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Fast interpolation and multiplication of unbalanced polynomials (2402.10139v2)

Published 15 Feb 2024 in cs.SC and cs.CC

Abstract: We consider the classical problems of interpolating a polynomial given a black box for evaluation, and of multiplying two polynomials, in the setting where the bit-lengths of the coefficients may vary widely, so-called unbalanced polynomials. Writing s for the total bit-length and D for the degree, our new algorithms have expected running time $\tilde{O}(s \log D)$, whereas previous methods for (resp.) dense or sparse arithmetic have at least $\tilde{O}(sD)$ or $\tilde{O}(s2)$ bit complexity.

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References (70)
  1. Noga Alon and Yishay Mansour. 1995. ϵitalic-ϵ\epsilonitalic_ϵ-Discrepancy Sets and Their Application for Interpolation of Sparse Polynomials. Inform. Process. Lett. 54, 6 (1995), 337–342. https://doi.org/10.1016/0020-0190(95)00032-8
  2. Andrew Arnold. 2016. Sparse polynomial interpolation and testing. Ph. D. Dissertation. University of Waterloo.
  3. Sparse interpolation over finite fields via low-order roots of unity. In ISSAC’14. 27–34. https://doi.org/10.1145/2608628.2608671
  4. Faster sparse multivariate polynomial interpolation of straight-line programs. J. Symb. Comput. 75 (2016), 4–24. https://doi.org/10.1016/j.jsc.2015.11.005
  5. Arnold Arnold and Daniel S. Roche. 2014. Multivariate sparse interpolation using randomized Kronecker substitutions. In ISSAC’14. 35–42. https://doi.org/10.1145/2608628.2608674
  6. Andrew Arnold and Daniel S. Roche. 2015. Output-sensitive algorithms for sumset and sparse polynomial multiplication. In ISSAC’15. 29–36. https://doi.org/10.1145/2755996.2756653
  7. Michael Ben-Or and Prasoon Tiwari. 1988. A Deterministic Algorithm for Sparse Multivariate Polynomial Interpolation. In STOC’88. 301–309. https://doi.org/10.1145/62212.62241
  8. A fast algorithm for computing the Smith normal form with multipliers for a nonsingular integer matrix. Journal of Symbolic Computation 116 (2023), 146–182. https://doi.org/10.1016/j.jsc.2022.09.002
  9. Markus Bläser and Gorav Jindal. 2014. A New Deterministic Algorithm for Sparse Multivariate Polynomial Interpolation. In Proceedings of the 39th International Symposium on Symbolic and Algebraic Computation (ISSAC ’14). Association for Computing Machinery, New York, NY, USA, 51–58. https://doi.org/10.1145/2608628.2608648
  10. Leo Bluestein. 1970. A linear filtering approach to the computation of discrete Fourier transform. IEEE Transactions on Audio and Electroacoustics 18, 4 (1970), 451–455. https://doi.org/10.1109/TAU.1970.1162132
  11. Marco Bodrato and Alberto Zanoni. 2020. Univariate Polynomials with Long Unbalanced Coefficients as Bivariate Balanced Ones: A Toom-Cook Multiplication Approach. In Computer Algebra in Scientific Computing, François Boulier, Matthew England, Timur M. Sadykov, and Evgenii V. Vorozhtsov (Eds.). 91–107. https://doi.org/10.1007/978-3-030-60026-6_6
  12. David G. Cantor and Erich Kaltofen. 1991. On fast multiplication of polynomials over arbitrary algebras. Acta Informatica 28 (1991), 693–701.
  13. Parallel Integer Polynomial Multiplication. In 18th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC). 72–80. https://doi.org/10.1109/SYNASC.2016.024
  14. Richard Cole and Ramesh Hariharan. 2002. Verifying candidate matches in sparse and wildcard matching. In STOC’02. 592–601. https://doi.org/10.1145/509907.509992
  15. Stephen A. Cook. 1966. On the Minimum Computation Time of Functions. Ph. D. Dissertation. Harvard.
  16. Annie Cuyt and Wen-shin Lee. 2011. Sparse Interpolation of Multivariate Rational Functions. Theoretical Computer Science 412, 16 (April 2011), 1445–1456. https://doi.org/10.1016/j.tcs.2010.11.050
  17. Richard J Fateman. 2010. Can You Save Time in Multiplying Polynomials by Encoding Them as Integers? (2010). https://people.eecs.berkeley.edu/~fateman/papers/polysbyGMP.pdf.
  18. Sanchit Garg and Éric Schost. 2009. Interpolation of polynomials given by straight-line programs. Theor. Comput. Sci. 410, 27 (2009), 2659–2662. https://doi.org/10.1016/j.tcs.2009.03.030
  19. Mark Giesbrecht and Daniel S. Roche. 2011. Diversification Improves Interpolation. In ISSAC’11. 123–130. https://doi.org/10.1145/1993886.1993909
  20. Essentially optimal sparse polynomial multiplication. In ISSAC’20. 202–209. https://doi.org/10.1145/3373207.3404026
  21. Polynomial Modular Product Verification and Its Implications. Journal of Symbolic Computation 116 (2023), 98–129. https://doi.org/10.1016/j.jsc.2022.08.011
  22. Sparse polynomial interpolation and division in soft-linear time. In Proceedings of the 2022 international symposium on symbolic and algebraic computation (ISSAC’22). ACM, 459–468. https://doi.org/10.1145/3476446.3536173 arXiv:2202.08106
  23. Sparse Polynomial Interpolation and Division in Soft-linear Time. In Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation (Villeneuve-d’Ascq, France) (ISSAC ’22). Association for Computing Machinery, New York, NY, USA, 459–468. https://doi.org/10.1145/3476446.3536173
  24. Fast Parallel Algorithms for Sparse Multivariate Polynomial Interpolation over Finite Fields. SIAM J. Comput. 19, 6 (Dec. 1990), 1059–1063. https://doi.org/10.1137/0219073
  25. David Harvey and Joris van der Hoeven. 2021. Integer Multiplication in Time O⁢(n⁢log⁡n)𝑂𝑛𝑛O(n\log n)italic_O ( italic_n roman_log italic_n ). Annals of Mathematics 193, 2 (2021). https://doi.org/10.4007/annals.2021.193.2.4
  26. David Harvey and Joris van der Hoeven. 2022. Polynomial Multiplication over Finite Fields in Time O⁢(n⁢log⁡n)𝑂𝑛𝑛O(n\log n)italic_O ( italic_n roman_log italic_n ). J. ACM 69, 2 (2022), 12:1–12:40. https://doi.org/10.1145/3505584
  27. Faster Polynomial Multiplication over Finite Fields. J. ACM 63, 6 (jan 2017). https://doi.org/10.1145/3005344
  28. Structured FFT and TFT: Symmetric and Lattice Polynomials. In ISSAC’13. ACM, 355–362. https://doi.org/10.1145/2465506.2465526
  29. Joris van der Hoeven and Grégoire Lecerf. 2012. On the Complexity of Multivariate Blockwise Polynomial Multiplication. In ISSAC’12. 211–218. https://doi.org/10.1145/2442829.2442861
  30. Joris van der Hoeven and Grégoire Lecerf. 2014. Sparse Polynomial Interpolation in Practice. ACM Commun. Comput. Algebra 48, 3/4 (2014), 187–191. https://doi.org/10.1145/2733693.2733721
  31. Joris van der Hoeven and Grégoire Lecerf. 2019. Sparse polynomial interpolation. Exploring fast heuristic algorithms over finite fields. (2019). https://hal.archives-ouvertes.fr/hal-02382117 preprint.
  32. Joris van der Hoeven and Grégoire Lecerf. 2021. On Sparse Interpolation of Rational Functions and Gcds. ACM Communications in Computer Algebra 55, 1 (May 2021), 1–12. https://doi.org/10.1145/3466895.3466896
  33. Joris van der Hoeven and Grégoire Lecerf. 2023. Fast interpolation of sparse multivariate polynomials. arXiv:2312.17664.
  34. Ming-Deh A Huang and Ashwin J Rao. 1999. Interpolation of Sparse Multivariate Polynomials over Large Finite Fields with Applications. Journal of Algorithms 33, 2 (Nov. 1999), 204–228. https://doi.org/10.1006/jagm.1999.1045
  35. Qiao-Long Huang. 2019. Sparse Polynomial Interpolation over Fields with Large or Zero Characteristic. In ISSAC’19. 219–226. https://doi.org/10.1145/3326229.3326250
  36. Qiao-Long Huang. 2021. Sparse Polynomial Interpolation Based on Diversification. Science China Mathematics (April 2021). https://doi.org/10.1007/s11425-020-1791-5
  37. Qiao-Long Huang. 2023. Sparse Polynomial Interpolation Based on Derivatives. Journal of Symbolic Computation 114 (Jan. 2023), 359–375. https://doi.org/10.1016/j.jsc.2022.06.002
  38. Qiao-Long Huang and Xiao-Shan Gao. 2019. Revisit Sparse Polynomial Interpolation Based on Randomized Kronecker Substitution. In CASC’19. Springer, 215–235. https://doi.org/10.1007/978-3-030-26831-2_15
  39. Qiao-Long Huang and Xiao-Shan Gao. 2020. Faster interpolation algorithms for sparse multivariate polynomials given by straight-line programs. J. Symb. Comput. 101 (2020), 367–386. https://doi.org/10.1016/j.jsc.2019.10.005
  40. (Nearly) Sample-Optimal Sparse Fourier Transform. 480–499. https://doi.org/10.1137/1.9781611973402.36
  41. Seyed Mohammad Mahdi Javadi and Michael Monagan. 2010. Parallel Sparse Polynomial Interpolation over Finite Fields. In Proceedings of the 4th International Workshop on Parallel and Symbolic Computation (PASCO ’10). Association for Computing Machinery, New York, NY, USA, 160–168. https://doi.org/10.1145/1837210.1837233
  42. Stephen C. Johnson. 1974. Sparse polynomial arithmetic. ACM SIGSAM Bulletin 8, 3 (1974), 63–71. https://doi.org/10.1145/1086837.1086847
  43. Modular Rational Sparse Multivariate Polynomial Interpolation. In Proceedings of the International Symposium on Symbolic and Algebraic Computation (ISSAC ’90). Association for Computing Machinery, New York, NY, USA, 135–139. https://doi.org/10.1145/96877.96912
  44. Erich Kaltofen and Wen-shin Lee. 2003. Early termination in sparse interpolation algorithms. J. Symb. Comput. 36, 3 (2003), 365–400. https://doi.org/10.1016/S0747-7171(03)00088-9
  45. Erich Kaltofen and Lakshman Yagati. 1989. Improved sparse multivariate polynomial interpolation algorithms. In ISSAC’88, P. Gianni (Ed.). Springer Berlin Heidelberg, Berlin, Heidelberg, 467–474.
  46. Erich Kaltofen and Zhengfeng Yang. 2007. On Exact and Approximate Interpolation of Sparse Rational Functions. In Proceedings of the 2007 International Symposium on Symbolic and Algebraic Computation (ISSAC ’07). Association for Computing Machinery, New York, NY, USA, 203–210. https://doi.org/10.1145/1277548.1277577
  47. Erich L. Kaltofen. 2010. Fifteen years after DSC and WLSS2: What parallel computations I do today [invited lecture at PASCO 2010]. In Proceedings of the 4th International Workshop on Parallel and Symbolic Computation (Grenoble, France) (PASCO ’10). ACM, 10–17. https://doi.org/10.1145/1837210.1837213
  48. Erich L. Kaltofen and Michael Nehring. 2011. Supersparse Black Box Rational Function Interpolation. In Proceedings of the 36th International Symposium on Symbolic and Algebraic Computation (ISSAC ’11). Association for Computing Machinery, New York, NY, USA, 177–186. https://doi.org/10.1145/1993886.1993916
  49. Anatolii Karatsuba and Yu Ofman. 1962. Multiplication of Multidigit Numbers on Automata. Soviet Physics Doklady 7 (12 1962), 595.
  50. Leopold Kronecker. 1882. Grundzüge einer arithmetischen Theorie der algebraischen Grössen. Journal für die reine und angewandte Mathematik 92 (1882), 1–122.
  51. Rank-Sensitive Computation of the Rank Profile of a Polynomial Matrix. In Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation (Villeneuve-d’Ascq, France) (ISSAC ’22). Association for Computing Machinery, New York, NY, USA, 351–360. https://doi.org/10.1145/3476446.3535495
  52. Fast, deterministic computation of the Hermite normal form and determinant of a polynomial matrix. Journal of Complexity 42 (2017), 44–71. https://doi.org/10.1016/j.jco.2017.03.003
  53. Yishay Mansour. 1995. Randomized Interpolation and Approximation of Sparse Polynomials. SIAM J. Comput. 24, 2 (1995), 357–368. https://doi.org/10.1137/S0097539792239291
  54. Michael Monagan and Roman Pearce. 2009. Parallel sparse polynomial multiplication using heaps. In ISSAC’09. 263–270. https://doi.org/10.1145/1576702.1576739
  55. Michael Monagan and Roman Pearce. 2011. Sparse polynomial division using a heap. J. Symb. Comput. 46, 7 (2011), 807–822. https://doi.org/10.1016/j.jsc.2010.08.014
  56. Hirokazu Murao and Tetsuro Fujise. 1996. Modular Algorithm for Sparse Multivariate Polynomial Interpolation and Its Parallel Implementation. Journal of Symbolic Computation 21, 4 (1996), 377–396. https://doi.org/10.1006/jsco.1996.0020
  57. Vasileios Nakos. 2020. Nearly Optimal Sparse Polynomial Multiplication. IEEE T. Inform. Theory 66, 11 (2020), 7231–7236. https://doi.org/10.1109/TIT.2020.2989385
  58. Henri Nussbaumer. 1980. Fast Polynomial Transform Algorithms for Digital Convolution. IEEE Transactions on Acoustics, Speech, and Signal Processing 28, 2 (1980), 205–215. https://doi.org/10.1109/TASSP.1980.1163372
  59. Armelle Perret du Cray. 2023. Algorithmes pour les polynômes creux: interpolation, arithmétique, test d’identité. Ph. D. Dissertation. L’Université de Montpellier.
  60. Deterministic sparse FFT for M-sparse vectors. Numerical Algorithms 78, 1 (01 May 2018), 133–159. https://doi.org/10.1007/s11075-017-0370-5
  61. Efficient Spectral Estimation by MUSIC and ESPRIT with Application to Sparse FFT. Frontiers in Applied Mathematics and Statistics 2 (2016). https://doi.org/10.3389/fams.2016.00001
  62. Gaspard Clair François Marie Riche de Prony. 1795. Essai expérimental et analytique sur les lois de la Dilatabilité de fluides élastique et sur celles de la Force expansive de la vapeur de l’eau et de la vapeur de l’alkool, à différentes températures. J. École Polytechnique 1, Floréal et Prairial III (1795), 24–76. https://gallica.bnf.fr/ark:/12148/bpt6k433661n/f32.item
  63. Daniel S. Roche. 2011. Chunky and equal-spaced polynomial multiplication. J. Symb. Comput. 46, 7 (2011), 791–806. https://doi.org/10.1016/j.jsc.2010.08.013
  64. J. Barkley Rosser and Lowell Schoenfeld. 1962. Approximate formulas for some functions of prime numbers. Illinois J. Math. 6, 1 (1962), 64–94. https://doi.org/10.1215/ijm/1255631807
  65. Arnold Schönhage. 1982. Asymptotically Fast Algorithms for the Numerical Muitiplication and Division of Polynomials with Complex Coefficients. In EUROCAM. Springer, 3–15. https://doi.org/10.1007/3-540-11607-9_1
  66. Arnold Schönhage and Volker Strassen. 1971. Schnelle Multiplikation großer Zahlen. Computing 7, 3 (1971), 281–292. https://doi.org/10.1007/BF02242355
  67. Andrei L. Toom. 1963. The Complexity of a Scheme of Functional Elements Realizing the Multiplication of Integers. Soviet Mathematics Doklady 3 (1963), 714–716.
  68. Wei Zhou and George Labahn. 2012. Efficient Algorithms for Order Basis Computation. 47, 7 (2012), 793–819.
  69. Richard Zippel. 1979. Probabilistic algorithms for sparse polynomials. In Symbolic and Algebraic Computation (EUROSAM’79, Vol. 72), G. Goos, J. Hartmanis, P. Brinch Hansen, D. Gries, C. Moler, G. Seegmüller, J. Stoer, N. Wirth, and Edward W. Ng (Eds.). Springer, 216–226. https://doi.org/10.1007/3-540-09519-5_73
  70. Richard Zippel. 1990. Interpolating Polynomials from Their Values. Journal of Symbolic Computation 9, 3 (March 1990), 375–403. https://doi.org/10.1016/S0747-7171(08)80018-1
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