Optimal Randomized Comparison Based Algorithms for Collision

  • Riko Jacob
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4708)

Abstract

We consider the well known problem of finding two identical elements, called a collision, in a list of n numbers. Here, the (very fast) comparison based algorithms are randomized and will only report existing collisions, and do this with (small) probability p, the success probability. We find a trade-off between p and the running time t, and show that this trade-off is optimal up to a constant factor. For worst-case running time t, the optimal success probability is \(p=\Theta\left(\min\{{t}/{n},1\}{t}/(n\log t)\right)\). For expected running time t, the success probability is \(p=\Theta\left(t/(n\log n)\right)\).

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ben-Or, M.: Lower bounds for algebraic computation trees. In: Proc. 15th Annual ACM Symposium on Theory of Computing, pp. 80–86. ACM Press, New York (1983)Google Scholar
  2. 2.
    Boppana, R.B.: The decision-tree complexity of element distinctness. Inf. Process. Lett. 52, 329–331 (1994)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Buhrman, H., Dürr, C., Heiligman, M., Høyer, P., Magniez, F., Santha, M., de Wolf, R.: Quantum algorithms for element distinctness. SIAM J. Comput. 34, 1324–1330 (2005) (electronic)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Aaronson, S., Shi, Y.: Quantum lower bounds for the collision and the element distinctness problems. J. ACM 51, 595–605 (2004) (electronic)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Maurer, U.M.: Abstract models of computation in cryptography. In: Smart, N.P. (ed.) Cryptography and Coding. LNCS, vol. 3796, pp. 1–12. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  6. 6.
    Yao, A.C.C.: Probabilistic computations: towards a unified measure of complexity. In: Proc. 18th FOCS, IEEE 1977, pp. 222–227. IEEE Computer Society Press, Los Alamitos (1977)Google Scholar
  7. 7.
    Motwani, R., Raghavan, P.: Randomized Algorithms. Cambridge University Press, Cambridge (1995)MATHGoogle Scholar
  8. 8.
    Snir, M.: Lower bounds on probabilistic linear decision trees. Theoret. Comput. Sci. 38, 69–82 (1985)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Grigoriev, D., Karpinski, M., Meyer auf der Heide, F.: A lower bound for randomized algebraic decision trees. Comput. Complexity 6, 357–375 (1996/1997)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Manber, U., Tompa, M.: Probabilistic, nondeterministic, and alternating decision trees (preliminary version). In: STOC 1982: Proceedings of the fourteenth annual ACM symposium on Theory of computing, New York, NY, USA, pp. 234–244. ACM Press, New York (1982)CrossRefGoogle Scholar
  11. 11.
    Manber, U., Tompa, M.: The complexity of problems on probabilistic, nondeterministic, and alternating decision trees. J. Assoc. Comput. Mach. 32, 720–732 (1985)MATHMathSciNetGoogle Scholar
  12. 12.
    Manber, U., Tompa, M.: The effect of number of Hamiltonian paths on the complexity of a vertex-coloring problem. SIAM J. Comput. 13, 109–115 (1984)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Chein, M., Habib, M.: The jump number of DAGs and posets: An introduction. Annals of Discrete Mathematics 9, 189–194 (1980)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Blum, M., Floyd, R., Pratt, V., Rivest, R., Tarjan, R.: Time bounds for selection. Journal of Computer and System Sciences 7, 448–461 (1973)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Riko Jacob
    • 1
  1. 1.Computer Science Department, Technische Universität München 

Personalised recommendations