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By Richard Bellman, Kenneth L. Cooke

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

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F 2 r f,) fi = min (tzl f,, tZ4) (2) + + h = min (t31 + fl, + t34) Next, we see that the second and third equations yield f 2 and f, in terms of fi. Substitution of these relations in the first equation results in the rela tion f,= min Ifl2 + min (f2i + fl, f2d, ti, + min (tSI+ fi, tM)l (3 involving only one unknown, fi. , Exercises 5 and 8 at the end of Applying (5) to ( 3 ) , we obtain 6 10). 34),)l Now applying (4), we get fl = min ( f l 2 + +fii 221 f12 + 213 f24, + + t3l fly f13 + f34) (7) This is simpler than (3) in that there is only one min symbol, and we have only one equation in one unknown, but unhappily the unknown f, still appears on both sides of the equation.

Consequently, even in this simple case it is far more efficient to determine by experiment the times associated with the links rather than with the set of complete routes. This will be increasingly the case as more streets are added to our map. In Table 7 , we list theoretical times for all possible links in a route, computed in the same way as in Table 3 . The number of these has increased from 15 to 3 0 , but the rate of increase is not as rapid as the increase in total number of routes. All that now remains to be done is to find the total time required to traverse each of the routes, and then to pick out the smallest of these.

We leave it as an exercise for the reader to complete this task and to compare 14 Commuting and Computing TABLE 6 ENUMERATION OF PATHSAND TIMES FOR FIG. 11 Times 0, I , 3, 2 , 4 , 5 , 7, 11 0,1,3,2,4,5,10,11 0,1,3,2,4,6,7,5,10,11 0,1,3,2,4,6,7,11 0, 1, 3, 5 , 7, 11 0, 1 , 3 , 5 , 4 , 6 , 7 , 11 0, 1, 3, 5, 10, 11 0, 1, 3, 9, 10, 11 0, 1, 3, 9, 10, 5, 7, 11 0, 1, 3, 9, 10, 5, 4, 6, 7, 11 0,1,8,9,3,2,4,5,7,11 0, 1, 8, 9, 3, 2, 4, 5 , 10, 11 0,1,8,9,3,2,4,6,7,5,10,11 0, 1, 8 , 9 , 3 , 2 , 4 , 6 , 7 , 11 0,1,8,9,3,5,4,6,7,11 0,1,8,9,3,5,7,11 0, 1, 8, 9, 3, 5 , 10, 11 0, 1, 8, 9, 10, 5 , 3, 2, 4, 6 , 7, 11 0, 1, 8, 9, 10, 5 , 4, 6, 7, 11 0, 1, 8, 9, 10, 5 , 7, 11 0, 1, 8, 9, 10, 11 0, 2, 3, 1, 8, 9, 10, 11 0, 2, 3, 1, 8, 9, 10, 5 , 7, 11 0, 2, 3, 1, 8, 9, 10, 5, 4, 6, 7, 11 0,2,3,5,4,6,7,11 0, 2, 3, 5 , 7, 11 0, 2, 3, 5 , 10, 11 0,2,3,9,10,5,4,6,7,11 0,2,3,9,10,5,7,11 0, 2, 3, 9, 10, 11 0, 2, 4, 5 , 3, 1, 8, 9, 10, 11 0, 2, 4, 5 , 3, 9, 10, 11 0, 2, 4, 5, 7 , 11 0, 2, 4, 5 , 10, 11 0, 2, 4, 6, 7, 5 , 3, 1, 8, 9, 10, 11 0, 2, 4, 6 , 7 , 5 , 3, 9, 10, 11 0, 2, 4, 6, 7 , 5 , 10, 11 0, 2 , 4, 6, 7, 11 274 280 496 346 226 340 232 225 283 397 355 36 1 577 427 421 307 313 462 392 27 8 220 315 373 487 343 229 235 400 286 228 373 286 223 229 589 502 445 295 An Improved Map 15 TABLE 7 TABLEOF TIMESFOR FIG.

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