Energy complexity and depth of threshold circuits

  • Authors:

  • Kei Uchizawa;Takao Nishizeki;Eiji Takimoto

  • Affiliations:

  • Graduate School of Information Sciences, Tohoku University, Sendai, Japan;Graduate School of Information Sciences, Tohoku University, Sendai, Japan;Department of Informatics, Graduate School of Information Science and Electrical Engineering, Kyushu University, Fukuoka, Japan

  • Venue:
  • FCT'09 Proceedings of the 17th international conference on Fundamentals of computation theory
  • Year:
  • 2009

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Abstract

In the paper we show that there is a close relationship between the energy complexity and the depth of threshold circuits computing any Boolean function although they have completely different physical meanings. Suppose that a Boolean function f can be computed by a threshold circuit C of energy complexity e and hence at most e threshold gates in C output "1" for any input to C. We then prove that the function f can be computed also by a threshold circuit C′ of depth 2e+1 and hence the parallel computation time of C′ is 2e+1. If the size of C is s, that is, there are s threshold gates in C, then the size s′ of C′ is s′ = 2es+1. Thus, if the size s of C is polynomial in the number n of input variables, then the size s′ of C′ is polynomial in n, too.