{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,26]],"date-time":"2025-12-26T11:26:05Z","timestamp":1766748365819},"publisher-location":"Berlin, Heidelberg","reference-count":36,"publisher":"Springer Berlin Heidelberg","isbn-type":[{"type":"print","value":"9783642348617"},{"type":"electronic","value":"9783642348624"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2012]]},"DOI":"10.1007\/978-3-642-34862-4_2","type":"book-chapter","created":{"date-parts":[[2012,11,26]],"date-time":"2012-11-26T04:00:11Z","timestamp":1353902411000},"page":"13-36","source":"Crossref","is-referenced-by-count":15,"title":["Graph Expansion Analysis for Communication Costs of Fast Rectangular Matrix Multiplication"],"prefix":"10.1007","author":[{"given":"Grey","family":"Ballard","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"James","family":"Demmel","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Olga","family":"Holtz","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Benjamin","family":"Lipshitz","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Oded","family":"Schwartz","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","reference":[{"issue":"3","key":"2_CR1","doi-asserted-by":"crossref","first-page":"319","DOI":"10.1017\/S0963548307008851","volume":"17","author":"N. Alon","year":"2008","unstructured":"Alon, N., Schwartz, O., Shapira, A.: An elementary construction of constant-degree expanders. Combinatorics, Probability & Computing\u00a017(3), 319\u2013327 (2008)","journal-title":"Combinatorics, Probability & Computing"},{"key":"2_CR2","doi-asserted-by":"publisher","first-page":"77","DOI":"10.1145\/2312005.2312021","volume-title":"Proceedings of the 24th ACM Symposium on Parallelism in Algorithms and Architectures, SPAA 2012","author":"G. Ballard","year":"2012","unstructured":"Ballard, G., Demmel, J., Holtz, O., Lipshitz, B., Schwartz, O.: Brief announcement: strong scaling of matrix multiplication algorithms and memory-independent communication lower bounds. In: Proceedings of the 24th ACM Symposium on Parallelism in Algorithms and Architectures, SPAA 2012, pp. 77\u201379. ACM, New York (2012)"},{"key":"2_CR3","doi-asserted-by":"publisher","first-page":"193","DOI":"10.1145\/2312005.2312044","volume-title":"Proceedings of the 24th ACM Symposium on Parallelism in Algorithms and Architectures, SPAA 2012","author":"G. Ballard","year":"2012","unstructured":"Ballard, G., Demmel, J., Holtz, O., Lipshitz, B., Schwartz, O.: Communication-optimal parallel algorithm for Strassen\u2019s matrix multiplication. In: Proceedings of the 24th ACM Symposium on Parallelism in Algorithms and Architectures, SPAA 2012, pp. 193\u2013204. ACM, New York (2012)"},{"issue":"3","key":"2_CR4","doi-asserted-by":"publisher","first-page":"866","DOI":"10.1137\/090769156","volume":"32","author":"G. Ballard","year":"2011","unstructured":"Ballard, G., Demmel, J., Holtz, O., Schwartz, O.: Minimizing communication in numerical linear algebra. SIAM J. Matrix Analysis Applications\u00a032(3), 866\u2013901 (2011)","journal-title":"SIAM J. Matrix Analysis Applications"},{"key":"2_CR5","doi-asserted-by":"crossref","unstructured":"Ballard, G., Demmel, J., Holtz, O., Schwartz, O.: Graph expansion and communication costs of fast matrix multiplication. J. ACM (accepted, 2012)","DOI":"10.1007\/978-3-642-34862-4_2"},{"key":"2_CR6","volume-title":"Proceedings of 2012 International Conference for High Performance Computing, Networking, Storage and Analysis, SC 2012","author":"G. Ballard","year":"2012","unstructured":"Ballard, G., Demmel, J., Lipshitz, B., Schwartz, O.: Communication-avoiding parallel Strassen: Implementation and performance. In: Proceedings of 2012 International Conference for High Performance Computing, Networking, Storage and Analysis, SC 2012, ACM, New York (2012)"},{"issue":"1-2","key":"2_CR7","doi-asserted-by":"publisher","first-page":"307","DOI":"10.1016\/S0304-3975(98)00003-6","volume":"205","author":"P. Beling","year":"1998","unstructured":"Beling, P., Megiddo, N.: Using fast matrix multiplication to find basic solutions. Theoretical Computer Science\u00a0205(1-2), 307\u2013316 (1998)","journal-title":"Theoretical Computer Science"},{"key":"2_CR8","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"47","DOI":"10.1007\/3-540-40064-8_6","volume-title":"Graph-Theoretic Concepts in Computer Science","author":"G. Bilardi","year":"2000","unstructured":"Bilardi, G., Pietracaprina, A., D\u2019Alberto, P.: On the Space and Access Complexity of Computation DAGs. In: Brandes, U., Wagner, D. (eds.) WG 2000. LNCS, vol.\u00a01928, pp. 47\u201358. Springer, Heidelberg (2000)"},{"issue":"5","key":"2_CR9","doi-asserted-by":"publisher","first-page":"1432","DOI":"10.1007\/s002240000131","volume":"32","author":"G. Bilardi","year":"1999","unstructured":"Bilardi, G., Preparata, F.: Processor-time tradeoffs under bounded-speed message propagation: Part II, lower boundes. Theory of Computing Systems\u00a032(5), 1432\u20134350 (1999)","journal-title":"Theory of Computing Systems"},{"key":"2_CR10","doi-asserted-by":"publisher","first-page":"87","DOI":"10.1007\/BF02575865","volume":"17","author":"D. Bini","year":"1980","unstructured":"Bini, D.: Relations between exact and approximate bilinear algorithms. applications. Calcolo\u00a017, 87\u201397 (1980), doi:10.1007\/BF02575865","journal-title":"Calcolo"},{"issue":"5","key":"2_CR11","doi-asserted-by":"publisher","first-page":"234","DOI":"10.1016\/0020-0190(79)90113-3","volume":"8","author":"D. Bini","year":"1979","unstructured":"Bini, D., Capovani, M., Romani, F., Lotti, G.: O(n\n                2.7799) complexity for n \u00d7n approximate matrix multiplication. Information Processing Letters\u00a08(5), 234\u2013235 (1979)","journal-title":"Information Processing Letters"},{"key":"2_CR12","doi-asserted-by":"crossref","unstructured":"B\u0171rgisser, P., Clausen, M., Shokrollahi, M.A.: Algebraic Complexity Theory. Grundlehren der mathematischen Wissenschaften, vol.\u00a0315. Springer (1997)","DOI":"10.1007\/978-3-662-03338-8"},{"issue":"3","key":"2_CR13","doi-asserted-by":"publisher","first-page":"467","DOI":"10.1137\/0211037","volume":"11","author":"D. Coppersmith","year":"1982","unstructured":"Coppersmith, D.: Rapid multiplication of rectangular matrices. SIAM Journal on Computing\u00a011(3), 467\u2013471 (1982)","journal-title":"SIAM Journal on Computing"},{"key":"2_CR14","doi-asserted-by":"publisher","first-page":"42","DOI":"10.1006\/jcom.1997.0438","volume":"13","author":"D. Coppersmith","year":"1997","unstructured":"Coppersmith, D.: Rectangular matrix multiplication revisited. J. Complex.\u00a013, 42\u201349 (1997)","journal-title":"J. Complex."},{"key":"2_CR15","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"413","DOI":"10.1007\/3-540-06841-4_78","volume-title":"Automata, Languages and Programming","author":"P. Fischer","year":"1974","unstructured":"Fischer, P., Probert, R.: Efficient Procedures for Using Matrix Algorithms. In: Loeckx, J. (ed.) ICALP 1974. LNCS, vol.\u00a014, pp. 413\u2013427. Springer, Heidelberg (1974)"},{"issue":"1","key":"2_CR16","doi-asserted-by":"publisher","first-page":"41","DOI":"10.1016\/0020-0190(89)90173-7","volume":"30","author":"Z. Galil","year":"1989","unstructured":"Galil, Z., Pan, V.: Parallel evaluation of the determinant and of the inverse of a matrix. Information Processing Letters\u00a030(1), 41\u201345 (1989)","journal-title":"Information Processing Letters"},{"key":"2_CR17","first-page":"326","volume-title":"STOC 1981: Proceedings of the Thirteenth Annual ACM Symposium on Theory of Computing","author":"J.W. Hong","year":"1981","unstructured":"Hong, J.W., Kung, H.T.: I\/O complexity: The red-blue pebble game. In: STOC 1981: Proceedings of the Thirteenth Annual ACM Symposium on Theory of Computing, pp. 326\u2013333. ACM, New York (1981)"},{"key":"2_CR18","doi-asserted-by":"publisher","first-page":"73","DOI":"10.1145\/800125.804038","volume-title":"Proceedings of the Fifth Annual ACM Symposium on Theory of Computing, STOC 1973","author":"J. Hopcroft","year":"1973","unstructured":"Hopcroft, J., Musinski, J.: Duality applied to the complexity of matrix multiplications and other bilinear forms. In: Proceedings of the Fifth Annual ACM Symposium on Theory of Computing, STOC 1973, pp. 73\u201387. ACM, New York (1973)"},{"issue":"1","key":"2_CR19","doi-asserted-by":"publisher","first-page":"30","DOI":"10.1137\/0120004","volume":"20","author":"J.E. Hopcroft","year":"1971","unstructured":"Hopcroft, J.E., Kerr, L.R.: On minimizing the number of multiplications necessary for matrix multiplication. SIAM Journal on Applied Mathematics\u00a020(1), 30\u201336 (1971)","journal-title":"SIAM Journal on Applied Mathematics"},{"key":"2_CR20","doi-asserted-by":"publisher","first-page":"11","DOI":"10.1145\/266670.266679","volume-title":"Proceedings of the Second International Symposium on Parallel Symbolic Computation, PASCO 1997","author":"X. Huang","year":"1997","unstructured":"Huang, X., Pan, V.Y.: Fast rectangular matrix multiplications and improving parallel matrix computations. In: Proceedings of the Second International Symposium on Parallel Symbolic Computation, PASCO 1997, pp. 11\u201323. ACM, New York (1997)"},{"key":"2_CR21","doi-asserted-by":"publisher","first-page":"257","DOI":"10.1006\/jcom.1998.0476","volume":"14","author":"X. Huang","year":"1998","unstructured":"Huang, X., Pan, V.Y.: Fast rectangular matrix multiplication and applications. J. Complex.\u00a014, 257\u2013299 (1998)","journal-title":"J. Complex."},{"issue":"9","key":"2_CR22","doi-asserted-by":"publisher","first-page":"1017","DOI":"10.1016\/j.jpdc.2004.03.021","volume":"64","author":"D. Irony","year":"2004","unstructured":"Irony, D., Toledo, S., Tiskin, A.: Communication lower bounds for distributed-memory matrix multiplication. J. Parallel Distrib. Comput.\u00a064(9), 1017\u20131026 (2004)","journal-title":"J. Parallel Distrib. Comput."},{"key":"2_CR23","doi-asserted-by":"publisher","first-page":"52","DOI":"10.1145\/1137856.1137866","volume-title":"Proceedings of the Twenty-Second Annual Symposium on Computational Geometry, SCG 2006","author":"H. Kaplan","year":"2006","unstructured":"Kaplan, H., Sharir, M., Verbin, E.: Colored intersection searching via sparse rectangular matrix multiplication. In: Proceedings of the Twenty-Second Annual Symposium on Computational Geometry, SCG 2006, pp. 52\u201360. ACM, New York (2006)"},{"key":"2_CR24","doi-asserted-by":"publisher","first-page":"389","DOI":"10.1007\/s11425-007-0169-2","volume":"51","author":"S. Ke","year":"2008","unstructured":"Ke, S., Zeng, B., Han, W., Pan, V.: Fast rectangular matrix multiplication and some applications. Science in China Series A: Mathematics\u00a051, 389\u2013406 (2008), doi:10.1007\/s11425-007-0169-2","journal-title":"Science in China Series A: Mathematics"},{"key":"2_CR25","doi-asserted-by":"publisher","first-page":"69","DOI":"10.1016\/0024-3795(93)00230-W","volume":"221","author":"P. Knight","year":"1995","unstructured":"Knight, P.: Fast rectangular matrix multiplication and QR decomposition. Linear Algebra and its Applications\u00a0221, 69\u201381 (1995)","journal-title":"Linear Algebra and its Applications"},{"key":"2_CR26","unstructured":"Koucky, M., Kabanets, V., Kolokolova, A.: Expanders made elementary (2007) (in preparation), \n                  \n                    http:\/\/www.cs.sfu.ca\/~kabanets\/papers\/expanders.pdf"},{"key":"2_CR27","first-page":"709","volume-title":"Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2003","author":"D. Kratsch","year":"2003","unstructured":"Kratsch, D., Spinrad, J.: Between O(nm) and O(n)?. In: Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2003, pp. 709\u2013716. Society for Industrial and Applied Mathematics, Philadelphia (2003)"},{"issue":"3","key":"2_CR28","doi-asserted-by":"publisher","first-page":"233","DOI":"10.1016\/0304-3975(83)90105-6","volume":"22","author":"G. Lev","year":"1983","unstructured":"Lev, G., Valiant, L.G.: Size bounds for superconcentrators. Theoretical Computer Science\u00a022(3), 233\u2013251 (1983)","journal-title":"Theoretical Computer Science"},{"issue":"2","key":"2_CR29","doi-asserted-by":"publisher","first-page":"171","DOI":"10.1016\/0304-3975(83)90054-3","volume":"23","author":"G. Lotti","year":"1983","unstructured":"Lotti, G., Romani, F.: On the asymptotic complexity of rectangular matrix multiplication. Theoretical Computer Science\u00a023(2), 171\u2013185 (1983)","journal-title":"Theoretical Computer Science"},{"key":"2_CR30","doi-asserted-by":"crossref","unstructured":"Mihail, M.: Conductance and convergence of Markov chains: A combinatorial treatment of expanders. In: Proceedings of the Thirtieth Annual IEEE Symposium on Foundations of Computer Science, pp. 526\u2013531 (1989)","DOI":"10.1109\/SFCS.1989.63529"},{"issue":"1","key":"2_CR31","doi-asserted-by":"publisher","first-page":"157","DOI":"10.2307\/3062153","volume":"155","author":"O. Reingold","year":"2002","unstructured":"Reingold, O., Vadhan, S., Wigderson, A.: Entropy waves, the zig-zag graph product, and new constant-degree expanders. Annals of Mathematics\u00a0155(1), 157\u2013187 (2002)","journal-title":"Annals of Mathematics"},{"key":"2_CR32","unstructured":"Savage, J.: Space-time tradeoffs in memory hierarchies. Technical report, Brown University, Providence, RI, USA (1994)"},{"key":"2_CR33","doi-asserted-by":"publisher","first-page":"354","DOI":"10.1007\/BF02165411","volume":"13","author":"V. Strassen","year":"1969","unstructured":"Strassen, V.: Gaussian elimination is not optimal. Numer. Math.\u00a013, 354\u2013356 (1969)","journal-title":"Numer. Math."},{"key":"2_CR34","first-page":"254","volume-title":"Proceedings of the Fifteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2004","author":"R. Yuster","year":"2004","unstructured":"Yuster, R., Zwick, U.: Detecting short directed cycles using rectangular matrix multiplication and dynamic programming. In: Proceedings of the Fifteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2004, pp. 254\u2013260. Society for Industrial and Applied Mathematics, Philadelphia (2004)"},{"issue":"1","key":"2_CR35","doi-asserted-by":"publisher","first-page":"2","DOI":"10.1145\/1077464.1077466","volume":"1","author":"R. Yuster","year":"2005","unstructured":"Yuster, R., Zwick, U.: Fast sparse matrix multiplication. ACM Trans. Algorithms\u00a01(1), 2\u201313 (2005)","journal-title":"ACM Trans. Algorithms"},{"key":"2_CR36","doi-asserted-by":"publisher","first-page":"289","DOI":"10.1145\/567112.567114","volume":"49","author":"U. Zwick","year":"2002","unstructured":"Zwick, U.: All pairs shortest paths using bridging sets and rectangular matrix multiplication. J. ACM\u00a049, 289\u2013317 (2002)","journal-title":"J. ACM"}],"container-title":["Lecture Notes in Computer Science","Design and Analysis of Algorithms"],"original-title":[],"link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/978-3-642-34862-4_2.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,5,4]],"date-time":"2021-05-04T09:06:48Z","timestamp":1620119208000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/978-3-642-34862-4_2"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012]]},"ISBN":["9783642348617","9783642348624"],"references-count":36,"URL":"https:\/\/doi.org\/10.1007\/978-3-642-34862-4_2","relation":{},"ISSN":["0302-9743","1611-3349"],"issn-type":[{"type":"print","value":"0302-9743"},{"type":"electronic","value":"1611-3349"}],"subject":[],"published":{"date-parts":[[2012]]}}}