Lower Bounds for Dynamic Algebraic Problems

Abstract. We consider dynamic evaluation of algebraic functions (matrix multiplication, determinant, convolution, Fourier transform, etc.) in the model of Reif and Tate; i.e., if f(x1, . . . , xn) = (y1, . . . , ym) is an algebraic problem, we consider serving on-line requests of the form “change input xi to value v” or “what is the value of output yi?”. We present techniques for showing lower bounds on the worst case time complexity per operation for such problems. The first gives lower bounds in a wide range of rather powerful models (for instance history dependent algebraic computation trees over any infinite subset of a field, the integer RAM, and the generalized real RAM model of Ben-Amram and Galil). Using this technique, we show optimal Ω(n) bounds for dynamic matrix-vector product, dynamic matrix multiplication and dynamic discriminant and an Ω( √ n) lower bound for dynamic polynomial multiplication (convolution), providing a good match with Reif and Tate’s O...

• Type: Conference
• Year: 1999
• Where: STACS
• Authors: Gudmund Skovbjerg Frandsen, Johan P. Hansen, Peter Bro Miltersen
• Keywords: Gives Lower Bounds, Lower Bound, Matrix Multiplication, STACS 1999, Theoretical Computer Science