Formulation of the addition-shift-sequence problem and its complexity

  • Akihiro Matsuura
  • Akira Nagoya
Session 2A
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1350)


Given a sequence of integers, what is the smallest number of additions and shifts needed to compute all integers starting with 1? This is a generalization of the addition-sequence problem which naturally appears in the multiplication of constants with a single variable and in its hardware implementation, and it will be called the addition-shift sequence problem. As a fundamental result on computational complexity, we show that the addition-shift-sequence problem is NP-complete. Then, we show lower and upper bounds of the number of operations for some particular sequence, where some techniques specific to our model are demonstrated.


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  1. 1.
    A. Scholz, “Aufgabe 253”, Jahresbericht der deutchen Mathematiker-Vereinigung, 47, pp. 41–42 (1937).Google Scholar
  2. 2.
    A. Brauer, “On addition chains”, Bull. Amer. Math. Soc., 45, pp. 736–739 (1939).Google Scholar
  3. 3.
    A. Schönhage, “A lower bound for the length of addition chains”, Theoret. Comp. Sci., 1, pp. 1–12 (1975).Google Scholar
  4. 4.
    A. Yao, “On the evaluation of powers”, SIAM J. Computing, vol. 5, no. 1, pp. 100–103 (1976).Google Scholar
  5. 5.
    P. Downey, B. Leong and R. Sethi, “Computing sequences with addition chains”, SIAM J. Computing, 3, pp. 638–696 (1981).Google Scholar
  6. 6.
    J. Bos and M. Coster, “Addition Chain Heuristics”, Proc. CRYPTO'89, pp. 400-407 (1989).Google Scholar
  7. 7.
    Y. Tsuruoka and K. Koyama, “Fast Exponentiation Algorithms Based on Batch-Processing and Precomputation”, IEICE Trans. Fundamentals, vol.E80-A, no. 1, pp. 34–39 (Jan. 1997).Google Scholar
  8. 8.
    M. Potkonjak, M. B. Srivastava and A. P. Chandrakasan, “Efficient substitution of multiple constant multiplications by shifts and additions using iterative pairwise matching”, ACM/IEEE Proc. 31st DAC, pp. 189–194 (1994).Google Scholar
  9. 9.
    M. Mehendale, S. D. Shelekar and G. Venkatesh, “Synthesis of Multiplier-less FIR Filters with Minimum Number of Additions", Proc. ICCAD'95, pp. 668–671 (1995).Google Scholar
  10. 10.
    A. Matsuura, M. Yukishita and A. Nagoya, ldAn Efficient Hierarchical Clustering Method for the Multiple Constant Multiplication Problem”, Proc. ASP-DAC'97, pp. 83–88 (1997).Google Scholar
  11. 11.
    R. M. Karp, “Reducibility among combinatorial problems”, In Complexity of Computer Computations, R. E. Miller and J. W. Thatcher, eds., Plenum Press, New York, pp. 85–103 (1972).Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Akihiro Matsuura
    • 1
  • Akira Nagoya
    • 1
  1. 1.NTT Communication Science LaboratoriesSoraku-gun, KyotoJapan

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