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Larger Lower Bounds on the OBDD Complexity of Integer Multiplication

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Book cover Language and Automata Theory and Applications (LATA 2009)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5457))

Abstract

Integer multiplication as one of the basic arithmetic functions has been in the focus of several complexity theoretical investigations. Ordered binary decision diagrams (OBDDs) are one of the most common dynamic data structures for Boolean functions. Only recently it has been shown that the OBDD complexity of the most significant bit of integer multiplication is exponential, answering an open question posed by Wegener (2000). In this paper a larger lower bound is presented, using a simpler proof. Moreover, the best known lower bound on the OBDD complexity for the so-called graph of integer multiplication is improved.

Ideas presented in this paper were obtained during the Dagstuhl seminar 08381 on computational complexity of discrete problems.

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References

  1. Bryant, R.E.: On the complexity of VLSI implementations and graph representations of boolean functions with application to integer multiplication. IEEE Trans. Computers 40(2), 205–213 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  2. Woelfel, P.: Bounds on the OBDD-size of integer multiplication via universal hashing. J. Comput. Syst. Sci. 71(4), 520–534 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bollig, B.: On the OBDD complexity of the most significant bit of integer multiplication. In: Agrawal, M., Du, D.-Z., Duan, Z., Li, A. (eds.) TAMC 2008. LNCS, vol. 4978, pp. 306–317. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  4. Wegener, I.: Branching programs and binary decision diagrams: theory and applications. Society for Industrial and Applied Mathematics, Philadelphia (2000)

    Book  MATH  Google Scholar 

  5. Bryant, R.E.: Graph-based algorithms for boolean function manipulation. IEEE Trans. Computers 35(8), 677–691 (1986)

    Article  MATH  Google Scholar 

  6. Sieling, D., Wegener, I.: NC-algorithms for operations on binary decision diagrams. Parallel Processing Letters 3, 3–12 (1993)

    Article  MathSciNet  Google Scholar 

  7. Gentilini, R., Piazza, C., Policriti, A.: Computing strongly connected components in a linear number of symbolic steps. In: Proc. SODA, pp. 573–582 (2003)

    Google Scholar 

  8. Gentilini, R., Piazza, C., Policriti, A.: Symbolic graphs: Linear solutions to connectivity related problems. Algorithmica 50(1), 120–158 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  9. Sawitzki, D.: Lower bounds on the OBDD size of graphs of some popular functions. In: Vojtáš, P., Bieliková, M., Charron-Bost, B., Sýkora, O. (eds.) SOFSEM 2005. LNCS, vol. 3381, pp. 298–309. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  10. Woelfel, P.: Symbolic topological sorting with OBDDs. J. Discrete Algorithms 4(1), 51–71 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bollig, B.: A note on the size of OBDDs for the graph of integer multiplication. Inf. Process. Lett. 109(2), 41–43 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  12. Amano, K., Maruoka, A.: Better upper bounds on the QOBDD size of integer multiplication. Discrete Applied Mathematics 155(10), 1224–1232 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  13. Bollig, B., Klump, J.: New results on the most significant bit of integer multiplication. In: Proc. ISAAC, pp. 883–894 (2008)

    Google Scholar 

  14. Bollig, B., Waack, S., Woelfel, P.: Parity graph-driven read-once branching programs and an exponential lower bound for integer multiplication. Theor. Comput. Sci. 362(1-3), 86–99 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  15. Sauerhoff, M., Woelfel, P.: Time-space tradeoff lower bounds for integer multiplication and graphs of arithmetic functions. In: Proc. STOC, pp. 186–195 (2003)

    Google Scholar 

  16. Bollig, B.: On the OBDD complexity of the most significant bit of integer multiplication (full version). Theoretical Computer Science (invited to special issue)

    Google Scholar 

  17. Hromkovič, J.: Communication complexity and parallel computing. Springer, Heidelberg (1997)

    Book  MATH  Google Scholar 

  18. Kushilevitz, E., Nisan, N.: Communication complexity. Cambridge University Press, New York (1997)

    Book  MATH  Google Scholar 

  19. Wegener, I.: Optimal lower bounds on the depth of polynomial-size threshold circuits for some arithmetic functions. Inf. Process. Lett. 46(2), 85–87 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  20. Gergov, J.: Time-space tradeoffs for integer multiplication on various types of input oblivious sequential machines. Inf. Process. Lett. 51(5), 265–269 (1994)

    Article  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. LS2 Informatik, TU Dortmund, 44221, Dortmund, Germany

    Beate Bollig

Authors
  1. Beate Bollig
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Editor information

Editors and Affiliations

  1. Research Group on Mathematical Linguistics, Rovira i Virgili University, Av. Catalunya, 35, 43002, Tarragona, Spain

    Adrian Horia Dediu

  2. Research Group on Mathematical Linguistics, Universitat Rovira i Virgili, Tarragona, Spain

    Armand Mihai Ionescu

  3. European Research Council Executive Agency, 16 Place Rogier, 1210, Brussels, Belgium

    Carlos Martín-Vide

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© 2009 Springer-Verlag Berlin Heidelberg

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Bollig, B. (2009). Larger Lower Bounds on the OBDD Complexity of Integer Multiplication. In: Dediu, A.H., Ionescu, A.M., Martín-Vide, C. (eds) Language and Automata Theory and Applications. LATA 2009. Lecture Notes in Computer Science, vol 5457. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-00982-2_18

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  • DOI: https://doi.org/10.1007/978-3-642-00982-2_18

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-00981-5

  • Online ISBN: 978-3-642-00982-2

  • eBook Packages: Computer ScienceComputer Science (R0)

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