Asymptotically Optimal Monte Carlo Sparse Multivariate Polynomial Interpolation Algorithms of Straight-Line Program
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In this paper, we propose new deterministic interpolation algorithms and Monte Carlo interpolation algorithms for sparse multivariate polynomials represented by straight-line programs. Let f be an n-variate polynomial with a degree bound D and and term bound T. Our deterministic algorithms have better complexities than existing deterministic interpolation algorithms in most cases. Our Monte Carlo interpolation algorithms are asymptotically optimal in the sense that their bit complexities are linear in nT and cubic in log D, when the coefficients of the polynomials are from a finite field. Since f has size nT, our algorithm implies that interpolating a straight-line program polynomial f is as easy as reading f, if the log D factor in the complexities is not considered. Based on the Monte Carlo interpolation algorithm, we give an asymptotically optimal algorithm for the multiplication of several multivariate polynomials, whose complexity is softly linear in the input size plus the output size, if the logarithm factors are ignored.
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