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# Principle: If No Solution Is Evident, Reduce Your Problem to Another Problem That Has a Solution

Problem reduction is one of the fundamental approaches used by computer scientists and mathematicians in solving problems. As a simple example, suppose you wanted an algorithm to locate the fourth largest element in a list. Instead of writing this special-purpose code, you could use any sorting algorithm to sort the list and then return the fourth element in the sorted list. Using this approach, you have defined an algorithm whose performance time is O(n log n); although this is not the most efficient way to solve the problem—see the `selectKth` method described in Chapter 4 instead—it is correct.

Chapter 8 presented a set of problems that all seemed related, but there didn't seem to be any easy way to tie them all together. It is possible to reduce all of these problems into linear programming (LP) and use commercially available software packages, such as Maple, to compute solutions, but the reductions are complicated; in addition, the general-purpose algorithms used to solve LP problems can be outperformed, often significantly, by the Ford-Fulkerson family of algorithms. We show in Chapter 8 how to solve a single problem type, namely computing the minimum-cost maximum flow in a flow network. With this algorithm in hand, the five other problems are immediately solved.

Table 11-5 shows the network flow algorithms described in Chapter 8.

Table 11-5. Chapter 8: Network flow algorithms

Algorithm ...

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