Skip to main content
SpringerLink
Log in
Book cover

Latin American Symposium on Theoretical Informatics

LATIN 1995: LATIN '95: Theoretical Informatics pp 112–130Cite as

Lower bounds for the matrix chain ordering problem

Extended abstract

  • Conference paper
  • First Online:

  • 174 Accesses

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 911))

Abstract

We show for any n-matrix instance of the Matrix Chain Ordering Problem (MCOP) to have certain parenthesizations of depth Θ(n) as solutions requires the matrix dimensions that are input to be exponential in n. That is, to ensure the MCOP can have any parenthesization as a solution, we must allow very expensive inputs. This exponential input lower bound implies a worst case bit complexity lower bound of Ώ(n 2). This lower bound is parameterized and, depending on the optimal product tree depth, it goes from Ώ(n 2) down to Ώ(n lg n). Also, this paper gives a very simple Ώ(n lg n) time lower bound for the MCOP for a class of algorithms on a comparison model with unit cost comparisons. This lower bound, to the authors' knowledge, captures all known algorithms for solving the matrix chain ordering problem, but does not consider bit operations. Finally, a trade-off is given between the input complexity lower bound and the atomic comparison based lower bound.

Please address correspondence to Phil Bradford.

This is a preview of subscription content, log in via an institution.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. A. Aho, J. E. Hopcroft, and J. D. Ullman, The Design and Analysis of Computer Algorithms, Addison-Wesley, 1974.

    Google Scholar 

  2. S. Baase, Computer Algorithms: Introduction to Design and Analysis, Addison-Wesley, 1988.

    Google Scholar 

  3. P. G. Bradford, “Efficient Parallel Dynamic Programming (Extended Abstract),” Proceedings of the 30-th Allerton Conference on Communication, Control and Computation, University of Illinois at Urbana-Champaign, 185–194, 1992.

    Google Scholar 

  4. G. Brassard and P. Bratley, Algorithmics: Theory and Practice, Prentice-Hall, 1988.

    Google Scholar 

  5. F. Y. Chin, “An O(n) Algorithm for Determining Near-Optimal Computation Order of Matrix Chain Products,” Communications of the ACM, Vol. 21, No. 7, 544–549, July 1978.

    Article  Google Scholar 

  6. T. H. Cormen, C. E. Leiserson, and R. L. Rivest, Introduction to Algorithms, McGraw Hill, 1990.

    Google Scholar 

  7. A. Czumaj, “An Optimal Parallel Algorithm for Computing a Near-Optimal Order of Matrix Multiplications,” Scandinavian Workshop on Algorithms Theory (SWAT), Springer Verlag, Lecture Nodes in Computer Science # 621, 62–72, 1992.

    Google Scholar 

  8. L. E. Deimel Jr. and T. A. Lampe, “An Invariance Theorem Concerning Optimal Computation of Matrix Chain Products,” North Carolina State Univ., TR79, No. 14, 1979.

    Google Scholar 

  9. A. Gibbons and W. Rytter, Introduction to Parallel Algorithms, Cambridge University Press, 1988.

    Google Scholar 

  10. E. Horowitz and S. Sahni, Fundamentals of Computer Algorithms, Computer Science Press, 1978.

    Google Scholar 

  11. T. C. Hu, Combinatorial Algorithms, Addison-Wesley, 1982.

    Google Scholar 

  12. T. C. Hu and M. T. Shing, “An O(n) Algorithm to Find a Near-Optimum Partition of a Convex Polygon,” Journal of Algorithms, Vol. 2, 122–138, 1981.

    Article  Google Scholar 

  13. T. C. Hu and M. T. Shing, “Some Theorems about Matrix Multiplication,” Proceedings of the 21-st Annual IEEE Symposium on the Foundations of Computer Science, IEEE Press, 28–35, 1980.

    Google Scholar 

  14. T. C. Hu and M. T. Shing, “Computation of Matrix Product Chains. Part I,” SIAM J. on Computing, Vo. 11, No. 3, 362–373, 1982.

    Google Scholar 

  15. T. C. Hu and M. T. Shing, “Computation of Matrix Product Chains. Part II,” SIAM J. on Computing, Vol. 13, No. 2, 228–251, 1984.

    Google Scholar 

  16. V. Kumar, A. Grama, A. Gupta, and G. Karypis, Introduction to Parallel Computing, Benjamin/Cummings, 1994.

    Google Scholar 

  17. M. Marcus, Introduction to Modern Algebra, Marcel Dekker, 1978.

    Google Scholar 

  18. C. Papadimitriou and K. Steiglitz, Combinatorial Optimization: Algorithms and Complexity, Prentice-Hall, 1982.

    Google Scholar 

  19. F. P. Preparata and M. I. Shamos, Computational Geometry—An Introduction, Springer Verlag, 1985.

    Google Scholar 

  20. P. W. Purdom Jr. and C. A. Brown, The Analysis of Algorithms, Holt, Rinehart and Winston, 1985.

    Google Scholar 

  21. P. Ramanan, “A New Lower Bound Technique and its Application: Tight Lower Bound for a Polygon Triangularization Problem,” Proceedings of the Second Annual ACM-SIAM Symposium on Discrete Algorithms, ACM Press, 281–290, 1991.

    Google Scholar 

  22. P. Ramanan, “A New Lower Bound Technique and its Application: Tight Lower Bound for a Polygon Triangularization Problem,” SIAM J. on Computing, Vol. 23, No. 4, 834–851, August 1994.

    Google Scholar 

  23. R. Sedgewick, Algorithms, Addision-Wesley, Second Edition, 1989.

    Google Scholar 

  24. A. C.-C. Yao, “Lower bounds for Algebraic Computation Trees with Integer Inputs,” SIAM J. on Computing, Vol. 20, No. 4, 655–668, 1991.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Max-Planck-Institut für Informatik, Im Stadtwald, D-66123, Saarbrücken, Germany

    Phillip G. Bradford

  2. Computer Science Department, Indiana University, 215 Lindley Hall, 47405, Bloomington, Indiana, USA

    Venkatesh Choppella & Gregory J. E. Rawlins

Authors
  1. Phillip G. Bradford
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Venkatesh Choppella
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Gregory J. E. Rawlins
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Phillip G. Bradford .

Editor information

Ricardo Baeza-Yates Eric Goles Patricio V. Poblete

Rights and permissions

Reprints and permissions

Copyright information

© 1995 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Bradford, P.G., Choppella, V., Rawlins, G.J.E. (1995). Lower bounds for the matrix chain ordering problem. In: Baeza-Yates, R., Goles, E., Poblete, P.V. (eds) LATIN '95: Theoretical Informatics. LATIN 1995. Lecture Notes in Computer Science, vol 911. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-59175-3_85

Download citation

  • DOI: https://doi.org/10.1007/3-540-59175-3_85

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-59175-7

  • Online ISBN: 978-3-540-49220-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation