SHOKTEST CONNECTION NETWORKS 1401 



VI. RELATED LITERATURE AND PROBLEMS 



J. B. Kruskal, Jr.' discusses the problem of constructing shortest 

 spanning subtrees for labelled graphs. He considers only distinct and 

 positive sets of edge lengths, and is primarily interested in establishing 

 imiciueness under these conditions. (This follows immediately from XCl 

 and NC2.) He also, however, gives three different constructions, or 

 algorithms, for finding SSS's. Two of these are contained as special 

 cases in PI — P2. The third is — "Perform the following step as many 

 times as possible: Among the edges not yet chosen, choose the longest 

 edge whose removal will not disconnect them" While this is not directly 

 a special case of PI — P2, it is an obvious corollary of the special case 

 in which the shortest of the edges permitted by PI — P2 is selected at 

 each stage. Kruskal refers to an obscure Czech paper- as giving a con- 

 struction and uniqueness proof inferior to his. 



The simplicity and power of the solution afforded by PI and P2 for 

 the Basic Problem of the present paper comes as something of a surprise, 

 because there are w^ell-known problems which seem quite similar in 

 nature for which no efficient solution procedure is known. 



One of these is Steiner's Problem : Find a shortest connection network 

 for a given terminal set, with freedom to add additional terminals 

 wherever desired. A number of necessary properties of these networks 

 are known^ but do not lead to an effective solution procedure. 



Another is the Traveling Salesman Problem : Find a closed path of 

 minimum length connecting a prescribed terminal set. Nothing even 

 approaching an effective solution procedure for this problem is now 

 known (early 1957). 



REFERENCES 



1. J. B. Kruskal, Jr., On the Shortest Spanning Subtree of a Graph and the Travel- 



ing Salesman Problem, Proc. Amer. ^lath. Soo. 7, pp. 48-50, 1956. 



2. Otakar Boruvka, On a Minimal Problem, Prdce Moravske Pridovedeck^ Spdec- 



nosti, 3, 1926. 



3. R. Courant and H. Robbins, What is Mathematics, 4th edition, Oxford Univ. 



Press, N. Y., 1941, pp. 374 et .seq. 



