Communication lower bounds for distributed-memory matrix multiplication
UNSPECIFIED. (2004) Communication lower bounds for distributed-memory matrix multiplication. JOURNAL OF PARALLEL AND DISTRIBUTED COMPUTING, 64 (9). pp. 1017-1026. ISSN 0743-7315Full text not available from this repository.
Official URL: http://dx.doi.org/10.1016/j.jpdc.2004.03.021
We present lower bounds on the amount of communication that matrix multiplication algorithms must perform on a distributed-memory parallel computer. We denote the number of processors by P and the dimension of square matrices by n. We show that the most widely used class of algorithms, the so-called two-dimensional (2D) algorithms, are optimal, in the sense that in any algorithm that only uses O(n(2)/p) words of memory per processor, at least one processor must send or receive Omega(n(2)/p(1/2)) words. We also show that algorithms from another class, the so-called three-dimensional (3D) algorithms, are also optimal. These algorithms use replication to reduce communication. We show that in any algorithm that uses Omega(n(2)/p(2/3)) words of memory per processor, at least one processor must send or receive Omega(n(2)/p(2/3)) words. Furthermore, we show a continuous tradeoff between the size of local memories and the amount of communication that must be performed. The 2D and 3D bounds are essentially instantiations of this tradeoff. We also show that if the input is distributed across the local memories of multiple nodes without replication, then Omega(n(2)) words must cross any bisection cut of the machine. All our bounds apply only to conventional o(n(3)) algorithms. They do not apply to Strassen's algorithm or other Theta(n(3)) algorithms. (C) 2004 Elsevier Inc. All rights reserved.
|Item Type:||Journal Article|
|Subjects:||Q Science > QA Mathematics > QA76 Electronic computers. Computer science. Computer software|
|Journal or Publication Title:||JOURNAL OF PARALLEL AND DISTRIBUTED COMPUTING|
|Publisher:||ACADEMIC PRESS INC ELSEVIER SCIENCE|
|Number of Pages:||10|
|Page Range:||pp. 1017-1026|
Actions (login required)