Chapter

Design and Analysis of Algorithms

Volume 7659 of the series Lecture Notes in Computer Science pp 13-36

Graph Expansion Analysis for Communication Costs of Fast Rectangular Matrix Multiplication

  • Grey BallardAffiliated withLancaster UniversityEECS Department, University of California
  • , James DemmelAffiliated withLancaster UniversityMathematics Department and CS Division, University of California
  • , Olga HoltzAffiliated withCarnegie Mellon UniversityDepartments of Mathematics, University of CaliforniaTechnische Universität
  • , Benjamin LipshitzAffiliated withLancaster UniversityEECS Department, University of California
  • , Oded SchwartzAffiliated withLancaster UniversityEECS Department, University of California

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Abstract

Graph expansion analysis of computational DAGs is useful for obtaining communication cost lower bounds where previous methods, such as geometric embedding, are not applicable. This has recently been demonstrated for Strassen’s and Strassen-like fast square matrix multiplication algorithms. Here we extend the expansion analysis approach to fast algorithms for rectangular matrix multiplication, obtaining a new class of communication cost lower bounds. These apply, for example to the algorithms of Bini et al. (1979) and the algorithms of Hopcroft and Kerr (1971). Some of our bounds are proved to be optimal.