CS 598: Lecture 4 notes
نویسنده
چکیده
1 Rectangular matrix multiplication These notes are based on the algorithm and analysis from this paper [1]. We would like to partition the m × n × k cube of matrix multiplications to minimize the surface area of the maximum partition. Using a P processors we can define a p 1 × p 2 × p 3 grid of processors with p 1 p 2 p 3 = P , that yields an optimal bandwidth cost W , p 1 p 2 p 3 =P mn p 1 p 2 + mk p 1 p 3 + kn p 2 p 3 − mn + mk + kn P. Figure 1 provides three examples of good partitions in which one, two, and three of p 1 , p 2 , and p 3 are greater than 1. We now sketch recursive algorithms that obtain the optimal costs in different scenarios, as well as combinations of these variants. The algorithms assume that m ≥ n ≥ k, so that C is the largest matrix and B is the smallest. It is straightforward to derive algorithms for the other cases. The algorithms assume that A and B are both distributed evenly among all processors at the start of execution and distribute C evenly among all processors at the end. Each processor performs mnk/P of the mnk multiplications and roughly as many additions. However, the exact layout is not specified, it is most natural to implement the approaches using a blocked or a cyclic distribution, but choosing a good starting processor grid is non-trivial. We will assume throughout the analysis that the available memory is unlimited. Subsequently, we will also not keep track of latency/synchronization cost, which will be S = O(log P) for all algorithms.
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Rotating notetaking: Notes are to be written up in LaTEX if you know it, or plain text if not, and sent by electronic mail to karp@cs, ruzzo@cs, or tompa@cs, as appropriate. (If in LaTEX, see the instructions in /homes/june/ruzzo/SHARED/590bi/lectures/DIRECTIONS. If in plain text, don’t worry much about mathematical formulas, figures, bibliography citations, etc.) Notes are due at noon the day ...
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