{"id":"https://openalex.org/W2158196818","doi":"https://doi.org/10.1142/s0218195911003597","title":"THE ALIGNED K-CENTER PROBLEM","display_name":"THE ALIGNED K-CENTER PROBLEM","publication_year":2011,"publication_date":"2011-04-01","ids":{"openalex":"https://openalex.org/W2158196818","doi":"https://doi.org/10.1142/s0218195911003597","mag":"2158196818"},"language":"en","primary_location":{"id":"doi:10.1142/s0218195911003597","is_oa":false,"landing_page_url":"https://doi.org/10.1142/s0218195911003597","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 & 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/A5021293894","display_name":"Peter Bra\u00df","orcid":"https://orcid.org/0009-0001-0247-8329"},"institutions":[{"id":"https://openalex.org/I125687163","display_name":"City College of New York","ror":"https://ror.org/00wmhkr98","country_code":"US","type":"education","lineage":["https://openalex.org/I125687163"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"PETER BRASS","raw_affiliation_strings":["Department of Computer Science, City College, New York, USA"],"affiliations":[{"raw_affiliation_string":"Department of Computer Science, City College, New York, USA","institution_ids":["https://openalex.org/I125687163"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5003621581","display_name":"Christian Knauer","orcid":null},"institutions":[{"id":"https://openalex.org/I54009628","display_name":"University of Bayreuth","ror":"https://ror.org/0234wmv40","country_code":"DE","type":"education","lineage":["https://openalex.org/I54009628"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"CHRISTIAN KNAUER","raw_affiliation_strings":["Institut f\u00fcr Informatik, Universit\u00e4t Bayreuth, Germany"],"affiliations":[{"raw_affiliation_string":"Institut f\u00fcr Informatik, Universit\u00e4t Bayreuth, Germany","institution_ids":["https://openalex.org/I54009628"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5059099525","display_name":"Hyeon-Suk Na","orcid":null},"institutions":[{"id":"https://openalex.org/I141371507","display_name":"Soongsil University","ror":"https://ror.org/017xnm587","country_code":"KR","type":"education","lineage":["https://openalex.org/I141371507"]}],"countries":["KR"],"is_corresponding":true,"raw_author_name":"HYEON-SUK NA","raw_affiliation_strings":["School of Computing, Soongsil University, Seoul, Korea"],"affiliations":[{"raw_affiliation_string":"School of Computing, Soongsil University, Seoul, Korea","institution_ids":["https://openalex.org/I141371507"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5042351684","display_name":"Chan-Su Shin","orcid":"https://orcid.org/0000-0003-3073-6863"},"institutions":[{"id":"https://openalex.org/I83436808","display_name":"Hankuk University of Foreign Studies","ror":"https://ror.org/051q2m369","country_code":"KR","type":"education","lineage":["https://openalex.org/I83436808"]}],"countries":["KR"],"is_corresponding":false,"raw_author_name":"CHAN-SU SHIN","raw_affiliation_strings":["School of Electronics and Information Engineering, Hankuk University of Foreign Studies, Korea"],"affiliations":[{"raw_affiliation_string":"School of Electronics and Information Engineering, Hankuk University of Foreign Studies, Korea","institution_ids":["https://openalex.org/I83436808"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5019937157","display_name":"Antoine Vigneron","orcid":"https://orcid.org/0000-0003-3586-3431"},"institutions":[{"id":"https://openalex.org/I71920554","display_name":"King Abdullah University of Science and Technology","ror":"https://ror.org/01q3tbs38","country_code":"SA","type":"education","lineage":["https://openalex.org/I71920554"]}],"countries":["SA"],"is_corresponding":false,"raw_author_name":"ANTOINE VIGNERON","raw_affiliation_strings":["Mathematical and Computer Sciences and Engineering Division, King Abdullah University of Science and Technolgy, Thuwal, Kingdom of Saudi Arabia"],"affiliations":[{"raw_affiliation_string":"Mathematical and Computer Sciences and Engineering Division, King Abdullah University of Science and Technolgy, Thuwal, Kingdom of Saudi Arabia","institution_ids":["https://openalex.org/I71920554"]}]}],"institutions":[],"countries_distinct_count":4,"institutions_distinct_count":5,"corresponding_author_ids":["https://openalex.org/A5059099525"],"corresponding_institution_ids":["https://openalex.org/I141371507"],"apc_list":null,"apc_paid":null,"fwci":0.7318,"has_fulltext":false,"cited_by_count":22,"citation_normalized_percentile":{"value":0.8086145,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":98},"biblio":{"volume":"21","issue":"02","first_page":"157","last_page":"178"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10996","display_name":"Computational Geometry and Mesh Generation","score":0.9998999834060669,"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":0.9998999834060669,"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/T11502","display_name":"Facility Location and Emergency Management","score":0.9941999912261963,"subfield":{"id":"https://openalex.org/subfields/1407","display_name":"Organizational Behavior and Human Resource Management"},"field":{"id":"https://openalex.org/fields/14","display_name":"Business, Management and Accounting"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},{"id":"https://openalex.org/T11830","display_name":"Point processes and geometric inequalities","score":0.9779000282287598,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"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/combinatorics","display_name":"Combinatorics","score":0.7216323018074036},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6857065558433533},{"id":"https://openalex.org/keywords/binary-logarithm","display_name":"Binary logarithm","score":0.6742000579833984},{"id":"https://openalex.org/keywords/center","display_name":"Center (category theory)","score":0.5846143960952759},{"id":"https://openalex.org/keywords/log-log-plot","display_name":"Log-log plot","score":0.5459597706794739},{"id":"https://openalex.org/keywords/running-time","display_name":"Running time","score":0.48161834478378296},{"id":"https://openalex.org/keywords/line","display_name":"Line (geometry)","score":0.4748696982860565},{"id":"https://openalex.org/keywords/randomized-algorithm","display_name":"Randomized algorithm","score":0.4706415832042694},{"id":"https://openalex.org/keywords/time-complexity","display_name":"Time complexity","score":0.42268872261047363},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.3258700966835022},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.3221628963947296},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.15291842818260193},{"id":"https://openalex.org/keywords/crystallography","display_name":"Crystallography","score":0.08473390340805054}],"concepts":[{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.7216323018074036},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6857065558433533},{"id":"https://openalex.org/C63553672","wikidata":"https://www.wikidata.org/wiki/Q581168","display_name":"Binary logarithm","level":2,"score":0.6742000579833984},{"id":"https://openalex.org/C2779463800","wikidata":"https://www.wikidata.org/wiki/Q5062222","display_name":"Center (category theory)","level":2,"score":0.5846143960952759},{"id":"https://openalex.org/C195292467","wikidata":"https://www.wikidata.org/wiki/Q2091879","display_name":"Log-log plot","level":3,"score":0.5459597706794739},{"id":"https://openalex.org/C3017489831","wikidata":"https://www.wikidata.org/wiki/Q2393193","display_name":"Running time","level":2,"score":0.48161834478378296},{"id":"https://openalex.org/C198352243","wikidata":"https://www.wikidata.org/wiki/Q37105","display_name":"Line (geometry)","level":2,"score":0.4748696982860565},{"id":"https://openalex.org/C128669082","wikidata":"https://www.wikidata.org/wiki/Q583461","display_name":"Randomized algorithm","level":2,"score":0.4706415832042694},{"id":"https://openalex.org/C311688","wikidata":"https://www.wikidata.org/wiki/Q2393193","display_name":"Time complexity","level":2,"score":0.42268872261047363},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3258700966835022},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.3221628963947296},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.15291842818260193},{"id":"https://openalex.org/C8010536","wikidata":"https://www.wikidata.org/wiki/Q160398","display_name":"Crystallography","level":1,"score":0.08473390340805054},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1142/s0218195911003597","is_oa":false,"landing_page_url":"https://doi.org/10.1142/s0218195911003597","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 & Applications","raw_type":"journal-article"},{"id":"pmh:oai:repository.kaust.edu.sa:10754/561744","is_oa":false,"landing_page_url":"http://hdl.handle.net/10754/561744","pdf_url":null,"source":{"id":"https://openalex.org/S4306401596","display_name":"King Abdullah University of Science and Technology Repository (King Abdullah University of Science and Technology)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I71920554","host_organization_name":"King Abdullah University of Science and Technology","host_organization_lineage":["https://openalex.org/I71920554"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Article"},{"id":"pmh:oai:scholarworks.unist.ac.kr:201301/19640","is_oa":false,"landing_page_url":"https://scholarworks.unist.ac.kr/handle/201301/19640","pdf_url":null,"source":{"id":"https://openalex.org/S4306401118","display_name":"Scholarworks@UNIST (Ulsan National Institute of Science and Technology)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I48566637","host_organization_name":"Ulsan National Institute of Science and Technology","host_organization_lineage":["https://openalex.org/I48566637"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"ARTICLE"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":31,"referenced_works":["https://openalex.org/W64781954","https://openalex.org/W92309384","https://openalex.org/W1488959752","https://openalex.org/W1502200255","https://openalex.org/W1986084080","https://openalex.org/W1993949040","https://openalex.org/W2004899468","https://openalex.org/W2007846378","https://openalex.org/W2013451793","https://openalex.org/W2015858208","https://openalex.org/W2018804864","https://openalex.org/W2034951808","https://openalex.org/W2035321239","https://openalex.org/W2098355410","https://openalex.org/W2112692787","https://openalex.org/W2119918499","https://openalex.org/W2125585783","https://openalex.org/W2133311553","https://openalex.org/W2133422013","https://openalex.org/W2133962824","https://openalex.org/W2139230981","https://openalex.org/W2139594840","https://openalex.org/W2145360688","https://openalex.org/W2157703255","https://openalex.org/W2161779238","https://openalex.org/W2243270611","https://openalex.org/W2911302472","https://openalex.org/W2911597709","https://openalex.org/W3137423196","https://openalex.org/W4234689394","https://openalex.org/W4285719527"],"related_works":["https://openalex.org/W2055932080","https://openalex.org/W4377371889","https://openalex.org/W2011106535","https://openalex.org/W1581210843","https://openalex.org/W2113270940","https://openalex.org/W1992470151","https://openalex.org/W1541068124","https://openalex.org/W1994598067","https://openalex.org/W1992104669","https://openalex.org/W2018076478"],"abstract_inverted_index":{"In":[0,115],"this":[1,107],"paper":[2],"we":[3,104,126,141,156,165,214],"study":[4],"several":[5],"instances":[6],"of":[7,18,50,61,93,122],"the":[8,12,22,47,51,62,68,90,94,97,116,120],"alignedk-center":[9],"problem":[10,56,65,95,108],"where":[11,96,118],"goal":[13],"is,":[14],"given":[15,101],"a":[16,25,37,58,75,80,82,85,143],"set":[17],"points":[19],"S":[20,45],"in":[21,66,74,102,109,180],"plane":[23],"and":[24,46,195,235],"parameter":[26],"k":[27,32,183,190,198,207],"\u2a7e":[28],"1,":[29],"to":[30,72],"find":[31],"disks":[33,52],"with":[34,146,219],"centers":[35,69],"on":[36],"line":[38,98],"\u2113":[39,99,123,136],"such":[40,78],"that":[41],"their":[42],"union":[43],"covers":[44],"maximum":[48],"radius":[49],"is":[53,57,100,124,137],"minimized.":[54],"This":[55],"constrained":[59,71],"version":[60,92],"well-known":[63],"k-center":[64],"which":[67],"are":[70],"lie":[73],"particular":[76],"region":[77],"as":[79],"segment,":[81],"line,":[83,140],"or":[84],"polygon.":[86],"We":[87],"first":[88],"consider":[89],"simplest":[91],"advance;":[103],"can":[105],"solve":[106],"time":[110,149],"O(n":[111,129,150,181,188,196,209,223,236],"log":[112,131,152,173,175,182,189,197,212,228,233,241],"2":[113,130,132,153,171,202,226],"n).":[114,154],"case":[117],"only":[119],"direction":[121],"fixed,":[125],"give":[127,142,216],"an":[128,138],"n)-time":[133],"algorithm.":[134],"When":[135,164],"arbitrary":[139],"randomized":[144,217],"algorithm":[145],"expected":[147,220],"running":[148,221],"4":[151,239],"Then":[155],"present":[157],"(1+\u03b5)-approximation":[158],"algorithms":[159,178,218],"for":[160],"these":[161,177],"three":[162],"problems.":[163],"denote":[166],"T(k,":[167,185,192,200],"\u03b5)":[168],"=":[169,208],"(k/\u03b5":[170,225,238],"+(k/\u03b5)":[172],"k)":[174],"(1/\u03b5),":[176],"run":[179],"+":[184,191,199,224,237],"\u03b5))":[186],"time,":[187,194,204],"\u03b5)/\u03b5)":[193],"\u03b5)/\u03b5":[201],")":[203,227,232,240],"respectively.":[205,243],"For":[206],"1/3":[210],"/":[211],"n),":[213],"also":[215],"times":[222],"(1/\u03b5)),":[229,234,242],"O(n+(k/\u03b5":[230],"3":[231]},"counts_by_year":[{"year":2025,"cited_by_count":2},{"year":2023,"cited_by_count":2},{"year":2022,"cited_by_count":3},{"year":2021,"cited_by_count":5},{"year":2019,"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":2014,"cited_by_count":1}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}