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Communication lower bounds and optimal algorithms for numerical linear algebra*

  • G. Ballard (a1), E. Carson (a2), J. Demmel (a2) (a3), M. Hoemmen (a4), N. Knight (a2) and O. Schwartz (a2)...

The traditional metric for the efficiency of a numerical algorithm has been the number of arithmetic operations it performs. Technological trends have long been reducing the time to perform an arithmetic operation, so it is no longer the bottleneck in many algorithms; rather, communication, or moving data, is the bottleneck. This motivates us to seek algorithms that move as little data as possible, either between levels of a memory hierarchy or between parallel processors over a network. In this paper we summarize recent progress in three aspects of this problem. First we describe lower bounds on communication. Some of these generalize known lower bounds for dense classical (O(n3)) matrix multiplication to all direct methods of linear algebra, to sequential and parallel algorithms, and to dense and sparse matrices. We also present lower bounds for Strassen-like algorithms, and for iterative methods, in particular Krylov subspace methods applied to sparse matrices. Second, we compare these lower bounds to widely used versions of these algorithms, and note that these widely used algorithms usually communicate asymptotically more than is necessary. Third, we identify or invent new algorithms for most linear algebra problems that do attain these lower bounds, and demonstrate large speed-ups in theory and practice.

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We acknowledge funding from Microsoft (award 024263) and Intel (award 024894), and matching funding by UC Discovery (award DIG07-10227). Additional support comes from ParLab affiliates National Instruments, Nokia, NVIDIA, Oracle and Samsung, as well as MathWorks. Research is also supported by DOE grants DE-SC0004938, DE-SC0005136, DE-SC0003959, DE-SC0008700, DE-SC0010200, DE-FC02-06-ER25786, AC02-05CH11231, and DARPA grant HR0011-12-2-0016. This research is supported by grant 3-10891 from the Ministry of Science and Technology, Israel, and grant 2010231 from the US-Israel Bi-National Science Foundation. This research was supported in part by an appointment to the Sandia National Laboratories Truman Fellowship in National Security Science and Engineering, sponsored by Sandia Corporation (a wholly owned subsidiary of Lockheed Martin Corporation) as Operator of Sandia National Laboratories under its US Department of Energy Contract DE-AC04-94AL85000.

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