Optimization I: Greedy Algorithms

In this chapter and the next, we consider algorithms for optimization problems. We have already seen an example of an optimization problem — the maximum subsequence sum problem from Chapter 1. We can characterize optimization problems as admitting a set of candidate solutions. In the maximum subsequence sum problem, the candidate solutions are the contiguous subsequences in the input array. An objective function then typically maps these candidate solutions to numeric values. The objective function for the maximum subsequence sum problem maps each contiguous subsequence to its sum. The goal is to find a candidate solution that either maximizes or minimizes, depending on the problem, the objective function. Thus, the goal of the maximum subsequence problem is to find a candidate solution that maximizes the objective function. In this chapter, we will examine optimization problems which admit greedy solutions. A greedy algorithm builds a specific candidate solution incrementally. The aspect of a greedy algorithm that makes it “greedy” is how it chooses from among the different ways of incrementing the current partial solution. In general, the different choices are ordered according to some criterion, and the best choice according to this criterion is taken. Thus, the algorithm builds the solution by always taking the step that appears to be most promising at that moment. Though there are many problems for which greedy strategies do not produce optimal solutions, when they do, they tend to be quite efficient. In the next chapter, we will examine a more general technique for solving optimization problems when greedy strategies fail.