# 4.1 Solving problems

Traditionally, a problem is an obstacle to overcome or some question to answer. Once the question is answered or the obstacle circumvented, the problem is solved and we can declare victory and move on to the next one.

When we talk about writing programs to solve problems, though, we have a larger goal. We don’t just want to solve one instance of a problem, we want an algorithm that can solve all instances of a problem. A problem is defined by its inputs and the desired property of the output. Recall from Chapter 1, that a procedure is a precise description of a process and a procedure is guaranteed to always finish is called an algorithm.algorithm The name algorithm is a Latinization of the name of the Persian mathematician and scientist, Muhammad ibn Mūsā al-Khwārizmi, who published a book in 825 on calculation with Hindu numerals. Although the name algorithm was adopted after al-Khwārizmi’s book, algorithms go back much further than that. The ancient Babylonians had algorithms for finding square roots more than 3500 years ago (see Exploration 4.1).

For example, we don’t just want to find the best route between New York and Washington, we want an algorithm that takes as inputs the map, start location, and end location, and outputs the best route. There are infinitely many possible inputs that each specify different instances of the problem; a general solution to the problem is a procedure that finds the best route for all possible inputs.1

To define a procedure that can solve a problem, we need to define a procedure that takes inputs describing the problem instance and produces a different information process depending on the actual values of its inputs. A procedure takes zero or more inputs, and produces one output or no outputs 2, as shown in Figure 4.1.

Figure 4.1: A procedure maps inputs to an output.

Our goal in solving a problem is to devise a procedure that takes inputs that define a problem instance, and produces as output the solution to that problem instance. The procedure should be an algorithm — this means every application of the procedure must eventually finish evaluating and produce an output value.

There is no magic wand for solving problems. But, most problem solving involves breaking problems you do not yet know how to solve into simpler and simpler problems until you find problems simple enough that you already know how to solve them. The creative challenge is to find the simpler subproblems that can be combined to solve the original problem. This approach of solving problems by breaking them into simpler parts is known as divide-and-conquer.

The following sections describe a two key forms of divide-and-conquer problem solving: composition and recursive problem solving. We will use these same problem-solving techniques in different forms throughout this book.

1. Actually finding a general algorithm that does without needing to essentially try all possible routes is a challenging and interesting problem, for which no efficient solution is known. Finding one (or proving no fast algorithm exists) would resolve the most important open problem in computer science!

2. Although procedures can produce more than one output, we limit our discussion here to procedures that produce no more than one output. In the next chapter, we introduce ways to construct complex data, so any number of output values can be packaged into a single output.