By Richard Bellman, Kenneth L. Cooke

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As n becomes very large, approximately what percentage of effort can one save? " See Chapter 4 of R. Bellman and S. Dreyfus, Applied Dynamic Programming, Princeton University Press, 1963. 4. Let M,(a,) = the second smallest of the n real numbers a , , a 2 , . What is the analogue of the relation in (9)? 5. Show that c + min ( a , b) = min (c + a , c + b) and, generally, c + m i n a , = min (a, + c) I 6. Under what conditions does rnin (ka,) = k rnin ai? 1 I - , a,. Equations f o r the f, 25 7. Show that la1 = max(a, - a ) .

7 , 8 , f;: = least time in which one can go from the intersection labelled i t o the destination, 8 (4) It is obvious that fs = 0. Also as before, we let t i j denote the time required to traverse the link from i to j . Here i and j are integers from 0 to 8, and t i j is defined when there is a street (link) connecting intersection i to intersection j . To obtain a set of simultaneous equations for the f;:, we proceed as above. Referring to Fig. 3, we see that starting from 0, the quickest route -goes either first to 1 or to 2, and thereafter follows the shortest route to 8.

0. Pollak, “Seiner Minimal Trees,” SZAM J. Appl. Math. 16 (1968) 1-29. For an interesting account of the application of these ideas to the theory of evolution, see A . W . F. Edwards and L. L. Cavalli-Sforza, Reconstruction of Systematics Association Publication No. 6, London, 1964. 7. Evolutionary Trees, Two vertices of a graph are adjacent if they are end points of the same edge. A graph is called 2-chromatic if its vertices can be painted with 2 colors in such a way that no two adjacent vertices have the same color.

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