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A possible extension is to conduct a binary, rather than sequential, search in the row and later the column of the matrix. Search in an unordered set. Although sequential scanning is by far inferior to the two other search methods described above, it is the only one which can also be used for a search in an unordered file. Its complexity does not change whether the file is ordered or not. In many applications, however, the probabilities P(a;) are not uniformly distributed. Consider, for example, a file of customer numbers.
Since it has been shown that all problems in NPC are polynomially equivalent, the existence of a polynomial algorithm for any problem in NPC would imply that P = NP. : NP :::> P u NPC. * * In general, one can distinguish between four classes of problems . , problems for which polynomially bounded algorithms exist. Problems in this category are in class P and examples of problems in this class are provided in the previous section. , problems for which at this point only exponentially bounded algorithms exist, but it is not known whether or not polynomial algorithms exist for these problems.
Observe that each column of the [n x m]-dimensional matrix I*(G) includes exactly two nonzero entries. a "+ 1" and a "-1" if the corresponding arc is directed Chapter c: Graph Theo ry 47 and two "+I"'s if the arc is undirected. As an example consider the mixed graph in Figure 16. Figure 16 The node-arc incidence matrix of the mixed graph G of Figure 16 is then 1* (G) : nt n2 n3 n4 at 1 -1 0 a2 1 -1 0 ° ° a3 I 0 -1 0 a4 0 1 -1 0 as 0 I -1° 06 0 0 1 1 Note that parallel arcs (given that they are both directed or both undirected) have identical columns in I*(G).