Time-space tradeoffs for matrix multiplication and the discrete fourier transform on any general sequential random-access computer

Abstract

On any general sequential model of computation with random-access input (e.g., a logarithmic cost RAM or a Turing machine with random-access input heads) the product time · space is: 1. (1) Not o( N 2), hence not o(( n log n ) 2) , for computing the discrete Fourier transform over finite prime fields, even when each entry in the input vector has length O(log N). Here N denotes the number of entries and n denotes the input length. 2. (2) Ω(M 3) , hence not o(( n log n ) 1,5) for M by M matrix multiplication over the integers or over finite prime fields, even when each entry in the matrices has length O(log M). For this range of entries length these lower bounds on time · space coincide, up to a log n o(1) factor, with the upper bounds achieved by the straightforward arithmetic algorithms. Time-space tradeoffs for the discrete Fourier transform and for matrix multiplication on the restricted model of a straight-line algorithm were previously obtained by Grigoryev (“Notes on Scientific Seminars 60,” pp. 38–48, Steklov Math. Inst., Leningrad, 1976. [Russian]) Ja'Ja' (“Proceedings 12th Annual ACM Sympos. Theory Comput., 1980,” pp. 339–349) and Tompa (University of Toronto, Dept. of Comput. Sci. Tech. Report 122 78 ). The model considered is general, meaning that it is not restricted to performing a sequence of arithmetic operations. Arbitrary bit processing and branching is allowed.