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Abstract

We study novel arithmetic algorithms on a canonical number representation based on the Catalan family of combinatorial objects.

For numbers corresponding to Catalan objects of low structural complexity our algorithms provide super-exponential gains while their average case complexity is within constant factors of their traditional counterparts.

Keywords

hereditary numbering systems arithmetic algorithms for Combinatorial objects structural complexity of natural numbers run-length compressed numbers Catalan families 

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References

  1. 1.
    Goodstein, R.: On the restricted ordinal theorem. Journal of Symbolic Logic 9(2), 33–41 (1944)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Knuth, D.E.: Mathematics and Computer Science: Coping with Finiteness. Science 194(4271), 1235–1242 (1976)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Knuth, D.E.: TCALC program (December 1994), http://www-cs-faculty.stanford.edu/~uno/programs/tcalc.w.gz
  4. 4.
    Salomaa, A.: Formal Languages. Academic Press, New York (1973)zbMATHGoogle Scholar
  5. 5.
    Sloane, N.J.A.: A000108, The On-Line Encyclopedia of Integer Sequences (2013), Published electronically at http://oeis.org/A000108
  6. 6.
    Stanley, R.P.: Enumerative Combinatorics. Wadsworth Publ. Co., Belmont (1986)CrossRefzbMATHGoogle Scholar
  7. 7.
    Tarau, P.: Declarative modeling of finite mathematics. In: PPDP 2010: Proceedings of the 12th International ACM SIGPLAN Symposium on Principles and Practice of Declarative Programming, pp. 131–142. ACM, New York (2010)CrossRefGoogle Scholar
  8. 8.
    Tarau, P.: Arithmetic Algorithms for Hereditarily Binary Natural Numbers (June 2013), http://arxiv.org/abs/1306.1128
  9. 9.
    Tarau, P., Haraburda, D.: On Computing with Types. In: Proceedings of SAC 2012, ACM Symposium on Applied Computing, PL track, pp. 1889–1896. Riva del Garda (Trento), Italy (March 2012)Google Scholar
  10. 10.
    Vuillemin, J.: Efficient Data Structure and Algorithms for Sparse Integers, Sets and Predicates. In: 19th IEEE Symposium on Computer Arithmetic, ARITH 2009, pp. 7–14 (June 2009)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Paul Tarau
    • 1
  1. 1.Department of Computer Science and EngineeringUniversity of North TexasUSA

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