Abstract
Integer multiplication as one of the basic arithmetic functions has been in the focus of several complexity theoretical investigations and ordered binary decision diagrams (OBDDs) are one of the most common dynamic data structures for Boolean functions. Only two years ago, the question whether the deterministic OBDD complexity of the most significant bit of integer multiplication is exponential has been answered affirmatively. Since probabilistic methods have turned out to be useful in almost all areas of computer science, one may ask whether randomization can help to represent the most significant bit of integer multiplication in smaller size. Here, it is proved that the randomized OBDD complexity is also exponential.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Ablayev, F.: Randomization and nondeterminism are incomparable for ordered read-once branching programs. In: Degano, P., Gorrieri, R., Marchetti-Spaccamela, A. (eds.) ICALP 1997. LNCS, vol. 1256, pp. 195–202. Springer, Heidelberg (1997)
Ablayev, F., Karpinski, M.: On the power of randomized ordered branching programs. In: Meyer auf der Heide, F., Monien, B. (eds.) ICALP 1996. LNCS, vol. 1099, pp. 348–356. Springer, Heidelberg (1996)
Ablayev, F., Karpinski, M.: A lower bound for integer multiplication on randomized ordered read-once branching programs. Information and Computation 186(1), 78–89 (2003)
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)
Bollig, B.: Integer multiplication and the complexity of binary decision diagrams. EATCS Bulletin 98, 78–106 (2009)
Bollig, B.: Larger lower bounds on the OBDD complexity of integer multiplication. In: Dediu, A.H., Ionescu, A.M., Martín-Vide, C. (eds.) LATA 2009. LNCS, vol. 5457, pp. 212–223. Springer, Heidelberg (2009)
Bollig, B.: A larger lower bound on the OBDD complexity of the most significant bit of multiplication. In: López-Ortiz, A. (ed.) LATIN 2010. LNCS, vol. 6034, pp. 255–266. Springer, Heidelberg (2010)
Bollig, B., Waack, S., Woelfel, P.: Parity graph-driven read-once branching programs and an exponential lower bound for integer multiplication. Theoretical Computer Science 362, 86–99 (2006)
Bollig, B., Woelfel, P.: A read-once branching program lower bound of Ω(2n/4) for integer multiplication using universal hashing. In: Proc. of 33rd STOC, pp. 419–424 (2001)
Bryant, R.E.: Graph-based algorithms for Boolean function manipulation. IEEE Trans. on Computers 35, 677–691 (1986)
Bryant, R.E.: On the complexity of VLSI implementations and graph representations of Boolean functions with application to integer multiplication. IEEE Trans. on Computers 40, 205–213 (1991)
De, A., Kurur, P., Saha, C., Sapthariski, R.: Fast integer multiplication using modular arithmetic. In: Proc.of 40th STOC, pp. 499–506 (2008)
Fürer, M.: Faster integer multiplication. In: Proc. of 39th STOC, pp. 57–66 (2007)
Hajnal, A., Maass, W., Pudlák, P., Szegedy, M., Turán, G.: Threshold circuits of bounded depth. In: Proc. 28th FOCS 1987, pp. 99–110 (1987)
Hromkovič, J.: Communication Complexity and Parallel Computing. Springer, Heidelberg (1997)
Kremer, I., Nisan, N., Ron, D.: On Randomized One-Round Communication Complexity. Computational Complexity 8(1), 21–49 (1999)
Kushilevitz, E., Nisan, N.: Communication Complexity. Cambridge University Press, Cambridge (1997)
Miltersen, P.B., Nisan, N., Safra, S., Wigderson, A.: On data structures and asymmetric communication complexity. Journal of Computer and System Science 57(1), 37–49 (1998)
Ponzio, S.: A lower bound for integer multiplication with read-once branching programs. SIAM Journal on Computing 28, 798–815 (1998)
Sauerhoff, M.: Lower bounds for randomized read-k-times branching programs. In: Meinel, C., Morvan, M. (eds.) STACS 1998. LNCS, vol. 1373, pp. 103–115. Springer, Heidelberg (1998)
Sauerhoff, M., Woelfel, P.: Time-space trade-off lower bounds for integer multiplication and graphs of arithmetic functions. In: Proc. of 35th STOC, pp. 186–195 (2003)
Schönhage, A., Strassen, V.: Schnelle Multiplikation großer Zahlen. Computing 7, 281–292 (1971)
Sen, P., Venkatesh, S.: Lower bounds for predecessor searching in the cell probe model. Journal of Computer and System Sciences 74(3), 364–385 (2008)
Smirnov, D.: Shannon’s information methods for lower bounds for probabilistic communication complexity. Master’s thesis, Moskow University (1988)
Wegener, I.: Branching Programs and Binary Decision Diagrams - Theory and Applications. SIAM Monographs on Discrete Mathematics and Applications (2000)
Woelfel, P.: New bounds on the OBDD-size of integer multiplication via universal hashing. Journal of Computer and System Science 71(4), 520–534 (2005)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Bollig, B., Gillé, M. (2011). Randomized OBDDs for the Most Significant Bit of Multiplication Need Exponential Size. In: Černá, I., et al. SOFSEM 2011: Theory and Practice of Computer Science. SOFSEM 2011. Lecture Notes in Computer Science, vol 6543. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-18381-2_11
Download citation
DOI: https://doi.org/10.1007/978-3-642-18381-2_11
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-18380-5
Online ISBN: 978-3-642-18381-2
eBook Packages: Computer ScienceComputer Science (R0)