Shortest Connection Networks 

 And Some Generalizations 



By R. C. PRIM 



(Manuscript received May 8, 1957) 



The basic problem considered is that of interconnecting a given set of 

 terminals with a shortest possible network of direct links. Simple and prac- 

 tical procedures are given for solving this problem both graphically and 

 computationally. It develops that these procedures also provide solutions 

 for a much broader class of problems, containing other examples of practical 

 interest. 



I. INTRODUCTION 



A problem of inherent interest in the planning of large-scale communi- 

 cation, distribution and transportation networks also arises in connec- 

 tion with the current rate structure for Bell System leased-line services. 

 It is the following: 



Basic Problem — Given a set of (point) terminals, connect them by a 

 network of direct terminal-to-terminal links having the smallest possible 

 total length (sum of the link lengths). (A set of terminals is "connected," 

 of course, if and only if there is an unbroken chain of links between every 

 two terminals in the set.) An example of such a Shortest Connection Net- 

 work is shown in Fig. 1. The prescribed terminal set here consists of 

 Washington and the forty-eight state capitals. The distances on the par- 

 ticular map used are accepted as true. 



Two simple construction principles will be established below which 

 provide simple, straight-forward and flexible procedures for solving the 

 basic problem. Among the several alternative algorithms whose validity 

 follows from the basic construction principles, one is particularly^ well 

 adapted for automatic computation. The nature of the construction 

 principles and of the demonstration of their validity leads quite naturally 

 to the consideration, and solution, of a broad class of minimization prob- 

 lems comprising a non-trivial abstraction and generalization of the basic 

 problem. This extended class of problems contains examples of practical 



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