{"id":"https://openalex.org/W2132958733","doi":"https://doi.org/10.1145/1542362.1542419","title":"k-means requires exponentially many iterations even in the plane","display_name":"k-means requires exponentially many iterations even in the plane","publication_year":2009,"publication_date":"2009-06-08","ids":{"openalex":"https://openalex.org/W2132958733","doi":"https://doi.org/10.1145/1542362.1542419","mag":"2132958733"},"language":"en","primary_location":{"id":"doi:10.1145/1542362.1542419","is_oa":false,"landing_page_url":"https://doi.org/10.1145/1542362.1542419","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Proceedings of the twenty-fifth annual symposium on Computational geometry","raw_type":"proceedings-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/A5057347957","display_name":"Andrea Vattani","orcid":null},"institutions":[{"id":"https://openalex.org/I36258959","display_name":"University of California, San Diego","ror":"https://ror.org/0168r3w48","country_code":"US","type":"education","lineage":["https://openalex.org/I36258959"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Andrea Vattani","raw_affiliation_strings":["University of California, San Diego, San Diego, CA, USA"],"affiliations":[{"raw_affiliation_string":"University of California, San Diego, San Diego, CA, USA","institution_ids":["https://openalex.org/I36258959"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5057347957"],"corresponding_institution_ids":["https://openalex.org/I36258959"],"apc_list":null,"apc_paid":null,"fwci":22.3896,"has_fulltext":false,"cited_by_count":196,"citation_normalized_percentile":{"value":0.99561466,"is_in_top_1_percent":true,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":91,"max":100},"biblio":{"volume":null,"issue":null,"first_page":"324","last_page":"332"},"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":0.9955999851226807,"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/T10720","display_name":"Complexity and Algorithms in Graphs","score":0.989799976348877,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"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/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.7841240763664246},{"id":"https://openalex.org/keywords/conjecture","display_name":"Conjecture","score":0.7625011205673218},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.7184499502182007},{"id":"https://openalex.org/keywords/exponential-function","display_name":"Exponential function","score":0.6234968900680542},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6096723675727844},{"id":"https://openalex.org/keywords/simple","display_name":"Simple (philosophy)","score":0.5903523564338684},{"id":"https://openalex.org/keywords/plane","display_name":"Plane (geometry)","score":0.5723304152488708},{"id":"https://openalex.org/keywords/polynomial","display_name":"Polynomial","score":0.5196301341056824},{"id":"https://openalex.org/keywords/exponential-growth","display_name":"Exponential growth","score":0.49617800116539},{"id":"https://openalex.org/keywords/space","display_name":"Space (punctuation)","score":0.4776019752025604},{"id":"https://openalex.org/keywords/pspace","display_name":"PSPACE","score":0.4682515561580658},{"id":"https://openalex.org/keywords/time-complexity","display_name":"Time complexity","score":0.44435733556747437},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.3867213726043701},{"id":"https://openalex.org/keywords/computational-complexity-theory","display_name":"Computational complexity theory","score":0.17313730716705322},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.1621454954147339},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.15460094809532166},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.13944602012634277},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.1349101960659027}],"concepts":[{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.7841240763664246},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.7625011205673218},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.7184499502182007},{"id":"https://openalex.org/C151376022","wikidata":"https://www.wikidata.org/wiki/Q168698","display_name":"Exponential function","level":2,"score":0.6234968900680542},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6096723675727844},{"id":"https://openalex.org/C2780586882","wikidata":"https://www.wikidata.org/wiki/Q7520643","display_name":"Simple (philosophy)","level":2,"score":0.5903523564338684},{"id":"https://openalex.org/C17825722","wikidata":"https://www.wikidata.org/wiki/Q17285","display_name":"Plane (geometry)","level":2,"score":0.5723304152488708},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.5196301341056824},{"id":"https://openalex.org/C75235859","wikidata":"https://www.wikidata.org/wiki/Q582659","display_name":"Exponential growth","level":2,"score":0.49617800116539},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.4776019752025604},{"id":"https://openalex.org/C197685441","wikidata":"https://www.wikidata.org/wiki/Q500716","display_name":"PSPACE","level":3,"score":0.4682515561580658},{"id":"https://openalex.org/C311688","wikidata":"https://www.wikidata.org/wiki/Q2393193","display_name":"Time complexity","level":2,"score":0.44435733556747437},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3867213726043701},{"id":"https://openalex.org/C179799912","wikidata":"https://www.wikidata.org/wiki/Q205084","display_name":"Computational complexity theory","level":2,"score":0.17313730716705322},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.1621454954147339},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.15460094809532166},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.13944602012634277},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.1349101960659027},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.0},{"id":"https://openalex.org/C111472728","wikidata":"https://www.wikidata.org/wiki/Q9471","display_name":"Epistemology","level":1,"score":0.0},{"id":"https://openalex.org/C111919701","wikidata":"https://www.wikidata.org/wiki/Q9135","display_name":"Operating system","level":1,"score":0.0}],"mesh":[],"locations_count":5,"locations":[{"id":"doi:10.1145/1542362.1542419","is_oa":false,"landing_page_url":"https://doi.org/10.1145/1542362.1542419","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Proceedings of the twenty-fifth annual symposium on Computational geometry","raw_type":"proceedings-article"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.250.1967","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.250.1967","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://arxiv.org/pdf/0812.0382v1.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.366.2467","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.366.2467","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://cseweb.ucsd.edu/users/avattani/papers/kmeans-journal.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.986.1447","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.986.1447","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"https://www.researchgate.net/profile/Andrea_Vattani/publication/23552590_k-means_requires_exponentially_many_iterations_even_in_the_plane/links/546641190cf2f5eb18016afe.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.999.3594","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.999.3594","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://cseweb.ucsd.edu/%7Eavattani/papers/kmeans.pdf","raw_type":"text"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":20,"referenced_works":["https://openalex.org/W1489597983","https://openalex.org/W1489608363","https://openalex.org/W1501500081","https://openalex.org/W1529871791","https://openalex.org/W1598917726","https://openalex.org/W1871632135","https://openalex.org/W1998739300","https://openalex.org/W2034053794","https://openalex.org/W2034380011","https://openalex.org/W2048239613","https://openalex.org/W2050761777","https://openalex.org/W2118858274","https://openalex.org/W2127218421","https://openalex.org/W2150593711","https://openalex.org/W2170648111","https://openalex.org/W2199495299","https://openalex.org/W2799061466","https://openalex.org/W3138330851","https://openalex.org/W4205687621","https://openalex.org/W4236385439"],"related_works":["https://openalex.org/W1634993130","https://openalex.org/W1602624896","https://openalex.org/W32300376","https://openalex.org/W1519862060","https://openalex.org/W1974979356","https://openalex.org/W2084061773","https://openalex.org/W2794917831","https://openalex.org/W2104564563","https://openalex.org/W2013108996","https://openalex.org/W3023224836"],"abstract_inverted_index":{"The":[0],"k-means":[1],"algorithm":[2],"is":[3,98],"a":[4,61,76,112],"well-known":[5],"method":[6],"for":[7,92],"partitioning":[8],"n":[9],"points":[10,50],"that":[11,118],"lie":[12],"in":[13,27,43,46,78,115],"the":[14,31,47,66,86,107,116,121],"d-dimensional":[15],"space":[16],"into":[17],"k":[18],"clusters.":[19],"Its":[20],"main":[21],"features":[22],"are":[23],"simplicity":[24],"and":[25,57,104],"speed":[26],"practice.":[28],"Theoretically,":[29],"however,":[30],"best":[32,67],"known":[33,68],"upper":[34],"bound":[35,70,124],"on":[36],"its":[37],"running":[38],"time":[39],"(i.e.":[40],"O(nkd))":[41],"is,":[42],"general,":[44],"exponential":[45,122],"number":[48],"of":[49,88],"(when":[51],"kd=\u03a9(n":[52],"log":[53],"n)).":[54],"Recently,":[55],"Arthur":[56],"Vassilvitskii":[58],"[2]":[59,82],"showed":[60],"super-polynomial":[62,89],"worst-case":[63],"analysis,":[64],"improving":[65],"lower":[69,90,108,123],"from":[71],"\u03a9(n)":[72],"to":[73,120],"2\u03a9(\u221an)":[74],"with":[75],"construction":[77,114],"d=\u03a9(\u221an)":[79],"dimensions.":[80],"In":[81],"they":[83],"also":[84],"conjectured":[85],"existence":[87],"bounds":[91],"any":[93],"d\u2265":[94],"2.":[95],"Our":[96],"contribution":[97],"twofold:":[99],"we":[100,105],"prove":[101],"this":[102],"conjecture":[103],"improve":[106],"bound,":[109],"by":[110],"presenting":[111],"simple":[113],"plane":[117],"leads":[119],"2\u03a9(n).":[125]},"counts_by_year":[{"year":2026,"cited_by_count":2},{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":4},{"year":2023,"cited_by_count":3},{"year":2022,"cited_by_count":3},{"year":2021,"cited_by_count":12},{"year":2020,"cited_by_count":15},{"year":2019,"cited_by_count":10},{"year":2018,"cited_by_count":18},{"year":2017,"cited_by_count":16},{"year":2016,"cited_by_count":22},{"year":2015,"cited_by_count":19},{"year":2014,"cited_by_count":19},{"year":2013,"cited_by_count":17},{"year":2012,"cited_by_count":16}],"updated_date":"2026-04-05T17:49:38.594831","created_date":"2025-10-10T00:00:00"}