1.1 Algorithms

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When we write a computer program, we are generally implementing a method that has been devised previously to solve some problem. This method is often independent of the particular computer to be used it is likely to be equally appropriate for many computers and many computer languages. It is the method, rather than the computer program itself, that we must study to learn how the problem is being attacked. The term algorithm is used in computer science to describe a problem-solving method suitable for implementation as a computer program. Algorithms are the stuff of computer science: They are central objects of study in many, if not most, areas of the field.

Most algorithms of interest involve methods of organizing the data involved in the computation. Objects created in this way are called data structures, and they also are central objects of study in computer science. Thus, algorithms and data structures go hand in hand. In this book we take the view that data structures exist as the byproducts or end products of algorithms and that we must therefore study them in order to understand the algorithms. Simple algorithms can give rise to complicated data structures and, conversely, complicated algorithms can use simple data structures. We shall study the properties of many data structures in this book; indeed, the book might well have been called Algorithms and Data Structures in Java.

When we use a computer to help us solve a problem, we typically are faced with a number of possible different approaches. For small problems, it hardly matters which approach we use, as long as we have one that solves the problem correctly. For huge problems (or applications where we need to solve huge numbers of small problems), however, we quickly become motivated to devise methods that use time or space as efficiently as possible.

The primary reason to learn about algorithm design is that this discipline gives us the potential to reap huge savings, even to the point of making it possible to do tasks that would otherwise be impossible. In an application where we are processing millions of objects, it is not unusual to be able to make a program millions of times faster by using a well-designed algorithm. We shall see such an example in Section 1.2 and on numerous other occasions throughout the book. By contrast, investing additional money or time to buy and install a new computer holds the potential for speeding up a program by perhaps a factor of only 10 or 100. Careful algorithm design is an extremely effective part of the process of solving a huge problem, whatever the applications area.

When a huge or complex computer program is to be developed, a great deal of effort must go into understanding and defining the problem to be solved, managing its complexity, and decomposing it into smaller subtasks that can be implemented easily. Often, many of the algorithms required after the decomposition are trivial to implement. In most cases, however, there are a few algorithms whose choice is critical because most of the system resources will be spent running those algorithms. Those are the types of algorithms on which we concentrate in this book. We shall study a variety of fundamental algorithms that are useful for solving huge problems in a broad variety of applications areas.

The sharing of programs in computer systems is becoming more widespread, so although we might expect to be using a large fraction of the algorithms in this book, we also might expect to have to implement only a small fraction of them. For example, the Java libraries contain implementations of a host of fundamental algorithms. However, implementing simple versions of basic algorithms helps us to understand them better and thus to more effectively use and tune advanced versions from a library. More important, the opportunity to reimplement basic algorithms arises frequently. The primary reason to do so is that we are faced, all too often, with completely new computing environments (hardware and software) with new features that old implementations may not use to best advantage. In other words, we often implement basic algorithms tailored to our problem, rather than depending on a system routine, to make our solutions more portable and longer lasting. Another common reason to reimplement basic algorithms is that, despite the advances embodied in Java, the mechanisms that we use for sharing software are not always sufficiently powerful to allow us to conveniently tailor library programs to perform effectively on specific tasks.

Computer programs are often overoptimized. It may not be worthwhile to take pains to ensure that an implementation of a particular algorithm is the most efficient possible unless the algorithm is to be used for an enormous task or is to be used many times. Otherwise, a careful, relatively simple implementation will suffice: We can have some confidence that it will work, and it is likely to run perhaps 5 or 10 times slower at worst than the best possible version, which means that it may run for an extra few seconds. By contrast, the proper choice of algorithm in the first place can make a difference of a factor of 100 or 1000 or more, which might translate to minutes, hours, or even more in running time. In this book, we concentrate on the simplest reasonable implementations of the best algorithms. We do pay careful attention to carefully coding the critical parts of the algorithms, and take pains to note where low-level optimization effort could be most beneficial.

The choice of the best algorithm for a particular task can be a complicated process, perhaps involving sophisticated mathematical analysis. The branch of computer science that comprises the study of such questions is called analysis of algorithms. Many of the algorithms that we study have been shown through analysis to have excellent performance; others are simply known to work well through experience. Our primary goal is to learn reasonable algorithms for important tasks, yet we shall also pay careful attention to comparative performance of the methods. We should not use an algorithm without having an idea of what resources it might consume, and we strive to be aware of how our algorithms might be expected to perform.

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