Lower bounds for dynamic algebraic problems

• Authors: Gudmund Skovbjerg Frandsen;Johan P. Hansen;Peter Bro Miltersen
• Venue: Information and Computation
• Year: 2002

We consider dynamic evaluation of algebraic functions (matrixmultiplication, determinant, convolution, Fourier transform, etc.)in the model of Reif and Tate; i.e., if f (x1,...,xn)= (y1, ..., ym) isan algebraic problem, we consider serving online requests of theform "change input xi to value v" or "whatis the value of output yi?" We present techniquesfor showing lower bounds on the worst case time complexity peroperation for such problems. The first gives lower bounds in a widerange of rather powerful models (for instance, history dependentalgebraic computation trees over any infinite subset of a field,the integer RAM, and the generalized real RAM model of Ben-Amramand Galil). Using this technique, we show optimal Ω(n)bounds for dynamic matrix-vector product, dynamic matrixmultiplication, and dynamic discriminant and an Ω(√n)lower bound for dynamic polynomial multiplication (convolution),providing a good match with Reif and Tate's O(√nlogn)upper bound. We also show linear lower bounds for dynamicdeterminant, matrix adjoint, and matrix inverse and anΩ(√n) lower bound for the elementary symmetricfunctions. The second technique is the communication complexitytechnique of Miltersen, Nisan, Safra, and Wigderson which we applyto the setting of dynamic algebraic problems, obtaining similarlower bounds in the word RAM model. The third technique gives lowerbounds in the weaker straight line program model. Using thistechnique, we show an Ω((log n)2/log logn) lower bound for dynamic discrete Fourier transform.Technical ingredients of our techniques are the incompressibilitytechnique of Ben-Amram and Galil and the lower bound for depth-twosuperconcentrators of Radhakrishnan and Ta-Shma. Theincompressibility technique is extended to arithmetic computationin arbitrary fields. 2001 Elsevier Science