberto M\u00e1rquez","orcid":"https://orcid.org/0000-0002-1589-1729"},"institutions":[{"id":"https://openalex.org/I79238269","display_name":"Universidad de Sevilla","ror":"https://ror.org/03yxnpp24","country_code":"ES","type":"education","lineage":["https://openalex.org/I79238269"]}],"countries":["ES"],"is_corresponding":false,"raw_author_name":"ALBERTO M\u00c1RQUEZ","raw_affiliation_strings":["Departamento de Matem\u00e1tica Aplicada I, Universidad de Sevilla, Avda. Reina Mercedes s/n, 41012 Sevilla, Spain"],"affiliations":[{"raw_affiliation_string":"Departamento de Matem\u00e1tica Aplicada I, Universidad de Sevilla, Avda. Reina Mercedes s/n, 41012 Sevilla, Spain","institution_ids":["https://openalex.org/I79238269"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5015247956","display_name":"Joseph S. B. Mitchell","orcid":"https://orcid.org/0000-0002-0152-2279"},"institutions":[{"id":"https://openalex.org/I59553526","display_name":"Stony Brook University","ror":"https://ror.org/05qghxh33","country_code":"US","type":"education","lineage":["https://openalex.org/I59553526"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"JOSEPH S. B. MITCHELL","raw_affiliation_strings":["Department of Applied Mathematics and Statistics, Stony Brook University, Stony Brook, New York 11794, USA"],"affiliations":[{"raw_affiliation_string":"Department of Applied Mathematics and Statistics, Stony Brook University, Stony Brook, New York 11794, USA","institution_ids":["https://openalex.org/I59553526"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5062044790","display_name":"Carlos Seara","orcid":"https://orcid.org/0000-0002-0095-1725"},"institutions":[{"id":"https://openalex.org/I9617848","display_name":"Universitat Polit\u00e8cnica de Catalunya","ror":"https://ror.org/03mb6wj31","country_code":"ES","type":"education","lineage":["https://openalex.org/I9617848"]}],"countries":["ES"],"is_corresponding":false,"raw_author_name":"CARLOS SEARA","raw_affiliation_strings":["Departament de Matem\u00e0tica Aplicada II, Universitat Polit\u00e8cnica de Catalunya, Jordi Girona 1, 08034 Barcelona, Spain"],"affiliations":[{"raw_affiliation_string":"Departament de Matem\u00e0tica Aplicada II, Universitat Polit\u00e8cnica de Catalunya, Jordi Girona 1, 08034 Barcelona, Spain","institution_ids":["https://openalex.org/I9617848"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":5,"corresponding_author_ids":["https://openalex.org/A5037943415"],"corresponding_institution_ids":["https://openalex.org/I59553526"],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":{"value":0.14017624,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"22","issue":"02","first_page":"143","last_page":"165"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10996","display_name":"Computational Geometry and Mesh Generation","score":0.9997000098228455,"subfield":{"id":"https://openalex.org/subfields/1704","display_name":"Computer Graphics and Computer-Aided Design"},"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/T10996","display_name":"Computational Geometry and Mesh Generation","score":0.9997000098228455,"subfield":{"id":"https://openalex.org/subfields/1704","display_name":"Computer Graphics and Computer-Aided Design"},"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/T12923","display_name":"Digital Image Processing Techniques","score":0.9980000257492065,"subfield":{"id":"https://openalex.org/subfields/1707","display_name":"Computer Vision and Pattern Recognition"},"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/T11164","display_name":"Remote Sensing and LiDAR Applications","score":0.9936000108718872,"subfield":{"id":"https://openalex.org/subfields/2305","display_name":"Environmental Engineering"},"field":{"id":"https://openalex.org/fields/23","display_name":"Environmental Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8512119054794312},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.8141871690750122},{"id":"https://openalex.org/keywords/partition","display_name":"Partition (number theory)","score":0.6509519219398499},{"id":"https://openalex.org/keywords/interval-tree","display_name":"Interval tree","score":0.6212409138679504},{"id":"https://openalex.org/keywords/binary-tree","display_name":"Binary tree","score":0.60458904504776},{"id":"https://openalex.org/keywords/range-tree","display_name":"Range tree","score":0.5938901305198669},{"id":"https://openalex.org/keywords/tree","display_name":"Tree (set theory)","score":0.5658217668533325},{"id":"https://openalex.org/keywords/k-ary-tree","display_name":"K-ary tree","score":0.5654435753822327},{"id":"https://openalex.org/keywords/time-complexity","display_name":"Time complexity","score":0.475180059671402},{"id":"https://openalex.org/keywords/point","display_name":"Point (geometry)","score":0.42570292949676514},{"id":"https://openalex.org/keywords/regular-polygon","display_name":"Regular polygon","score":0.41168013215065},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.3836579918861389},{"id":"https://openalex.org/keywords/tree-structure","display_name":"Tree structure","score":0.2634280323982239},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.11637872457504272}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8512119054794312},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.8141871690750122},{"id":"https://openalex.org/C42812","wikidata":"https://www.wikidata.org/wiki/Q1082910","display_name":"Partition (number theory)","level":2,"score":0.6509519219398499},{"id":"https://openalex.org/C84927040","wikidata":"https://www.wikidata.org/wiki/Q6057306","display_name":"Interval tree","level":4,"score":0.6212409138679504},{"id":"https://openalex.org/C197855036","wikidata":"https://www.wikidata.org/wiki/Q380172","display_name":"Binary tree","level":2,"score":0.60458904504776},{"id":"https://openalex.org/C68775195","wikidata":"https://www.wikidata.org/wiki/Q818944","display_name":"Range tree","level":5,"score":0.5938901305198669},{"id":"https://openalex.org/C113174947","wikidata":"https://www.wikidata.org/wiki/Q2859736","display_name":"Tree (set theory)","level":2,"score":0.5658217668533325},{"id":"https://openalex.org/C100560664","wikidata":"https://www.wikidata.org/wiki/Q3608019","display_name":"K-ary tree","level":4,"score":0.5654435753822327},{"id":"https://openalex.org/C311688","wikidata":"https://www.wikidata.org/wiki/Q2393193","display_name":"Time complexity","level":2,"score":0.475180059671402},{"id":"https://openalex.org/C28719098","wikidata":"https://www.wikidata.org/wiki/Q44946","display_name":"Point (geometry)","level":2,"score":0.42570292949676514},{"id":"https://openalex.org/C112680207","wikidata":"https://www.wikidata.org/wiki/Q714886","display_name":"Regular polygon","level":2,"score":0.41168013215065},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3836579918861389},{"id":"https://openalex.org/C163797641","wikidata":"https://www.wikidata.org/wiki/Q2067937","display_name":"Tree structure","level":3,"score":0.2634280323982239},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.11637872457504272}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.1142/s0218195912500021","is_oa":false,"landing_page_url":"https://doi.org/10.1142/s0218195912500021","pdf_url":null,"source":{"id":"https://openalex.org/S25827807","display_name":"International Journal of Computational Geometry & Applications","issn_l":"0218-1959","issn":["0218-1959","1793-6357"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319815","host_organization_name":"World Scientific","host_organization_lineage":["https://openalex.org/P4310319815"],"host_organization_lineage_names":["World Scientific"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"International Journal of Computational Geometry &amp; Applications","raw_type":"journal-article"},{"id":"pmh:oai:idus.us.es:11441/38751","is_oa":false,"landing_page_url":"https://idus.us.es/handle/11441/38751","pdf_url":null,"source":{"id":"https://openalex.org/S4306400333","display_name":"idUS (Universidad de Sevilla)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I79238269","host_organization_name":"Universidad de Sevilla","host_organization_lineage":["https://openalex.org/I79238269"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"","raw_type":"info:eu-repo/semantics/article"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.298.2216","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.298.2216","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www-ma2.upc.edu/seara/data/publications/articles/level_tree_paper.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.649.8387","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.649.8387","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www-ma2.upc.edu/seara/data/publications/articles/level_tree.pdf","raw_type":"text"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":13,"referenced_works":["https://openalex.org/W976710","https://openalex.org/W1983741890","https://openalex.org/W2005314985","https://openalex.org/W2010348345","https://openalex.org/W2017921773","https://openalex.org/W2107423006","https://openalex.org/W2112823227","https://openalex.org/W2117862225","https://openalex.org/W2139594840","https://openalex.org/W2142766864","https://openalex.org/W2752885492","https://openalex.org/W3041834803","https://openalex.org/W4251391716"],"related_works":["https://openalex.org/W2117536196","https://openalex.org/W2107381954","https://openalex.org/W4244673516","https://openalex.org/W2382364329","https://openalex.org/W3022865202","https://openalex.org/W3169685561","https://openalex.org/W1999216101","https://openalex.org/W1510719821","https://openalex.org/W2125650843","https://openalex.org/W2022598760"],"abstract_inverted_index":{"Let":[0],"R":[1,32],"and":[2,8,33,47,120],"B":[3],"be":[4,77,100,124,134],"sets":[5],"of":[6,21,50,58,113],"red":[7,64],"blue":[9,67],"points":[10],"in":[11,14,82,102,126,136],"the":[12,19,37,51,59,110],"plane":[13],"general":[15],"position.":[16],"We":[17,87],"study":[18],"problem":[20],"computing":[22],"a":[23,40,55,73,90,130],"k-level":[24],"binary":[25],"space":[26],"partition":[27,60],"(BSP)":[28],"tree":[29,38,52,75,98,132],"to":[30,54],"classify/separate":[31],"B,":[34],"such":[35],"that":[36,61,72,89,118],"defines":[39],"linear":[41],"decision":[42],"at":[43],"each":[44,48],"internal":[45],"node":[46],"leaf":[49],"corresponds":[53],"(convex)":[56],"cell":[57],"contains":[62],"only":[63,66],"or":[65],"points.":[68],"Specifically,":[69],"we":[70,116],"show":[71,88,117],"2-level":[74,119],"can":[76,99,123,133],"computed,":[78],"if":[79],"one":[80],"exists,":[81],"time":[83,103,127,137],"O(n":[84,138],"2":[85],").":[86,140],"minimum-level":[91,131],"(3":[92],"\u2264":[93,95],"k":[94],"log":[96,106],"n)":[97,107],"computed":[101,125,135],"n":[104],"O(":[105],".":[108],"In":[109],"special":[111],"case":[112],"axis-parallel":[114],"partitions,":[115],"3-level":[121],"trees":[122],"O(n),":[128],"while":[129],"5":[139]},"counts_by_year":[],"updated_date":"2026-04-05T17:49:38.594831","created_date":"2025-10-10T00:00:00"}
