, Volume 6, Issue 2, pp 151–177 | Cite as

Nonlinearity of davenport—Schinzel sequences and of generalized path compression schemes

  • Sergiu Hart
  • Micha Sharir


Davenport—Schinzel sequences are sequences that do not contain forbidden subsequences of alternating symbols. They arise in the computation of the envelope of a set of functions. We show that the maximal length of a Davenport—Schinzel sequence composed ofn symbols is Θ (nα(n)), where α(n) is the functional inverse of Ackermann’s function, and is thus very slowly increasing to infinity. This is achieved by establishing an equivalence between such sequences and generalized path compression schemes on rooted trees, and then by analyzing these schemes.

AMS subject classification (1980)

05 A 99 05 C 35 68 B 15 


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  1. [1]
    W. Ackermann, Zum Hilbertschen Aufbau der reellen Zahlen,Math. Ann. 99 (1928), 113–133.CrossRefMathSciNetGoogle Scholar
  2. [2]
    M. Atallah, Dynamic Computational Geometry,Proc. 24th Symp. on Foundations of Computer Science, 1983, 92–99.Google Scholar
  3. [3]
    H. Davenport, A Combinatorial Problem Connected with Differential Equations, II,Acta Arithmetica 17 (1971), 363–372.zbMATHMathSciNetGoogle Scholar
  4. [4]
    H. Davenport andA. Schinzel, A Combinatorial Problem Connected with Differential Equations,Amer. J. Math. 87 (1965), 684–694.zbMATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    M. J. Fisher, Efficiency of Equivalence Algorithms,in: Complexity of Computer Computations, (R. E. Miller and J. W. Thatcher, Eds.), Plenum Press, New York, 1972, 153–168.Google Scholar
  6. [6]
    R. L. Graham, B. L. Rothschild andJ. H. Spencer,Ramsey Theory, Wiley-Interscience, New York, 1980.zbMATHGoogle Scholar
  7. [7]
    G. Kreisel, On the Interpretation of Nonfinitistic Proofs, II,J. Symbolic Logic 17 (1952), 43–58.zbMATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    J. Ketonen andR. M. Solovay, Rapidly Growing Ramsey Functions,Ann. of Math. 113 (1981), 267–314.CrossRefMathSciNetGoogle Scholar
  9. [9]
    R. Livne andM. Sharir, On Minima of Functions, Intersection Patterns of Curves, and Davenport—Schinzel Sequences,Proc. 26th IEEE Symposium on Foundations of Computer Science, Portland, Ore., October 1985, 312–320.Google Scholar
  10. [10]
    J. Paris andL. Harrington, A Mathematical Incompleteness in Peano Arithmetic,in: Handbook of Mathematical Logic, (ed: J. Barwise), North-Holland 1977, 1133–1142.Google Scholar
  11. [11]
    H. Rogers,Theory of Recursive Functions and Effective Computability, McGraw-Hill, 1967.Google Scholar
  12. [12]
    D. P. Roselle andR. G. Stanton, Some Properties of Davenport—Schinzel Sequences,Acta Arithmetica 17 (1971), 355–362.zbMATHMathSciNetGoogle Scholar
  13. [13]
    M. Sharir, Almost Linear Upper Bounds on the Length of General Davenport—Schinzel Sequences,Combinatorica 7 (1987),to appear.Google Scholar
  14. [14]
    M. Sharir, On the Two-dimensional Davenport—Schinzel Problem,Techn. Rep. 193, Comp. Sci. Dept., Courant Institute, 1985.Google Scholar
  15. [15]
    E. Szemerédi, On a Problem by Davenport and Schinzel,Acta Arithmetica 25 (1974), 213–224.zbMATHGoogle Scholar
  16. [16]
    R. E. Tarjan, Efficiency of a Good but not Linear Set-union Algorithm,J. Assoc. Computing Machinery 22 (1975), 215–225.zbMATHMathSciNetGoogle Scholar

Copyright information

© Akadémiai Kiadó 1986

Authors and Affiliations

  • Sergiu Hart
    • 1
  • Micha Sharir
    • 1
  1. 1.School of Mathematical SciencesTel Aviv UniversityTel AvivIsrael

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