Skip to main content
SpringerLink
Log in

On Algebraic Expressions of Series-Parallel and Fibonacci Graphs

  • Conference paper
  • First Online:
Discrete Mathematics and Theoretical Computer Science (DMTCS 2003)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2731))

Abstract

The paper investigates relationship between algebraic expressions and graphs. Through out the paper we consider two kinds of digraphs: series-parallel graphs and Fibonacci graphs (which give a generic example of non-series-parallel graphs). Motivated by the fact that the most compact expressions of series-parallel graphs are read-once formulae, and, thus, of O(n) length, we propose an algorithm generating expressions of O(n 2) length for Fibonacci graphs.A serious effort was made to prove that this algorithm yields expressions with a minimum number of terms. Using an interpretation of a shortest path algorithm as an algebraic expression, a symbolic approach to the shortest path problem is proposed.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. W.W. Bein, J. Kamburowski, and M.F.M. Stallmann, Optimal Reduction of Two-Terminal Directed Acyclic Graphs, SIAM Journal of Computing, Vol. 21, No 6, 1992, 1112–1129.

    Article  MATH  MathSciNet  Google Scholar 

  2. P. Butkovic, Simple Image Set of (max, +) Linear Mappings, Discrete Applied Mathematics 105, 2000, 237–246.

    Article  MathSciNet  Google Scholar 

  3. Th.H. Cormen, Ch.E. Leiseron, and R.L. Rivest, Introduction to Algorithms, The MIT Press, Cambridge, Massachusetts, 1994.

    Google Scholar 

  4. Hr.N. Djidjev, Efficient Algorithms for Shortest Path Queries in Planar Digraphs, in: Graph-Theoretic Concepts in Computer Science, Proc. 22nd Int. Workshop, WG’ 96, LNCS 1197, Springer, 1996, 151–165.

    Google Scholar 

  5. R.J. Duffin, Topology of Series-Parallel Networks, Journal of Mathematical Analysis and Applications 10, 1965, 303–318.

    Article  MATH  MathSciNet  Google Scholar 

  6. M.Ch. Golumbic and A. Mintz, Factoring Logic Functions Using Graph Partitioning, in: Proc. IEEE/ACM Int. Conf. Computer Aided Design, November 1999, 109–114.

    Google Scholar 

  7. M.Ch. Golumbic, A. Mintz, and U. Rotics, Factoring and Recognition of Read-Once Functions using Cographs and Normality, in: Proc. 38th Design Automation Conf., June 2001, 195–198.

    Google Scholar 

  8. M.Ch. Golumbic and Y. Perl, Generalized Fibonacci Maximum Path Graphs, Discrete Mathematics 28, 1979, 237–245.

    Article  MATH  MathSciNet  Google Scholar 

  9. V.E. Levit and M. Korenblit, Symbolic PERT and its Generalization, in: Intelligent Scheduling of Robots and Flexible Manufacturing Systems, Proc. Int. Workshop held at the Center for Technological Education Holon, Israel, July 1995, 65–80.

    Google Scholar 

  10. V.E. Levit and M. Korenblit, A Symbolic Approach to Scheduling of Robotic Lines, in: Intelligent Scheduling of Robots and Flexible Manufacturing Systems, The Center for Technological Education Holon, Israel, 1996, 113–125.

    Google Scholar 

  11. D. Mundici, Functions Computed by Monotone Boolean Formulas with no Repeated Variables, Theoretical Computer Science 66, 1989, 113–114.

    Article  MATH  MathSciNet  Google Scholar 

  12. D. Mundici, Solution of Rota’s Problem on the Order of Series-Parallel Networks, Advances in Applied Mathematics 12, 1991, 455–463.

    Article  MATH  MathSciNet  Google Scholar 

  13. V. Naumann, Measuring the Distance to Series-Parallelity by Path Expressions, in: Graph-Theoretic Concepts in Computer Science, Proc. 20th Int. Workshop, WG’ 94, LNCS 903, Springer, 1994, 269–281.

    Google Scholar 

  14. P. Savicky and A.R. Woods, The Number of Boolean Functions Computed by Formulas of a Given Size, Random Structures and Algorithms 13, 1998, 349–382.

    Article  MATH  MathSciNet  Google Scholar 

  15. B. Schoenmakers, A New Algorithm for the Recognition of Series Parallel Graphs, Report CS-R9504, ISSN 0169-118X, CWI, Amsterdam, The Netherlands, 1995.

    Google Scholar 

  16. J. Valdes, R.E. Tarjan, and E.L. Lawler, The Recognition of Series Parallel Digraphs, SIAM Journal of Computing, Vol. 11, No 2, 1982, 298–313.

    Article  MATH  MathSciNet  Google Scholar 

  17. A.R.R. Wang, Algorithms for Multilevel Logic Optimization, Ph.D. Thesis, University of California, Berkeley, 1989.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, Holon Academic Institute of Technology, Holon, Israel

    Mark Korenblit & Vadim E. Levit

Authors
  1. Mark Korenblit
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Vadim E. Levit
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Computer Science, University of Auckland, Maths-Physics Building 303, 38 Princes St., Auckland, New Zealand

    Cristian S. Calude  & Michael J. Dinneen  & 

  2. Université de Bourgogne, LE2I, B.P. 47870, 21078, Dijon Cedex, France

    Vincent Vajnovszki

Rights and permissions

Reprints and permissions

Copyright information

© 2003 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Korenblit, M., Levit, V.E. (2003). On Algebraic Expressions of Series-Parallel and Fibonacci Graphs. In: Calude, C.S., Dinneen, M.J., Vajnovszki, V. (eds) Discrete Mathematics and Theoretical Computer Science. DMTCS 2003. Lecture Notes in Computer Science, vol 2731. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45066-1_17

Download citation

  • DOI: https://doi.org/10.1007/3-540-45066-1_17

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-40505-4

  • Online ISBN: 978-3-540-45066-5

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation