By Radford D.E., Souza F.J.O., Yetter D.N. (eds.)
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Additional resources for Diagrammatic Morphisms and Applications
5. A BINARY SEARCH PROCEDURE A third method (see Lawler [8 J) for solving the Periodic Optimization problem is now presented. ) the weights cij is defined as the sign of ~ cij . Moreover, suppose one knows an algorithm for detecting a negative cycle (if any) in a fmite graph (for a recent review on this subject see Yen (15) ). Now, let C be a guessed value for the minimum average cost CO and associate a new cost to each arc (i,j) E' A. Then, one of the following three situations must arise. Case 1.
N in the iterative procedure above. 0 I I It follows that in the worst possible case eqs. 6) must be solved for m = 1, ... , n-l. Since each equation is solved by a minimization over about n/2 alternatives, on the average, the number of additions and comparisons required is approximately 1/2 n 3 . In conclusion, if we are concerned with the population of problems with similar cost and time parameter values, in the sense that ~ and T are invariant with n, then the number of negative cycles computations is simply proportional to logl n (see eq.
Consider the finite graph with n = 5 described in Fig. 1. and suppose r = 1 (the only refueling node is node 1) and T = 23. Since T = 5, eq. 1) is satisfied; therefore, the first method is used for the computation of f 11 (C). Moreover, in this case Step 6 of the algorithm reduces to test for the sign of f 11 (T). Of course, if f 11 (t) < 0 for some t < T, the recursive equations of Method 1 can be stopped since there is a feasible negative cycle of length t passing through node 1. In the problem at hand, a first interval of uncertainty is, for example, a = 0, b = 5.
Diagrammatic Morphisms and Applications by Radford D.E., Souza F.J.O., Yetter D.N. (eds.)