Advertisement

Lower bounds for dynamic transitive closure, planar point location, and parentheses matching

  • Thore Husfeldt
  • Theis Rauhe
  • Søren Skyum
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1097)

Abstract

We give a number of new lower bounds in the cell probe model with logarithmic cell size, which entails the same bounds on the random access computer with logarithmic word size and unit cost operations.

We study the signed prefix sum problem: given a string of length n of Os and signed 1s, compute the sum of its ith prefix during updates. We show a lower bound of Ω(log n/log log n) time per operations, even if the prefix sums are bounded by log n/log log n during all updates. We also show that if the update time is bounded by the product of the worst-case update time and the answer to the query, then the update time must be Ω(√(log n/log log n)).

These results allow us to prove lower bounds for a variety of seemingly unrelated dynamic problems. We give a lower bound for the dynamic planar point location in monotone subdivisions of Ω(logn/log log n) per operation. We give a lower bound for dynamic transitive closure on upward planar graphs with one source and one sink of Ω(logn/(log log n)2) per operation. We give a lower bound of Ω(√/(log n/loglogn)) for the dynamic membership problem of any Dyck language with two or more letters. This implies the same lower bound for the dynamic word problem for the free group with k generators. We also give lower bounds for the dynamic prefix majority and prefix equality problems.

Keywords

Query Time Membership Problem Query Operation Random Access Machine Polylogarithmic Time 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Arne Andersson. Sublogarithmic searching without multiplications. In Proc. 36th FOCS, pages 655–663. IEEE Computer Society, 1995.Google Scholar
  2. 2.
    Arne Andersson, Torben Hagerup, Stefan Nilsson, and Rajeev Raman. Sorting in linear time? In Proc 27thSTOC, pages 427–436, 1995.Google Scholar
  3. 3.
    Paul Beame and Faith Fich, 1994. Personal communication, reported by Peter Bro Miltersen.Google Scholar
  4. 4.
    Yi-Jen Chiang and Roberto Tamassia. Dynamic algorithms in Computational Geometry. Technical Report CS-91-24, Dept. of Comp. Sc., Brown University, 1991.Google Scholar
  5. 5.
    Giuseppe Di Battista, Peter Eades, Roberto Tamassia, and loannis G. Tollis. Algorithms for drawing graphs: an annotated bibliography. Available via anonymous ftp from wilma.cs.brown.edu in /pub/papers/compgeo/gdbiblio.ps.Z, 1994.Google Scholar
  6. 6.
    Paul F. Dietz. Optimal algorithms for list indexing and subset rank. In Proc. First Workshop on Algorithms and Data Structures (WADS), volume 382 of Lecture Notes in Computer Science, pages 39–46. Springer Verlag, Berlin, 1989.Google Scholar
  7. 7.
    Gudmund Skovbjerg Frandsen, Thore Husfeldt, Peter Bro Miltersen, Theis Rauhe, and Søren Skyum. Dynamic algorithms for the Dyck languages. In Proc. 4th WADS, volume 955 of Lecture Notes in Computer Science, pages 98–108. Springer, 1995.Google Scholar
  8. 8.
    Gudmund Skovbjerg Frandsen, Peter Bro Miltersen, and Sven Skyum. Dynamic word problems. In Proc 34th FOCS, pages 470–479, 1993.Google Scholar
  9. 9.
    Michael L. Fredman and Monika Rauch Henzinger. Lower bounds for fully dynamic connectivity problems in graphs. Manuscript, preliminary version in STOC 94.Google Scholar
  10. 10.
    Michael L. Fredman and Michael E. Saks. The cell probe complexity of dynamic data structures. In Proc. 21st STOC, pages 345–354, 1989.Google Scholar
  11. 11.
    Michael A. Harrison. Introduction to Formal Language Theory. Addison-Wesley, 1978.Google Scholar
  12. 12.
    P. B. Miltersen, S. Subramanian, J. S. Vitter, and R. Tamassia. Complexity models for incremental computation. Theoretical Computer Science, 130:203–236, 1994.CrossRefGoogle Scholar
  13. 13.
    Peter Bro Miltersen. Lower bounds for union-split-find related problems on random access machines. In Proc. 26th STOC, pages 625–634. ACM, 1994.Google Scholar
  14. 14.
    Peter Bro Miltersen, Noam Nisan, Shmuel Safra, and Avi Wigderson. On data structures and asymmetric communication complexity. In Proc. 27th STOC, pages 103–111. ACM, 1995.Google Scholar
  15. 15.
    Rajeev Motwani and Prabhakar Raghavan. Randomized Algorithms. Cambridge University Press, 1995.Google Scholar
  16. 16.
    Franco P. Preparata and Roberto Tamassia. Fully dynamic point location in a monotone subdivision. SIAM Journal of Computing, 18(4): 811–830, 1989.Google Scholar
  17. 17.
    Roberto Tamassia and Franco P. Preparata. Dynamic maintenance of planar digraphs, with applications. Algorithmica, 5:509–527, 1990.CrossRefGoogle Scholar
  18. 18.
    Mikkel Thorup. On RAM priority queue. In Proc 7th Ann. Symp. on Discrete Algorithms (SODA), pages 59–67, 1996.Google Scholar
  19. 19.
    Andrew Chi-Chih Yao. Should tables be sorted? Journal of the ACM, 28(3): 615–628, July 1981.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1996

Authors and Affiliations

  • Thore Husfeldt
    • 1
  • Theis Rauhe
    • 1
  • Søren Skyum
    • 1
  1. 1.BRICS, Department of Computer ScienceUniversity of Aarhus, Ny MunkegadeÅrhus CDenmark

Personalised recommendations