{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,10]],"date-time":"2026-06-10T07:54:26Z","timestamp":1781078066184,"version":"3.54.1"},"reference-count":20,"publisher":"Association for Computing Machinery (ACM)","issue":"2","license":[{"start":{"date-parts":[[2010,1,1]],"date-time":"2010-01-01T00:00:00Z","timestamp":1262304000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.acm.org\/publications\/policies\/copyright_policy#Background"}],"content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"short-container-title":["J. ACM"],"published-print":{"date-parts":[[2010,1]]},"abstract":"<jats:p>\n            We present a general approach for designing approximation algorithms for a fundamental class of geometric clustering problems in arbitrary dimensions. More specifically, our approach leads to simple randomized algorithms for the\n            <jats:italic>k<\/jats:italic>\n            -means,\n            <jats:italic>k<\/jats:italic>\n            -median and discrete\n            <jats:italic>k<\/jats:italic>\n            -means problems that yield (1+\u03b5) approximations with probability \u2265 1\/2 and running times of\n            <jats:italic>O<\/jats:italic>\n            (2\n            <jats:sup>\n              (\n              <jats:italic>k<\/jats:italic>\n              \/\u03b5)\n              <jats:sup>\n                <jats:italic>O<\/jats:italic>\n                (1)\n              <\/jats:sup>\n            <\/jats:sup>\n            <jats:italic>dn<\/jats:italic>\n            ). These are the first algorithms for these problems whose running times are linear in the size of the input (\n            <jats:italic>nd<\/jats:italic>\n            for\n            <jats:italic>n<\/jats:italic>\n            points in\n            <jats:italic>d<\/jats:italic>\n            dimensions) assuming\n            <jats:italic>k<\/jats:italic>\n            and \u03b5 are fixed. Our method is general enough to be applicable to clustering problems satisfying certain simple properties and is likely to have further applications.\n          <\/jats:p>","DOI":"10.1145\/1667053.1667054","type":"journal-article","created":{"date-parts":[[2010,8,24]],"date-time":"2010-08-24T13:16:40Z","timestamp":1282655800000},"page":"1-32","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":88,"title":["Linear-time approximation schemes for clustering problems in any dimensions"],"prefix":"10.1145","volume":"57","author":[{"given":"Amit","family":"Kumar","sequence":"first","affiliation":[{"name":"Indian Institute of Technology, New Delhi, India"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yogish","family":"Sabharwal","sequence":"additional","affiliation":[{"name":"IBM Research - India, New Delhi, India"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Sandeep","family":"Sen","sequence":"additional","affiliation":[{"name":"Indian Institute of Technology, New Delhi, India"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"320","published-online":{"date-parts":[[2010,2,8]]},"reference":[{"key":"e_1_2_1_1_1","doi-asserted-by":"publisher","DOI":"10.1145\/290179.290180"},{"key":"e_1_2_1_2_1","doi-asserted-by":"publisher","DOI":"10.1145\/276698.276718"},{"key":"e_1_2_1_3_1","doi-asserted-by":"publisher","DOI":"10.1145\/509907.509947"},{"key":"e_1_2_1_4_1","unstructured":"Bern M. and Eppstein D. 1996. Approximation algorithms for geometric problems. In Approximation Algorithms for Geometric Problems PWS Publishing 296--345.   Bern M. and Eppstein D. 1996. Approximation algorithms for geometric problems. In Approximation Algorithms for Geometric Problems PWS Publishing 296--345."},{"key":"e_1_2_1_5_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0169-7552(97)00031-7"},{"key":"e_1_2_1_6_1","doi-asserted-by":"publisher","DOI":"10.5555\/1109557.1109687"},{"key":"e_1_2_1_7_1","volume-title":"Tech. Rep. CS2007-0890","author":"Dasgupta S.","year":"2007"},{"key":"e_1_2_1_8_1","doi-asserted-by":"publisher","DOI":"10.1145\/780542.780550"},{"key":"e_1_2_1_9_1","doi-asserted-by":"publisher","DOI":"10.1002\/(SICI)1097-4571(199009)41:6<391::AID-ASI1>3.0.CO;2-9"},{"key":"e_1_2_1_10_1","unstructured":"Duda R. O. Hart P. E. and Stork D. G. 2000. Pattern Classification (2nd Edition) 2 ed. Wiley-Interscience New York.   Duda R. O. Hart P. E. and Stork D. G. 2000. Pattern Classification (2nd Edition) 2 ed. Wiley-Interscience New York."},{"key":"e_1_2_1_11_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF00962238"},{"key":"e_1_2_1_12_1","volume-title":"Proceedings of the Symposium on Discrete Algorithms, ACM","author":"Guruswami V."},{"key":"e_1_2_1_13_1","doi-asserted-by":"publisher","DOI":"10.1145\/1007352.1007400"},{"key":"e_1_2_1_14_1","doi-asserted-by":"publisher","DOI":"10.1145\/177424.178042"},{"key":"e_1_2_1_15_1","volume-title":"Symposium, CA.","author":"Indyk P.","year":"2004"},{"key":"e_1_2_1_16_1","doi-asserted-by":"publisher","DOI":"10.1137\/S0097539702404055"},{"key":"e_1_2_1_17_1","doi-asserted-by":"publisher","DOI":"10.1007\/s004540010019"},{"key":"e_1_2_1_18_1","doi-asserted-by":"publisher","DOI":"10.1109\/FOCS.2006.75"},{"key":"e_1_2_1_19_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.comgeo.2005.01.003"},{"key":"e_1_2_1_20_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF00130487"}],"container-title":["Journal of the ACM"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/1667053.1667054","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/dl.acm.org\/doi\/pdf\/10.1145\/1667053.1667054","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,18]],"date-time":"2025-06-18T12:41:30Z","timestamp":1750250490000},"score":1,"resource":{"primary":{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/1667053.1667054"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,1]]},"references-count":20,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2010,1]]}},"alternative-id":["10.1145\/1667053.1667054"],"URL":"https:\/\/doi.org\/10.1145\/1667053.1667054","relation":{},"ISSN":["0004-5411","1557-735X"],"issn-type":[{"value":"0004-5411","type":"print"},{"value":"1557-735X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2010,1]]},"assertion":[{"value":"2007-01-01","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2009-10-01","order":1,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2010-02-08","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}]}}