{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,9,15]],"date-time":"2024-09-15T19:52:37Z","timestamp":1726429957826},"reference-count":44,"publisher":"Association for Computing Machinery (ACM)","issue":"5","funder":[{"DOI":"10.13039\/501100002920","name":"Research Grants Council, University Grants Committee, Hong Kong","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100002920","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"short-container-title":["J. ACM"],"published-print":{"date-parts":[[2013,10]]},"abstract":"\n We consider the problem of computing the rank of an\n m<\/jats:italic>\n \u00d7\n n<\/jats:italic>\n matrix\n A<\/jats:italic>\n over a field. We present a randomized algorithm to find a set of\n r<\/jats:italic>\n = rank(\n A<\/jats:italic>\n ) linearly independent columns in\n \u00d5<\/jats:italic>\n (|\n A<\/jats:italic>\n | +\n r<\/jats:italic>\n \u03c9<\/jats:sup>\n ) field operations, where |\n A<\/jats:italic>\n | denotes the number of nonzero entries in\n A<\/jats:italic>\n and \u03c9 < 2.38 is the matrix multiplication exponent. Previously the best known algorithm to find a set of\n r<\/jats:italic>\n linearly independent columns is by Gaussian elimination, with deterministic running time\n O<\/jats:italic>\n (\n mnr<\/jats:italic>\n \u03c9-2<\/jats:sup>\n ). Our algorithm is faster when\n r<\/jats:italic>\n < max{\n m<\/jats:italic>\n ,\n n<\/jats:italic>\n }, for instance when the matrix is rectangular. We also consider the problem of computing the rank of a matrix dynamically, supporting the operations of rank one updates and additions and deletions of rows and columns. We present an algorithm that updates the rank in\n \u00d5<\/jats:italic>\n (\n mn<\/jats:italic>\n ) field operations. We show that these algorithms can be used to obtain faster algorithms for various problems in exact linear algebra, combinatorial optimization and dynamic data structure.\n <\/jats:p>","DOI":"10.1145\/2528404","type":"journal-article","created":{"date-parts":[[2013,10,23]],"date-time":"2013-10-23T15:29:17Z","timestamp":1382542157000},"page":"1-25","update-policy":"http:\/\/dx.doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":47,"title":["Fast matrix rank algorithms and applications"],"prefix":"10.1145","volume":"60","author":[{"given":"Ho Yee","family":"Cheung","sequence":"first","affiliation":[{"name":"The Chinese University of Hong Kong"}]},{"given":"Tsz Chiu","family":"Kwok","sequence":"additional","affiliation":[{"name":"The Chinese University of Hong Kong"}]},{"given":"Lap Chi","family":"Lau","sequence":"additional","affiliation":[{"name":"The Chinese University of Hong Kong"}]}],"member":"320","published-online":{"date-parts":[[2013,10,28]]},"reference":[{"key":"e_1_2_1_1_1","doi-asserted-by":"publisher","DOI":"10.1109\/18.850663"},{"key":"e_1_2_1_2_1","doi-asserted-by":"publisher","DOI":"10.1145\/1132516.1132597"},{"key":"e_1_2_1_3_1","volume-title":"Proceedings of the 22nd Annual ACM-SIAM Symposium on Discrete Algorithms. 185--191","author":"Ailon N.","unstructured":"Ailon , N. and Liberty , E . 2011. 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