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Paper Abstract and Keywords
Presentation 2012-09-03 15:25
Faster Algorithms for Rectangular Matrix Multiplication
Francois Le Gall (Univ. of Tokyo) COMP2012-32
Abstract (in Japanese) (See Japanese page) 
(in English) Let $\alpha$ be the maximal value such that the product of an $n\times n^\alpha$ matrix by an $n^\alpha\times n$ matrix can be computed with $n^{2+o(1)}$ arithmetic operations. In this paper we show that $\alpha>0.30298$, which improves the previous record $\alpha>0.29462$ by Coppersmith (Journal of Complexity, 1997). More generally, we construct a new algorithm for multiplying an $n\times n^k$ matrix by an $n^k\times n$ matrix, for any value $k\neq 1$. The complexity of this algorithm is better than all known algorithms for rectangular matrix multiplication. In the case of square matrix multiplication (i.e., for $k=1$), we recover exactly the complexity of the algorithm
by Coppersmith and Winograd (Journal of Symbolic Computation, 1990).

These new upper bounds can be used to improve the time complexity of several known algorithms that rely on rectangular matrix multiplication. For example, we directly obtain a $O(n^{2.5302})$-time algorithm for the all-pairs shortest paths problem over directed graphs with small integer weights, improving over the $O(n^{2.575})$-time algorithm by Zwick (JACM 2002), and also improve the time complexity of sparse square matrix multiplication.
Keyword (in Japanese) (See Japanese page) 
(in English) Matrix multiplication / Rectangular matrices / Algorithms / / / / /  
Reference Info. IEICE Tech. Rep., vol. 112, no. 199, COMP2012-32, pp. 41-48, Sept. 2012.
Paper # COMP2012-32 
Date of Issue 2012-08-27 (COMP) 
ISSN Print edition: ISSN 0913-5685    Online edition: ISSN 2432-6380
Copyright
and
reproduction		 All rights are reserved and no part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. Notwithstanding, instructors are permitted to photocopy isolated articles for noncommercial classroom use without fee. (License No.: 10GA0019/12GB0052/13GB0056/17GB0034/18GB0034)
Download PDF COMP2012-32

Conference Information
Committee COMP  
Conference Date 2012-09-03 - 2012-09-03 
Place (in Japanese) (See Japanese page) 
Place (in English) Hosei University 
Topics (in Japanese) (See Japanese page) 
Topics (in English)  
Paper Information
Registration To COMP 
Conference Code 2012-09-COMP 
Language English 
Title (in Japanese) (See Japanese page) 
Sub Title (in Japanese) (See Japanese page) 
Title (in English) Faster Algorithms for Rectangular Matrix Multiplication 
Sub Title (in English)  
Keyword(1) Matrix multiplication  
Keyword(2) Rectangular matrices  
Keyword(3) Algorithms  
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1st Author's Name Francois Le Gall  
1st Author's Affiliation The University of Tokyo (Univ. of Tokyo)
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Speaker Author-1 
Date Time 2012-09-03 15:25:00 
Presentation Time 35 minutes 
Registration for COMP 
Paper # COMP2012-32 
Volume (vol) vol.112 
Number (no) no.199 
Page pp.41-48 
#Pages 8 
Date of Issue 2012-08-27 (COMP) 

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