1394 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1957 



from successive applications (A^ — 1 for N terminals) of PI and P2 is 

 independent of which nearest neighbor is chosen for connection at a 

 stage when more than one nearest neigh))or to an isolated terminal or 

 t is available. This is a consequence of the following considera- 

 tions: For a prescribed terminal set there are only a finite number of 

 connection networks (certainly fewer than Cn'^i~^''''^ — the number of 

 distinct ways of choosing A^ — 1 links from the total of A^(A^ — l)/2 possible 

 links) . The length of each one of this finite set of connection networks is 

 a continuous function of the individual interterminal distances, dij (as a 

 matter of f ct, it is a linear function with coefficients and 1). With 

 the dij specified, the length, L, of a shortest connection network is 

 simply the smallest length in this finite set of connection network 

 lengths. Therefore L is unicjuely determined. (It may, of course, be 

 associated with more than one of the connection networks.) Now, if at 

 each stage of construction employing PI and P2 at which a choice is to 

 be made among two or more nearest neighbors rii , ■ • • , n^ of an isolated 

 terminal (or fragment) t, a small positive quantity, e, is subtracted from 

 any specific one of the distances d,„j , • • •, dm, — say from dm^ — the 

 construction will be uniciuely determined. The total length, L', of the 

 resulting SCN for the modified problem will now depend on e, as well 

 as on the dtj of the original terminal set. The dependence on e will be 

 continuous, however, because the minimum of a finite set of continuous 

 functions of e (the set of lengths of all connection networks of the modi- 

 fied problem) is itself a continuous function of e. Hence, as e is made 

 vanishingly small, L' approaches L, regardless of which "nearest neigh- 

 bor" links were chosen for shortening to decide the construction. 



IV. ABSTRACTION AND GENERALIZATION 



In the examples of Figs. 1 and 2, the terminal set to be connected was 

 represented by points on a distance-true map. The decisions involved 

 in applying PI and P2 could then be based on visual judgements of 

 relative distances, perhaps augmented by application of a pair of di- 

 viders in a few close instances. These direct geometric comparisons can 

 of course, be replaced by numerical ones if the inter-terminal distances 

 are entered on the terminal plot, as in Fig. 4(a). The application of PI 

 and P2 goes through as before, with the relevant "nearest neighbors" 

 determined bj^ a comparison of numerical labels, rather than by a 

 geometric scanning process. For example, Pi applied to Terminal 5 of 

 Fig. 4(a) yields the Link 5-6 of the SCN of Fig. 4(b), because 4.6 is 

 less than 5.6, 8.0, 8.5, and 5.1, and so on. 



