Intro to Algorithms: PART I: Mathematical Foundations

The analysis of algorithms often requires us to draw upon a body of mathematical tools. Some of these tools are as simple as high-school algebra, but others, such as solving recurrences, may be new to you. This part of the book is a compendium of the methods and tools we shall use to analyze algorithms. It is organized primarily for reference, with a tutorial flavor to some of the topics.

We suggest that you not try to digest all of this mathematics at once. Skim the chapters in this part to see what material they contain. You can then proceed directly to the chapters that focus on algorithms. As you read those chapters, though, refer back to this part whenever you need a better understanding of the tools used in the mathematical analyses. At some point, however, you will want to study each of these chapters in its entirety, so that you have a firm foundation in the mathematical techniques.

Chapter 2 precisely defines several asymptotic notations, an example of which is the -notation that you met in Chapter 1. The rest of Chapter 2 is primarily a presentation of mathematical notation. Its purpose is more to ensure that your use of notation matches that in this book than to teach you new mathematical concepts.

Chapter 3 offers methods for evaluating and bounding summations, which occur frequently in the analysis of algorithms. Many of the formulas in this chapter can be found in any calculus text, but you will find it convenient to have these methods compiled in one place.

Methods for solving recurrences, which we used to analyze merge sort in Chapter 1 and which we shall see many times again, are given in Chapter 4. One powerful technique is the "master method," which can be used to solve recurrences that arise from divide-and-conquer algorithms. Much of Chapter 4 is devoted to proving the correctness of the master method, though this proof may be skipped without harm.

Chapter 5 contains basic definitions and notations for sets, relations, functions, graphs, and trees. This chapter also gives some basic properties of these mathematical objects. This material is essential for an understanding of this text but may safely be skipped if you have already had a discrete mathematics course.

Chapter 6 begins with elementary principles of counting: permutations, combinations, and the like. The remainder of the chapter contains definitions and properties of basic probability. Most of the algorithms in this book require no probability for their analysis, and thus you can easily omit the latter sections of the chapter on a first reading, even without skimming them. Later, when you encounter a probabilistic analysis that you want to understand better, you will find Chapter 6 well organized for reference purposes.

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