1398 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1957 



The longest spanning subtree of a connected labelled graph 

 also maximizes all increasing symmetric functions, and mini- 

 mizes all decreasing symmetric functions, of the edge "lengths." 



For example, with positive ''lengths" the same spanning subtree that 

 minimizes the sum of the edge "lengths" also minimizes the product and 

 the square root of the sum of the squares. At the same time, it maximizes 

 the sum of the reciprocals and the product of the arc cotangents. 



It seems likely that these extensions of the original class of problems 

 soluble by PI and P2 contain many examples of practical interest in 

 quite different contexts from the original connection networks. A not 

 entirely facetious example is the following: A message is to be passed 

 to all members of a certain underground organization. Each member 

 knows some of the other members and has procedures for arranging a 

 rendezvous with anyone he knows. Associated with each such possible 

 rendezvous — say between member "^" and member "/" — is a certain 

 probability, pij , that the message will fall into hostile hands. How is 

 the message to be distributed so as to minimize the over-all chances of 

 its being compromised? If members are represented as vertices, possible 

 rendezvous as edges, and compromise probabilities as "length" labels 

 in a labelled graph, the problem is to find a spanning subtree for which 

 1 — n(l — Pij) is minimized. Since this is an increasing symmetric 

 function of the p,:/'s, this is the same as the spanning subtree minimiz- 

 ing 2 Pij , and this is easily found by PI and P2. 



Another application, closer to the original one, is that of minimizing 

 the lengths of wire used in cabling panels of electrical equipment. Re- 

 strictions on the permitted wiring patterns lead to shortest connection 

 network problems in which the effective distances between terminals 

 are not the direct terminal-to-terminal distances. Thus the more general 

 viewpoint of the present section is applicable. 



V. COMPUTATIONAL TECHNIQUE 



Return now to the exemplary shortest connection network problem 

 of Figs. 4(a) and 5(a) which served as the center for discussion of the 

 arithmetizing of the metric factors in the Basic Problem. It is evident 

 that the actual drawing and labelling of all the edges of a complete 

 graph will get very cumbersome as the number of vertices increases — 

 an A^-vertex graph has (l/2)(N^ — N) edges. For large N, it is convenient 

 to organize the numerical metric information in the form of a distance 

 table, such as Fig. 9, which is equivalent in content to Fig. 4(a) or Fig. 



