Towards Soft Exact Computation (Invited Talk)

  • Chee YapEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11661)


Exact geometric computation (EGC) is a general approach for achieving robust numerical algorithms that satisfy geometric constraints. At the heart of EGC are various Zero Problems, some of which are not-known to be decidable and others have high computational complexity. Our current goal is to introduce notions of “soft- Open image in new window correctness” in order to avoid Zero Problems. We give a bird’s eye view of our recent work with collaborators in two principle areas: computing zero sets and robot path planning. They share a common Subdivision Framework. Such algorithms (a) have adaptive complexity, (b) are practical, and (c) are effective. Here, “effective algorithm” means it is easily and correctly implementable from standardized algorithmic components. Our goals are to outline these components and to suggest new components to be developed. We discuss a systematic pathway to go from the abstract algorithmic description to an effective algorithm in the subdivision framework.



The author is deeply grateful for the feedback and bug reports from Michael Burr, Matthew England, Rémi Imbach, Juan Xu and Bo Huang.


  1. 1.
    Abbott, J.: Quadratic interval refinement for real roots. ACM Commun. Comput. Algebra 48(1/2), 3–12 (2014). Scholar
  2. 2.
    Agrawal, A., Requicha, A.: A paradigm for the robust design of algorithms for geometric modeling. In: Computer Graphics Forum, vol. 13, no. 3, pp. 33–44 (1994). 15th Annual Conference and Exhibition. EUROGRAPHICS 1994CrossRefGoogle Scholar
  3. 3.
    Becker, R.: The Bolzano method to isolate the real roots of a bitstream polynomial. Bachelor thesis, University of Saarland, Saarbruecken, Germany, May 2012Google Scholar
  4. 4.
    Becker, R., Sagraloff, M., Sharma, V., Xu, J., Yap, C.: Complexity analysis of root clustering for a complex polynomial. In: 41st International Symposium on Symbolic and Algebraic Computation, iSSAC 2016, Wilfrid Laurier University, Waterloo, Canada, 20–22 July, pp. 71–78 (2016)Google Scholar
  5. 5.
    Becker, R., Sagraloff, M., Sharma, V., Yap, C.: A near-optimal subdivision algorithm for complex root isolation based on Pellet test and Newton iteration. J. Symbolic Comput. 86, 51–96 (2018)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bennett, H., Papadopoulou, E., Yap, C.: Planar minimization diagrams via subdivision with applications to anisotropic Voronoi diagrams. In: Eurographics Symposium on Geometric Processing, SGP 2016, Berlin, Germany, 20–24 June 2016, vol. 35, no. 5 (2016)CrossRefGoogle Scholar
  7. 7.
    Bennett, H., Yap, C.: Amortized analysis of smooth quadtrees in all dimensions. Comput. Geom. Theory Appl. 63, 20–39 (2017). Also, in Proceedings SWAT 2014MathSciNetCrossRefGoogle Scholar
  8. 8.
    de Berg, M., Cheong, O., van Kreveld, M., Overmars, M.: Computational Geometry: Algorithms and Applications, 3rd edn. Springer, Berlin (2008). Scholar
  9. 9.
    Boissonnat, J.D., Teillaud, M. (eds.): Effective Computational Geometry for Curves and Surfaces. Springer, Heidelberg (2007). Scholar
  10. 10.
    Brauße, F., et al.: Semantics, logic, and verification of “exact real computation” (2019)Google Scholar
  11. 11.
    Brent, R.P.: Algorithms for Minimization Without Derivatives. Prentice Hall, Englewood Cliffs (1973)zbMATHGoogle Scholar
  12. 12.
    Burr, M., Krahmer, F.: SqFreeEVAL: an (almost) optimal real-root isolation algorithm. J. Symbolic Comput. 47(2), 153–166 (2012)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Burr, M., Krahmer, F., Yap, C.: Continuous amortization: a non-probabilistic adaptive analysis technique. Electronic Colloquium on Computational Complexity (ECCC) TR09(136), December 2009.
  14. 14.
    Burr, M.A.: Continuous amortization and extensions: with applications to bisection-based root isolation. J. Symb. Comput. 77, 78–126 (2016). MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Burr, M.A., Gao, S., Tsigaridas, E.: The complexity of an adaptive subdivision method for approximating real curves. In: 42nd International Symposium on Symbolic and Algebraic Computation (ISSAC), ISSAC 2017, pp. 61–68. ACM, New York (2017).
  16. 16.
    Burr, M.A., Gao, S., Tsigaridas, E.: The complexity of subdivision for diameter-distance tests. J. Symbolic Computation (2019, to appear)Google Scholar
  17. 17.
    Cucker, F., Ergür, A.A., Tonelli-Cueto, J.: Plantinga-vegter algorithm takes average polynomial time. arXiv:1901.09234 [cs.CG] (2019)
  18. 18.
    Du, Z., Eleftheriou, M., Moreira, J., Yap, C.: Hypergeometric functions in exact geometric computation. In: Brattka, V., Schoeder, M., Weihrauch, K. (eds.) Proceedings of 5th Workshop on Computability and Complexity in Analysis, Malaga, Spain, 12–13 July 2002, pp. 55–66 (2002). In Electronic Notes in Theoretical Computer Science 66:1 (2002). Scholar
  19. 19.
    Du, Z., Yap, C.: Uniform complexity of approximating hypergeometric functions with absolute error. In: Pae, S., Park, H. (eds.) Proceedings of 7th Asian Symposium on Computer Mathematics, ASCM 2005, pp. 246–249 (2006)Google Scholar
  20. 20.
    Emiris, I.Z., Pan, V.Y., Tsigaridas, E.P.: Algebraic algorithms. In: Gonzalez, T., Diaz-Herrera, J., Tucker, A. (eds.) Computing Handbook: Computer Science and Software Engineering, 3rd edn., pp. 10: 1–30. Chapman and Hall/CRC, Boca Raton (2014)Google Scholar
  21. 21.
    Higham, N.J.: Accuracy and Stability of Numerical Algorithms, 2nd edn. Society for Industrial and Applied Mathematics, Philadelphia (2002) CrossRefGoogle Scholar
  22. 22.
    Hsu, C.H., Chiang, Y.J., Yap, C.: Rods and rings: soft subdivision planner for \({ R}^{3}\) x \({S}^{2}\). In: Proceedings of 35th International Symposium on Computational Geometry, SoCG 2019, 18–21 June 2019. CG Week 2019, Portland Oregon. Also in arXiv:1903.09416
  23. 23.
    Imbach, R., Pan, V.Y., Yap, C.: Implementation of a near-optimal complex root clustering algorithm. In: Davenport, J.H., Kauers, M., Labahn, G., Urban, J. (eds.) ICMS 2018. LNCS, vol. 10931, pp. 235–244. Springer, Cham (2018). Scholar
  24. 24.
    Imbach, R., Pouget, M., Yap, C.: Effective subdivision algorithm for isolating zeros of real systems of equations, with complexity analysis, 21st CASC, Moscow (2019, to appear)Google Scholar
  25. 25.
    Kearfott, R.B.: Rigorous Global Search: Continuous Problems, vol. 13. Springer, Dordrecht (2013). Scholar
  26. 26.
    Kerber, M., Sagraloff, M.: Efficient real root approximation. In: Schost, É., Emiris, I.Z. (eds.) ISSAC, pp. 209–216. ACM (2011)Google Scholar
  27. 27.
    Kincaid, D., Cheney, W.: Numerical Analysis: Mathematics of Scientific Computing, 3rd edn. Brooks/Cole, Boston (2002)zbMATHGoogle Scholar
  28. 28.
    Ko, K.I.: Complexity Theory of Real Functions. Progress in Theoretical Computer Science. Birkhäuser, Boston (1991)CrossRefGoogle Scholar
  29. 29.
    Lien, J.-M., Sharma, V., Vegter, G., Yap, C.: Isotopic arrangement of simple curves: an exact numerical approach based on subdivision. In: Hong, H., Yap, C. (eds.) ICMS 2014. LNCS, vol. 8592, pp. 277–282. Springer, Heidelberg (2014). Scholar
  30. 30.
    Lin, L., Yap, C.: Adaptive isotopic approximation of nonsingular curves: the parameterizability and nonlocal isotopy approach. Discrete Comp. Geom. 45(4), 760–795 (2011)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Lin, L., Yap, C., Yu, J.: Non-local isotopic approximation of nonsingular surfaces. Comput. Aided Des. 45(2), 451–462 (2012). Symposium on Solid and Physical Modeling (SPM). U. of Burgundy, Dijon, France, 29–31 October 2012Google Scholar
  32. 32.
    Mehlhorn, K., Schirra, S.: Exact computation with leda\_real - theory and geometric applications. In: Alefeld, G., Rohn, J., Rump, S., Yamamoto, T. (eds.) Symbolic Algebraic Methods and Verification Methods, pp. 163–172. Springer, Vienna (2001). Scholar
  33. 33.
    Moore, R.E.: Interval Analysis. Prentice Hall, Englewood Cliffs (1966)Google Scholar
  34. 34.
    Ó’Dúnlaing, C., Yap, C.K.: A “retraction” method for planning the motion of a disc. J. Algorithms 6, 104–111 (1985). Also, Chapter 6 in Planning, Geometry, and Complexity, eds. Schwartz, Sharir and Hopcroft, Ablex Pub. Corp., Norwood, NJ 1987Google Scholar
  35. 35.
    Pedersen, P.: Counting real zeros. In: Proceedings of Conference on Algebraic Algorithms and Error Correcting Codes. LNCS, vol. 539, pp. 318–332. Springer (1991)Google Scholar
  36. 36.
    Pedersen, P.: Counting real zeros. Ph.D. thesis, New York University (1991). Also, Courant Institute Computer Science Technical Report 545 (Robotics Report R243)Google Scholar
  37. 37.
    Pedersen, P., Roy, M.-F., Szpirglas, A.: Counting real zeros in the multivariate case. In: Eyssette, F., Galligo, A. (eds.) Computational Algebraic Geometry. PM, vol. 109, pp. 203–224. Birkhäuser, Boston (1993). Scholar
  38. 38.
    Plantinga, S., Vegter, G.: Isotopic approximation of implicit curves and surfaces. In: Proceedings of Eurographics Symposium on Geometry Processing, pp. 245–254. ACM Press, New York (2004)Google Scholar
  39. 39.
    Preparata, F.P., Shamos, M.I.: Computational Geometry. Springer, New York (1985). Scholar
  40. 40.
    Riley, K., Hopson, M., Bence, S.: Mathematical Methods for Physics and Engineering, 3rd edn. Cambridge University Press, New York (2006)CrossRefGoogle Scholar
  41. 41.
    Sagraloff, M., Yap, C.K.: A simple but exact and efficient algorithm for complex root isolation. In: Emiris, I.Z. (ed.) 36th International Symposium on Symbolic and Algebraic Computing, San Jose, California, 8–11 June, pp. 353–360 (2011)Google Scholar
  42. 42.
    Sagraloff, M., Yap, C.K.: An efficient exact subdivision algorithm for isolating complex roots of a polynomial and its complexity analysis, July 2009, submitted. Full paper from or
  43. 43.
    Schuurman, P., Woeginger, G.: Approximation schemes: a tutorial. In: Möhring, R., Potts, C., Schulz, A., Woeginger, G., Wolsey, L. (eds.) Lectures in Scheduling (2007, to appear)Google Scholar
  44. 44.
    Sharma, V., Yap, C.: Near optimal tree size bounds on a simple real root isolation algorithm. In: 37th International Symposium on Symbolic and Algebraic Computing, ISSAC 2012, Grenoble, France, 22–25 July 2012, pp. 319–326 (2012)Google Scholar
  45. 45.
    Sharma, V., Yap, C.K.: Robust geometric computation. In: Goodman, J.E., O’Rourke, J., Tóth, C. (eds.) Handbook of Discrete and Computational Geometry, 3rd edn., chap. 45, pp. 1189–1224. Chapman & Hall/CRC, Boca Raton (2017)Google Scholar
  46. 46.
    Snyder, J.: Generative Modeling for Computer Graphics and CAD. Symbolic Shape Design Using Interval Analysis. Academic Press Professional Inc., San Diego (1992)zbMATHGoogle Scholar
  47. 47.
    Trefethen, L.N., Bau, D.: Numerical Linear Algebra. Society for Industrial and Applied Mathematics, Philadelphia (1997)CrossRefGoogle Scholar
  48. 48.
    Ueberhuber, C.W.: Numerical Computation 2: Methods, Software, and Analysis. Springer, Berlin (1997)CrossRefGoogle Scholar
  49. 49.
    Wang, C., Chiang, Y.J., Yap, C.: On soft predicates in subdivision motion planning. Comput. Geom. Theory Appl. 48(8), 589–605 (2015). (Special Issue for SoCG 2013)MathSciNetCrossRefGoogle Scholar
  50. 50.
    Weihrauch, K.: Computable Analysis. Springer, Berlin (2000). Scholar
  51. 51.
    Xu, J., Yap, C.: Effective subdivision algorithm for isolating zeros of real systems of equations, with complexity analysis. In: 44th International Symposium Symbolic and Algebraic Computing, Beihang University, Beijing, 15–18 July (2019)Google Scholar
  52. 52.
    Yap, C., Sagraloff, M., Sharma, V.: Analytic root clustering: a complete algorithm using soft zero tests. In: Bonizzoni, P., Brattka, V., Löwe, B. (eds.) CiE 2013. LNCS, vol. 7921, pp. 434–444. Springer, Heidelberg (2013). Scholar
  53. 53.
    Yap, C.K.: Soft subdivision search in motion planning, II: axiomatics. In: Wang, J., Yap, C. (eds.) FAW 2015. LNCS, vol. 9130, pp. 7–22. Springer, Cham (2015). Scholar
  54. 54.
    Yap, C., Luo, Z., Hsu, C.H.: Resolution-exact planner for thick non-crossing 2-link robots. In: Proceedings of 12th International Workshop on Algorithmic Foundations of Robotics, WAFR 2016, San Francisco, 13–16 December 2016 (2016). The appendix in the full paper (and arXiv from (and arXiv:1704.05123 [cs.CG]) contains proofs and additional experimental data
  55. 55.
    Yap, C.K.: On guaranteed accuracy computation. In: Chen, F., Wang, D. (eds.) Geometric Computation, chap. 12, pp. 322–373. World Scientific Publishing Co., Singapore (2004)Google Scholar
  56. 56.
    Yap, C.K.: In praise of numerical computation. In: Albers, S., Alt, H., Näher, S. (eds.) Efficient Algorithms. LNCS, vol. 5760, pp. 380–407. Springer, Heidelberg (2009). Scholar
  57. 57.
    Yu, J., Yap, C., Du, Z., Pion, S., Brönnimann, H.: The design of core 2: a library for exact numeric computation in geometry and algebra. In: Fukuda, K., Hoeven, J., Joswig, M., Takayama, N. (eds.) ICMS 2010. LNCS, vol. 6327, pp. 121–141. Springer, Heidelberg (2010). Scholar

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Authors and Affiliations

  1. 1.Department of Computer ScienceCourant Institute, NYUNew YorkUSA

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