{"id":"https://openalex.org/W2160487122","doi":"https://doi.org/10.1007/s00453-009-9281-8","title":"Algorithmic Aspects of Proportional Symbol Maps","display_name":"Algorithmic Aspects of Proportional Symbol Maps","publication_year":2009,"publication_date":"2009-02-11","ids":{"openalex":"https://openalex.org/W2160487122","doi":"https://doi.org/10.1007/s00453-009-9281-8","mag":"2160487122"},"language":"en","primary_location":{"id":"doi:10.1007/s00453-009-9281-8","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00453-009-9281-8","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00453-009-9281-8.pdf","source":{"id":"https://openalex.org/S89324355","display_name":"Algorithmica","issn_l":"0178-4617","issn":["0178-4617","1432-0541"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by-nc","license_id":"https://openalex.org/licenses/cc-by-nc","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Algorithmica","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://link.springer.com/content/pdf/10.1007/s00453-009-9281-8.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5034923438","display_name":"Sergio Cabello","orcid":"https://orcid.org/0000-0002-3183-4126"},"institutions":[{"id":"https://openalex.org/I4210106342","display_name":"Institute of Mathematics, Physics, and Mechanics","ror":"https://ror.org/01eb3qa50","country_code":"SI","type":"facility","lineage":["https://openalex.org/I4210106342"]}],"countries":["SI"],"is_corresponding":false,"raw_author_name":"Sergio Cabello","raw_affiliation_strings":["Department of Mathematics, Institute for Mathematics, Physics and Mechanics, Ljubljana, Slovenia"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Institute for Mathematics, Physics and Mechanics, Ljubljana, Slovenia","institution_ids":["https://openalex.org/I4210106342"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5083767409","display_name":"Herman Haverkort","orcid":null},"institutions":[{"id":"https://openalex.org/I83019370","display_name":"Eindhoven University of Technology","ror":"https://ror.org/02c2kyt77","country_code":"NL","type":"education","lineage":["https://openalex.org/I83019370"]}],"countries":["NL"],"is_corresponding":false,"raw_author_name":"Herman Haverkort","raw_affiliation_strings":["Department of Mathematics and Computer Science, TU Eindhoven, Eindhoven, The Netherlands"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics and Computer Science, TU Eindhoven, Eindhoven, The Netherlands","institution_ids":["https://openalex.org/I83019370"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5053874521","display_name":"Marc van Kreveld","orcid":"https://orcid.org/0000-0001-8208-3468"},"institutions":[{"id":"https://openalex.org/I193662353","display_name":"Utrecht University","ror":"https://ror.org/04pp8hn57","country_code":"NL","type":"education","lineage":["https://openalex.org/I193662353"]}],"countries":["NL"],"is_corresponding":false,"raw_author_name":"Marc van Kreveld","raw_affiliation_strings":["Department of Information and Computing Sciences, Utrecht University, Utrecht, The Netherlands"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Information and Computing Sciences, Utrecht University, Utrecht, The Netherlands","institution_ids":["https://openalex.org/I193662353"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5000420564","display_name":"Bettina Speckmann","orcid":"https://orcid.org/0000-0002-8514-7858"},"institutions":[{"id":"https://openalex.org/I83019370","display_name":"Eindhoven University of Technology","ror":"https://ror.org/02c2kyt77","country_code":"NL","type":"education","lineage":["https://openalex.org/I83019370"]}],"countries":["NL"],"is_corresponding":true,"raw_author_name":"Bettina Speckmann","raw_affiliation_strings":["Department of Mathematics and Computer Science, TU Eindhoven, Eindhoven, The Netherlands"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics and Computer Science, TU Eindhoven, Eindhoven, The Netherlands","institution_ids":["https://openalex.org/I83019370"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5000420564"],"corresponding_institution_ids":["https://openalex.org/I83019370"],"apc_list":{"value":2890,"currency":"USD","value_usd":2890},"apc_paid":{"value":2890,"currency":"USD","value_usd":2890},"fwci":4.2658,"has_fulltext":true,"cited_by_count":27,"citation_normalized_percentile":{"value":0.93968423,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":98},"biblio":{"volume":"58","issue":"3","first_page":"543","last_page":"565"},"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/T10481","display_name":"Computer Graphics and Visualization Techniques","score":0.9968000054359436,"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":0.9950000047683716,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/symbol","display_name":"Symbol (formal)","score":0.9022643566131592},{"id":"https://openalex.org/keywords/square","display_name":"Square (algebra)","score":0.6305869817733765},{"id":"https://openalex.org/keywords/stacking","display_name":"Stacking","score":0.6006386280059814},{"id":"https://openalex.org/keywords/point","display_name":"Point (geometry)","score":0.578372597694397},{"id":"https://openalex.org/keywords/boundary","display_name":"Boundary (topology)","score":0.49220481514930725},{"id":"https://openalex.org/keywords/theory-of-computation","display_name":"Theory of computation","score":0.4865371286869049},{"id":"https://openalex.org/keywords/opacity","display_name":"Opacity","score":0.4853900671005249},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.48274725675582886},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.44991469383239746},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.41813042759895325},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.4179689586162567},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.35147714614868164},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.13115975260734558},{"id":"https://openalex.org/keywords/optics","display_name":"Optics","score":0.0748671293258667}],"concepts":[{"id":"https://openalex.org/C134400042","wikidata":"https://www.wikidata.org/wiki/Q2372244","display_name":"Symbol (formal)","level":2,"score":0.9022643566131592},{"id":"https://openalex.org/C135692309","wikidata":"https://www.wikidata.org/wiki/Q111124","display_name":"Square (algebra)","level":2,"score":0.6305869817733765},{"id":"https://openalex.org/C33347731","wikidata":"https://www.wikidata.org/wiki/Q285210","display_name":"Stacking","level":2,"score":0.6006386280059814},{"id":"https://openalex.org/C28719098","wikidata":"https://www.wikidata.org/wiki/Q44946","display_name":"Point (geometry)","level":2,"score":0.578372597694397},{"id":"https://openalex.org/C62354387","wikidata":"https://www.wikidata.org/wiki/Q875399","display_name":"Boundary (topology)","level":2,"score":0.49220481514930725},{"id":"https://openalex.org/C24858836","wikidata":"https://www.wikidata.org/wiki/Q844718","display_name":"Theory of computation","level":2,"score":0.4865371286869049},{"id":"https://openalex.org/C60056205","wikidata":"https://www.wikidata.org/wiki/Q691914","display_name":"Opacity","level":2,"score":0.4853900671005249},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.48274725675582886},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.44991469383239746},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.41813042759895325},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.4179689586162567},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.35147714614868164},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.13115975260734558},{"id":"https://openalex.org/C120665830","wikidata":"https://www.wikidata.org/wiki/Q14620","display_name":"Optics","level":1,"score":0.0748671293258667},{"id":"https://openalex.org/C46141821","wikidata":"https://www.wikidata.org/wiki/Q209402","display_name":"Nuclear magnetic resonance","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},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0}],"mesh":[],"locations_count":10,"locations":[{"id":"doi:10.1007/s00453-009-9281-8","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00453-009-9281-8","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00453-009-9281-8.pdf","source":{"id":"https://openalex.org/S89324355","display_name":"Algorithmica","issn_l":"0178-4617","issn":["0178-4617","1432-0541"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by-nc","license_id":"https://openalex.org/licenses/cc-by-nc","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Algorithmica","raw_type":"journal-article"},{"id":"pmh:oai:pure.tue.nl:openaire_cris_publications/d3e28ea6-3e31-4b75-8604-c8933db5fd44","is_oa":false,"landing_page_url":"https://research.tue.nl/en/publications/d3e28ea6-3e31-4b75-8604-c8933db5fd44","pdf_url":null,"source":{"id":"https://openalex.org/S4406922641","display_name":"TU/e Research Portal","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Cabello, S, Haverkort, H J, Kreveld, van, M J & Speckmann, B 2010, 'Algorithmic aspects of proportional symbol maps', Algorithmica, vol. 58, no. 3, pp. 543-565. https://doi.org/10.1007/s00453-009-9281-8","raw_type":"info:eu-repo/semantics/article"},{"id":"pmh:651200","is_oa":false,"landing_page_url":"http://library.tue.nl/csp/dare/LinkToRepository.csp?recordnumber=651200","pdf_url":null,"source":{"id":"https://openalex.org/S4406922641","display_name":"TU/e Research Portal","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"ISSN:0178-4617","raw_type":"Article / Letter to the editor"},{"id":"pmh:661051","is_oa":false,"landing_page_url":"http://library.tue.nl/csp/dare/LinkToRepository.csp?recordnumber=661051","pdf_url":null,"source":{"id":"https://openalex.org/S4406922641","display_name":"TU/e Research Portal","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":null},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.140.3499","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.140.3499","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.cs.uu.nl/research/techreps/repo/CS-2008/2008-008.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.418.95","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.418.95","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.imfm.si/preprinti/PDF/01042.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.99.7849","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.99.7849","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.fmf.uni-lj.si/~cabello/publications/cabello+haverkort+kreveld+speckmann+symbols-maps+esa2006.pdf","raw_type":"text"},{"id":"pmh:oai:library.tue.nl:651200","is_oa":false,"landing_page_url":"http://repository.tue.nl/651200","pdf_url":null,"source":{"id":"https://openalex.org/S4406922641","display_name":"TU/e Research Portal","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"ISSN:0178-4617","raw_type":"Article / Letter to the editor"},{"id":"pmh:oai:library.tue.nl:661051","is_oa":false,"landing_page_url":"http://repository.tue.nl/661051","pdf_url":null,"source":{"id":"https://openalex.org/S4406922641","display_name":"TU/e Research Portal","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":null},{"id":"pmh:tue:oai:pure.tue.nl:publications/d3e28ea6-3e31-4b75-8604-c8933db5fd44","is_oa":false,"landing_page_url":"https://research.tue.nl/nl/publications/d3e28ea6-3e31-4b75-8604-c8933db5fd44","pdf_url":null,"source":{"id":"https://openalex.org/S4306401843","display_name":"Data Archiving and Networked Services (DANS)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1322597698","host_organization_name":"Royal Netherlands Academy of Arts and Sciences","host_organization_lineage":["https://openalex.org/I1322597698"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Algorithmica, 58(3), 543 - 565. Springer","raw_type":"info:eu-repo/semantics/article"}],"best_oa_location":{"id":"doi:10.1007/s00453-009-9281-8","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00453-009-9281-8","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00453-009-9281-8.pdf","source":{"id":"https://openalex.org/S89324355","display_name":"Algorithmica","issn_l":"0178-4617","issn":["0178-4617","1432-0541"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by-nc","license_id":"https://openalex.org/licenses/cc-by-nc","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Algorithmica","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G6468083095","display_name":"Algorithms for locational analysis","funder_award_id":"J1-7218","funder_id":"https://openalex.org/F4320322554","funder_display_name":"Javna Agencija za Raziskovalno Dejavnost RS"}],"funders":[{"id":"https://openalex.org/F4320322554","display_name":"Javna Agencija za Raziskovalno Dejavnost RS","ror":"https://ror.org/059bp8k51"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2160487122.pdf","grobid_xml":"https://content.openalex.org/works/W2160487122.grobid-xml"},"referenced_works_count":19,"referenced_works":["https://openalex.org/W1512977524","https://openalex.org/W1585686737","https://openalex.org/W1946253038","https://openalex.org/W1980092773","https://openalex.org/W2009683649","https://openalex.org/W2013451793","https://openalex.org/W2020473451","https://openalex.org/W2035057147","https://openalex.org/W2060737356","https://openalex.org/W2061750047","https://openalex.org/W2066577300","https://openalex.org/W2070310809","https://openalex.org/W2116268321","https://openalex.org/W2121601978","https://openalex.org/W2147106864","https://openalex.org/W2149906774","https://openalex.org/W2624264009","https://openalex.org/W4242657188","https://openalex.org/W4251391716"],"related_works":["https://openalex.org/W1978042415","https://openalex.org/W2046492811","https://openalex.org/W2100408123","https://openalex.org/W1965272439","https://openalex.org/W2320007222","https://openalex.org/W4381661963","https://openalex.org/W2964019025","https://openalex.org/W1974680193","https://openalex.org/W1996160459","https://openalex.org/W4372280079"],"abstract_inverted_index":{"Proportional":[0],"symbol":[1,30,45],"maps":[2],"visualize":[3],"numerical":[4,35],"data":[5,168],"associated":[6,37],"with":[7,38],"point":[8,22],"locations":[9],"by":[10],"placing":[11],"a":[12,24,61],"scaled":[13],"symbol\u2014typically":[14],"an":[15],"opaque":[16],"disk":[17],"or":[18],"square\u2014at":[19],"the":[20,34,64,90,95,104,144,161,167],"corresponding":[21],"on":[23,166],"map.":[25],"The":[26],"area":[27],"of":[28,77,100],"each":[29,101],"is":[31],"proportional":[32,44],"to":[33,56,139,154,160],"value":[36],"its":[39],"location.":[40],"Every":[41],"visually":[42],"meaningful":[43],"map":[46],"will":[47],"contain":[48],"at":[49],"least":[50],"some":[51],"overlapping":[52],"symbols.":[53,111],"These":[54],"need":[55],"be":[57,125,137],"drawn":[58],"in":[59,127],"such":[60],"way":[62],"that":[63,114],"user":[65],"can":[66,124,136],"still":[67],"judge":[68],"their":[69],"relative":[70],"sizes":[71],"accurately.":[72],"We":[73,112,149],"identify":[74],"two":[75,92],"types":[76],"suitable":[78],"drawings:":[79,157],"physically":[80,120],"realizable":[81,121],"drawings":[82],"and":[83],"stacking":[84,133,156],"drawings.":[85,122],"For":[86],"these":[87],"we":[88],"study":[89],"following":[91],"problems:":[93],"Max-Min\u2014maximize":[94],"minimum":[96],"visible":[97,106],"boundary":[98,107],"length":[99,108],"symbol\u2014and":[102],"Max-Total\u2014maximize":[103],"total":[105],"over":[109],"all":[110],"show":[113],"both":[115],"problems":[116],"are":[117],"NP-hard":[118],"for":[119,132],"Max-Min":[123,162],"solved":[126],"O(n":[128],"2log":[129],"n)":[130,141],"time":[131,142],"drawings,":[134],"which":[135],"improved":[138],"O(nlog":[140],"when":[143],"input":[145],"has":[146],"certain":[147],"properties.":[148],"also":[150],"implemented":[151],"several":[152],"methods":[153],"compute":[155],"our":[158],"solution":[159],"problem":[163],"performs":[164],"best":[165],"sets":[169],"considered.":[170]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":1},{"year":2021,"cited_by_count":3},{"year":2020,"cited_by_count":1},{"year":2019,"cited_by_count":3},{"year":2017,"cited_by_count":1},{"year":2016,"cited_by_count":1},{"year":2015,"cited_by_count":3},{"year":2014,"cited_by_count":1},{"year":2013,"cited_by_count":5},{"year":2012,"cited_by_count":2}],"updated_date":"2026-07-28T07:46:37.118299","created_date":"2025-10-10T00:00:00"}