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Let us examine the route of least time from 1 to 4, which we shall call R. Evidently, it begins by going to one of the available next points, either 2 or 3 . If the next point on R is 2, the route R thereafter must follow the quickest route from 2 to 4. The proof of this “commonsense” statement is by contradiction: otherwise, there would be a quicker route from 1 to 4 passing through 2. 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.

Consequently, the path for which the sum of the numbers t i j is least is the quickest path. On the other hand, we might let the tij be the lengths of the edges, and then the path for which the sum of the t i , is least will be the shortest path. In other applications the numbers tij may have a different meaning. ” - + 17. The Problem of the Konigsberg Bridges It should be mentioned that the formulation of the shortest route problem involves two essential aspects. First, there is the graph itself.

2. Show that m a x a , = max (al, m a x a , ) . 3. Suppose that the quantities a, first strictly decrease, then strictly increase as i = 1, 2, . . , n . Is there a more efficient way of finding rnin ai than by use of the step-by-step sequential procedure discussed Ili

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