By Richard Bellman, Kenneth L. Cooke
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Extra info for Algorithms, Graphs, and Computers, Vol. 62
26 Commuting and Computing If, on the other hand, the next point in the path is 3 , the route R thereafter must follow the quickest route from 3 to 4. Thus, we see that f, is either tI2 f 2 or t I 3 f,. Since f l is the least time, we have the equation + + + f l = min (tt2 f2, tl3 + f3) (2) We see now the convenience of the minimum function in writing our equations. In the same way, we can deduce that This reasoning readily applies t o any map, not just to the one in Fig. 12. For example, consider Fig.
For the sake of abbreviation the entire matrix is frequently denoted by a symbol ( t , , ) ,in which parentheses enclose the symbol for the ijth element. 18 Commuting and Computing One place where the reader may have encountered the concept of a matrix previously is in the study of systems of linear algebraic equations. For example, in connection with equations 3x 2x - + 7z = 0 y y + 42 = 4 z =1 x - y + the square matrix 3 2 1 7 4 1 -1 -1 -1 is usually called the matrix of coeficients and the rectangular matrix 3 2 1 7 4 1 -1 -1 - 1 0 8 1 is often called the augmented matrix.
Minimum Time Functions In the previous section, we indicated that it might be expedient to enlarge the original problem to the following more general problem: “Find the least time from any intersection point on the map to the fixed destination. This idea of imbedding a particular problem within a family of similar problems is one of the most important and powerful in mathematics. It is often the case that the family of problems can be solved simultaneously and elegantly despite the fact that the individual problem remains obdurate.
Algorithms, Graphs, and Computers, Vol. 62 by Richard Bellman, Kenneth L. Cooke