By Lichy C.

Lichy, Carsten. 1998. A multidirectional and multifunctional gateway among GIS and hydrodynamic versions. In court cases of the 3rd foreign convention on Hydroinformatics, Vol. 1, Cophenagen, Denmark, 24-26 August 1998, pp. 395-399.

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Definition 20: A problem M I is called NP -complete, if M o: M 1 for every M E NP, indicating that M 1 is at least as hard as any problem in NP . The class of all NPcomplete problems is called NPC. If any decision problem is in NPC, then the corresponding optimization problem is NP-hard because the existence of a polynomial algorithm for the optimization problem would imply that the decision problem can also be solved in polynomial time. The relations between P, NP, and NPC can be visualized as follows (assuming that P '1= NP) 29 Chapter b: Analysis of Algorithms Figure 6 Problems in the class NPC are the hardest in NP.

Even though in the next step the search is only in an [n x (n - I)]-dimensional matrix (and after that in an [n x (n - 2)] or in an [(n - 1) x (n - I)]-dimensional matrix and so forth), the required number of comparisons is still of order O(n 2) . This procedure is repeated (2n - 1) = O(n) times, and thus its computational complexity is O(n\ In Vogel's Approximation Method, the calculation of the I-minimum and the 2minimum requires nand (n - I) comparisons respectively. Since the penalty for each row and column is the difference between the l-min and the 2-min, n + (n I) + 1 = 2n = O(n) elementary operations are needed to determine the penalty for each row and column.

But even without the problem of nonuniqueness of the graph representation, the construction of the graph from a given arc-path incidence matrix is not a trivial one. All of the above methods of capturing the structure of a graph in a matrix have in common that they are---at least in the form outlined here---rather inefficient. In other words, we will be needing O(nz), O(mn), and O(nr) units of storage space. , graphs with m « n Z• Two of the more popular of these types are described below. Arc list.

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