Energy and depth of threshold circuits

Authors:
Kei Uchizawa;Takao Nishizeki;Eiji Takimoto
Affiliations:
Graduate School of Information Sciences, Tohoku University, 6-6-05, Aramaki aza Aoba, Aoba-ku, Sendai 980-8579, Japan;Graduate School of Information Sciences, Tohoku University, 6-6-05, Aramaki aza Aoba, Aoba-ku, Sendai 980-8579, Japan;Department of Informatics, Graduate School of Information Science and Electrical Engineering, Kyushu University, 744 Motooka, Nishi-ku, Fukuoka 819-0395, Japan
Venue:
Theoretical Computer Science
Year:
2010

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Abstract

In this 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 prove that the function f can also be computed by a threshold circuit C^' of the 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.