computational complexity

, Volume 5, Issue 2, pp 132–154 | Cite as

PI k mass production and an optimal circuit for the Nečiporuk slice

  • Alain P. Hiltgen
  • Mike S. Paterson


Letf: {0,1} n →{0,1} m be anm-output Boolean function inn variables.f is called ak-slice iff(x) equals the all-zero vector for allx with Hamming weight less thank andf(x) equals the all-one vector for allx with Hamming weight more thank. Wegener showed that “PI k -set circuits” (set circuits over prime implicants of lengthk) are at the heart of any optimum Boolean circuit for ak-slicef. We prove that, in PI k -set circuits, savings are possible for the mass production of anyFX, i.e., any collectionF ofm output-sets given any collectionX ofn input-sets, if their PI k -set complexity satisfiesSC m (FX)≥3n+2m. This PI k mass production, which can be used in monotone circuits for slice functions, is then exploited in different ways to obtain a monotone circuit of complexity 3n+o(n) for the Nečiporuk slice, thus disproving a conjecture by Wegener that this slice has monotone complexity θ(n3/2). Finally, the new circuit for the Nečiporuk slice is proven to be asymptotically optimal, not only with respect to monotone complexity, but also with respect to combinational complexity.

Key words

Combinational complexity mass production slice functions set circuits upper bounds 

Subject classifications

68Q15 94C10 


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Copyright information

© Birkhäuser Verlag 1995

Authors and Affiliations

  • Alain P. Hiltgen
    • 1
  • Mike S. Paterson
    • 2
  1. 1.EDP Security SystemsCrypto AGZug
  2. 2.Department of Computer ScienceUniversity of WarwickCoventryUK

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