Optimization II: Dynamic Programming

In the last chapter, we saw that greedy algorithms are efficient solutions to certain optimization problems. However, there are optimization problems for which no greedy algorithm exists. In this chapter, we will examine a more general technique, known as dynamic programming, for solving optimization problems. Dynamic programming is a technique of implementing a top-down solution using bottom-up computation. We have already seen several examples of how top-down solutions can be implemented bottom-up. Dynamic programming extends this idea by saving the results of many subproblems in order to solve the desired problem. As a result, dynamic programming algorithms tend to be more costly, in terms of both time and space, than greedy algorithms. On the other hand, they are often much more efficient than straightforward recursive implementations of the top-down solution. Thus, when greedy algorithms are not possible, dynamic programming algorithms are often the most appropriate.