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Randomized complexity of linear arrangements and polyhedra?

  • Marek Karpinski
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1684)

Abstract

We survey some of the recent results on the complexity of recognizing n-dimensional linear arrangements and convex polyhedra by randomized algebraic decision trees. We give also a number of concrete applications of these results. In particular, we derive first nontrivial, in fact quadratic, randomized lower bounds on the problems like Knapsack and Bounded Integer Programming. We formulate further several open problems and possible directions for future research.

Keywords

Knapsack Problem Betti Number Linear Arrangement Randomize Complexity Random Access Machine 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [AHU74]
    A.V. Aho, J.E. Hopcroft and J.D. Ullman, The Design and Analysis of Computer Algorithms, Addison-Wesley, 1974.Google Scholar
  2. [B-O83]
    M. Ben-Or, Lower Bounds for Algebraic Computation Trees, Proc. 15th ACM STOC (1983), pp. 80–86.Google Scholar
  3. [BLY92]
    A. Björner, L. Lov’asz and A. Yao, Linear Decision Trees: Volume Estimates and Topological Bounds, Proc. 24th ACM STOC (1992), pp. 170–177.Google Scholar
  4. [B93]
    A. Borodin, Time Space Tradeoffs (Getting Closer to the Barrier?), Proc. ISAAC’93, LNCS 762 (1993), Springer, 1993, pp. 209–220.Google Scholar
  5. [BOR99]
    A. Borodin, R. Ostrovsky and Y. Rabani, Lower Bounds for High Dimensional Nearest Neighbour Search and Related Problems, Proc. 31st ACM STOC (1999), pp. 312–321.Google Scholar
  6. [BKL93]
    P. Bürgisser, M. Karpinski and T. Lickteig, On Randomized Algebraic Test Complexity, J. of Complexity 9 (1993), pp. 231–251.zbMATHCrossRefGoogle Scholar
  7. [CKKLW95]
    F. Cucker, M. Karpinski, P. Koiran, T. Lickteig, K. Werther, On Real Turing Machines that Toss Coins, Proc. 27th ACM STOC (1995), pp. 335–342.Google Scholar
  8. [DL78]
    D.P. Dobkin and R. J. Lipton, A Lower Bound of 1/2n 2 on Linear Search Programs for the Knapsack Problem, J. Compt. Syst. Sci. 16 (1978), pp. 413–417.zbMATHCrossRefMathSciNetGoogle Scholar
  9. [E87]
    H. Edelsbrunner, Algorithms in Computational Geometry, Springer, 1987.Google Scholar
  10. [FK95]
    R. Freivalds and M. Karpinski, Lower Time Bounds for Randomized Computation, Proc. 22nd ICALP’95, LNCS 944, Springer, 1995, pp. 183–195.Google Scholar
  11. [G67]
    B. Grünbaum, Convex Polytopes, John Wiley, 1967.Google Scholar
  12. [GK93]
    D. Grigoriev and M. Karpinski, Lower Bounds on Complexity of Testing Membership to a Polygon for Algebraic and Randomized Computation Trees, Technical Report TR-93-042, International Computer Science Institute, Berkeley, 1993.Google Scholar
  13. [GK94]
    D. Grigoriev and M. Karpinski, Lower Bound for Randomized Linear Decision Tree Recognizing a Union of Hyperplanes in a Generic Position, Research Report No. 85114-CS, University of Bonn, 1994.Google Scholar
  14. [GK97]
    D. Grigoriev and M. Karpinski, Randomized μ(n2) Lower Bound for Knapsack, Proc. 29th ACM STOC (1997), pp. 76–85.Google Scholar
  15. [GKMS97]
    D. Grigoriev, M. Karpinski, F. Meyer auf der Heide and R. Smolensky, A Lower Bound for Randomized Algebraic Decision Trees, Comput. Complexity 6 (1997), pp. 357–375.CrossRefGoogle Scholar
  16. [GKS97]
    D. Grigoriev, M. Karpinski, and R. Smolensky, Randomization and the Computational Power of Analytic and Algebraic Decision Trees, Comput. Complexity 6 (1997), pp. 376–388.CrossRefMathSciNetGoogle Scholar
  17. [GKV97]
    D. Grigoriev, M. Karpinski and N. Vorobjov, Lower Bound on Testing Membership to a Polyhedron by Algebraic Decision Trees, Discrete Comput. Geom. 17 (1997), pp. 191–215.zbMATHMathSciNetCrossRefGoogle Scholar
  18. [GKY98]
    D. Grigoriev, M. Karpinski and A. C. Yao, An Exponential Lower Bound on the Size of Algebraic Decision Trees for MAX, Computational Complexity 7 (1998), pp. 193–203.zbMATHCrossRefMathSciNetGoogle Scholar
  19. [G98]
    D. Grigoriev, Randomized Complexity Lower Bounds, Proc. 30th ACM STOC (1998), pp. 219–223.Google Scholar
  20. [K98a]
    M. Karpinski, On the Computational Power of Randomized Branching Programs, Proc. Randomized Algorithms 1998, Brno, 1998, pp. 1–12.Google Scholar
  21. [K98b]
    M. Karpinski, Randomized OBDDs and the Model Checking, Proc. Probabilistic Methods in Verification, PROBMIV’98, Indianapolis, 1998, pp. 35–38.Google Scholar
  22. [KM90]
    M. Karpinski and F. Meyer auf der Heide, On the Complexity of Genuinely Polynomial Computation, Proc. MFCS’90, LNCS 452, Springer, 1990, pp. 362–368.Google Scholar
  23. [KV88]
    M. Karpinski and R. Verbeek, Randomness, Provability, and the Separation of Monte Carlo Time and Space, LNCS 270 (1988), Springer, 1988, pp. 189–207.Google Scholar
  24. [L84]
    S. Lang, Algebra, Addison-Wesley, New York, 1984.zbMATHGoogle Scholar
  25. [MT85]
    U. Manber and M. Tompa, Probabilistic, Nondeterministic and Alternating Decision Trees, J. ACM 32 (1985), pp. 720–732.zbMATHCrossRefMathSciNetGoogle Scholar
  26. [M93]
    S. Meiser, Point Location in Arrangements of Hyperplanes, Information and Computation 106 (1993), pp. 286–303.zbMATHCrossRefMathSciNetGoogle Scholar
  27. [M84]
    F. Meyer auf der Heide, A Polynomial Linear Search Algorithm for the n-Dimensional Knapsack Problem, J. ACM 31 (1984), pp. 668–676.zbMATHCrossRefGoogle Scholar
  28. [M85a]
    F. Meyer auf der Heide, Nondeterministic versus Probabilistic Linear Search Algorithms, Proc. IEEE FOCS (1985a), pp. 65–73.Google Scholar
  29. [M85b]
    F. Meyer auf der Heide, Lower Bounds for Solving Linear Diophantine Equations on Random Access Machines, J. ACM 32 (1985), pp. 929–937.zbMATHCrossRefGoogle Scholar
  30. [M85c]
    F. Meyer auf der Heide, Simulating Probabilistic by Deterministic Algebraic Computation Trees, Theoretical Computer Science 41 (1985c), pp. 325–330.zbMATHCrossRefMathSciNetGoogle Scholar
  31. [M64]
    J. Milnor, On the Betti Numbers of Real Varieties, Proc. Amer. Math. Soc. 15 (1964), pp. 275–280.zbMATHCrossRefMathSciNetGoogle Scholar
  32. [R72]
    M.O. Rabin, Proving Simultaneous Positivity of Linear Forms, J. Comput. Syst. Sciences 6 (1972), pp. 639–650.Google Scholar
  33. [R91]
    A. Razborov, Lower Bounds for Deterministic and Nondeterministic Branching Programs, Proc. FCT’91, LNCS 529, Springer, 1991, pp. 47–60.Google Scholar
  34. [SP82]
    J. Simon and W.J. Paul, Decision Trees and Random Access Machines, L’Enseignement Mathematique. Logic et Algorithmic, Univ. Geneva, 1982, pp. 331–340.Google Scholar
  35. [S85]
    M. Snir, Lower Bounds for Probabilistic Linear Decision Trees, Theor. Comput. Sci. 38 (1985), pp. 69–82.zbMATHCrossRefMathSciNetGoogle Scholar
  36. [SY82]
    J.M. Steele and A.C. Yao, Lower Bounds for Algebraic Decision Trees, J. of Algorithms 3 (1982), pp. 1–8.zbMATHCrossRefMathSciNetGoogle Scholar
  37. [T51]
    A. Tarski, A Decision Method for Elementary Algebra and Geometry, University of California Press, 1951.Google Scholar
  38. [T98]
    J. S. Thathachar, On Separating the Read-k-Times Branching Program Hierarchy, Proc. 30th ACM STOC (1998), pp. 653–662.Google Scholar
  39. [T65]
    R. Thom, Sur L’Homologie des Varièetès Algèbriques Rèelles, Princeton University Press, Princeton, 1965.Google Scholar
  40. [TY94]
    H.F. Ting and A.C. Yao, Randomized Algorithm for finding Maximum with O((log n)2) Polynomial Tests, Information Processing Letters 49 (1994), pp. 39–43.zbMATHCrossRefMathSciNetGoogle Scholar
  41. [WY98]
    A. Wigderson and A.C. Yao, A Lower Bound for Finding Minimum on Probabilistic Decision Trees, to appear.Google Scholar
  42. [Y81]
    A.C. Yao, A Lower Bound to Finding Convex Hulls, J. ACM 28 (1981), pp. 780–787.zbMATHCrossRefGoogle Scholar
  43. [Y82]
    A.C. Yao, On the Time-Space Tradeoff for Sorting with Linear Queries, Theoretical Computer Science 19 (1982), pp. 203–218.zbMATHCrossRefMathSciNetGoogle Scholar
  44. [Y92]
    A.C. Yao, Algebraic Decision Trees and Euler Characteristics, Proc. 33rd IEEE FOCS (1992), pp. 268–277.Google Scholar
  45. [Y94]
    A.C. Yao, Decision Tree Complexity and Betti Numbers, Proc. 26th ACM STOC (1994), pp. 615–624.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Marek Karpinski
    • 1
  1. 1.The Institute for Advanced Study, Princeton, and Dept. of Computer ScienceUniversity of BonnBonnUSA

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