iV-TERMINAL SWITCHING CIRCUITS 669 



M 



^ = Z log pi 

 t=l 



bits (in this paper "log" stands for "logarithm to base 2"). The results to 



follow include an estimate of the minimum number of contacts needed for 



almost all iV-terminal switching functions and a network synthesis method 



which uses a number of contacts of the same order of magnitude as the 



minimum number. The number of contacts needed for almost all iV- terminal 



N log N l" 



switching functions is about „ . , — whenZf andiV are large. The words 



±? + log iV 



"almost all" are used here in the sense that the fraction of switching func- 

 tions which can be synthesized using fewer contacts than the given number 

 tends to zero as R and/or N increases. The number of contacts used by the 



synthesis method is about — — — where P is the number of positions on the 



largest switch. The factor P can be reduced in most cases. The analogous ex- 



pressions found in Shannon's paper are — and where n is the number of 



n n 



switching variables. 



One of the most surprising facts about switching functions is that, if H 

 is moderately large, almost none of them can be synthesized without using 

 fantastically many contacts. This is already true of Shannon's two-terminal 

 networks, and for iV-terminal networks the situation is even worse. The 

 reader may first turn to page 685 where a numerical example illustrating 

 this phenomenon is given. 



These paradoxical results are explained by noting that switching functions 

 in general are much different from the usual kinds of switching functions 

 which have practical applications. One concludes that the invention of better 

 methods for synthesizing any imaginable function whatsoever will be of 

 little help in practice. Almost all these functions are impossible to build 

 (because of contact cost) and would be of no use if built. Instead one must 

 try to isolate classes of useful switching functions which are easy to build. 



Part I: Two-Terminal Networks 



Selector Switches 



A typical selector switch is shown in Fig. 1. It consists of a number of 

 rotors turned by a shaft which can be set in any one of p positions. In each 

 position of the shaft, certain of the rotors touch contacts, thereby closing 

 those branches in the network containing the touched contacts. However, 

 the only kinds of switches to be considered here are those with the property 



