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By Richard Bellman, Kenneth L. Cooke

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Additional info for Algorithms, Graphs, and Computers, Vol. 62

Example text

Review,” J . Math.

Therefore, fi = ttk + (time for Since, by definition, the time for RkNis no less than fi 2 trk (4) RkN) +h f k , we get (5) for some k value. Since both ( 5 ) and (3) hold for this particular k , it follows that fi = t z k f f k ( 6 1. + Finally, this sum t , k f k must be the minimum of all of the quantities t,, f , , j f i , since otherwise we should have + + fi = h for some j , contrary to (3). Thus, fi = Irk + fk -k f, = min +f,) (7) (8) which is precisely (1). Exercise 1. Suppose that we wish to find the quickest routes from one fixed vertex to every other vertex, instead of the quickest route from each vertex to one fixed terminal vertex.

Press, New York, A . F. Veinott, Jr. and H. M. Wagner, “Optimal Capacity Scheduling,” RM-3021-PR, 1962, The R A N D Corp. J . M. Hamrnersley, “On Steiner’s Network Problem,” Mathematika 8 (1961) 131-132. 6. A variation of Steiner’s problem occurs in printed circuit technology. Suppose that n electrical junctions are to be connected with the shortest possible length of wire and moreover that the wires must run in the horizontal and vertical directions. Solve this problem for n = 3 and n = 4. See M .

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