1392 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1957 



(2) add link 2-9 (PI applied to Term. 2) 



(3) add link 4-8 (PI applied to Term. 4) 



(4) add link 8-4 (P2 applied to frag. 3-8) 



(5) add link 1-9 (P2 applied to frag. 1-6-7-5). 



One possible sequence for completing this construction is: 4-8 (Pi), 8-2 

 (P2), 9-2 (PI), and 1-9 (P2). Another is: 1-9 (P2), 9-2 (P2), 2-8 (P2), 

 and 8-4 (P2). 



As a second example, the construction of the network of Fig. 1 could 

 have proceeded as follows: Olympia-Salem (PI), Salem-Boise (P2), Boise- 

 Salt Lake City (P2), Helena-Boise (PI), Sacramento-Carson City (PI), 

 Carson City-Boise (P2), Salt Lake City-Denver (P2), PhoenLx-Santa Fe 

 (PI), Santa Fe-Denver (P2), and so on. 



The kind of intermixture of apphcations of PI and P2 demonstrated 

 here is very efficient when the shortest connection network is actually 

 being laid out on a map on which the given terminal set is plotted to 

 scale. With only a few minutes of practice, an example as complex as 

 that of Fig. 1 can be solved in less than 10 minutes. Another mode of 

 procecku'e, making less use of the flexibihty permitted by the construc- 

 tion principles, involves using PI only once to produce a single frag- 

 ment, which is then extended by successive applications of P2 until the 

 network is completed. This highly systematic variant, as will emerge 

 later, has advantages for computer mechanization of the solution proc- 

 ess. As applied to the example of Fig. 1, this algorithm would proceed 

 as follows if Sacramento were the indicated initial terminal: Sacramento- 

 Carson City, Carson City-Boise, Boise-Salt Lake City, Boise-Helena, 

 Boise-Salem, Salem-Olympia, Salt Lake City-Denver, Denver-Cheyenne, 

 Denver-Santa Fe, and so on. 



Since each application of either PI or P2 reduces the total number 

 of isolated terminals and fragments by one, it is evident that an A^-ter- 

 minal network is connected by N-1 applications. 



III. VALIDATION OF CONSTRUCTION PRINCIPLES 



The validity of PI and P2 depends essentially on the establishment 

 of two necessary conditions (NCl and NC2) for a shortest connection 

 network (SCN) : 



Necessary Condition 1 — Every terminal in a SCN is directly con- 

 nected to at least one nearest neighbor. 



Necessary Condition 2 — Every fragment in a SCN is connected to at 

 least one nearest neighbor by a shortest available path. 



To simplify the argument, it will at first be assumed that all distances 



