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Introduction
What’s in This Book
The book is structured as follows:
Chapter 1: Introduction. You’ve already gotten through most of this. It gives an overview of the book.
Chapter 2: The Basics. This covers the basic concepts and terminology, as well as some fundamental math. Among other things, you learn how to be sloppier with your formulas than ever before, and still get the right results, with asymptotic notation.
Chapter 3: Counting 101. More math—but it’s really fun math, I promise! There’s some basic combinatorics for analyzing the running time of algorithms, as well as a gentle introduction to recursion and recurrence relations.
Chapter 4: Induction and Recursion … and Reduction. The three terms in the title are crucial, and they are closely related. Here we work with induction and recursion, which are virtually mirror images of each other, both for designing new algorithms and for proving correctness. We also have a somewhat briefer look at the idea of reduction, which runs as a common thread through almost all algorithmic work.
Chapter 5: Traversal: A Skeleton Key to Algorithmics. Traversal can be understood using the ideas of induction and recursion, but it is in many ways a more concrete and specific technique. Several of the algorithms in this book are simply augmented traversals, so mastering traversal will give you a real jump start.
Chapter 6: Divide, Combine, and Conquer. When problems can be decomposed into independent subproblems, you can recursively solve these subproblems and usually get efficient, correct algorithms as a result. This principle has several applications, not all of which are entirely obvious, and it is a mental tool well worth acquiring.
Chapter 7: Greed is Good? Prove It! Greedy algorithms are usually easy to construct. One can even formulate a general scheme that most, if not all, greedy algorithms follow, yielding a plug-and-play solution. Not only are they easy to construct, but they are usually very efficient. The problem is, it can be hard to show that they are correct (and often they aren’t). This chapter deals with some well-known examples and some more general methods for constructing correctness proofs.
Chapter 8: Tangled Dependencies and Memoization. This chapter is about the design method (or, historically, the problem) called, somewhat confusingly, dynamic programming. It is an advanced technique that can be hard to master but that also yields some of the most enduring insights and elegant solutions in the field.
Chapter 9: From A to B with Edsger and Friends. Rather than the design methods of the previous three chapters, we now focus on a specific problem, with a host of applications: finding shortest paths in networks, or graphs. There are many variations of the problem, with corresponding (beautiful) algorithms.
Chapter 10: Matchings, Cuts, and Flows. How do you match, say, students with colleges so you maximize total satisfaction? In an online community, how do you know whom to trust? And how do you find the total capacity of a road network? These, and several other problems, can be solved with a small class of closely related algorithms and are all variations of the maximum flow problem, which is covered in this chapter.
Chapter 11: Hard Problems and (Limited) Sloppiness. As alluded to in the beginning of the introduction, there are problems we don’t know how to solve efficiently and that we have reasons to think won’t be solved for a long time—maybe never. In this chapter, you learn how to apply the trusty tool of reduction in a new way: not to solve problems but to show that they are hard. Also, we have a look at how a bit of (strictly limited) sloppiness in our optimality criteria can make problems a lot easier to solve.
Appendix A: Pedal to the Metal: Accelerating Python. The main focus of this book is asymptotic efficiency—making your programs scale well with problem size. However, in some cases, that may not be enough. This appendix gives you some pointers to tools that can make your Python programs go faster. Sometimes a lot (as in hundreds of times) faster.
Appendix B: List of Problems and Algorithms. This appendix gives you an overview of the algorithmic problems and algorithms discussed in the book, with some extra information to help you select the right algorithm for the problem at hand.
Appendix C: Graph Terminology and Notation. Graphs are a really useful structure, both in describing real-world systems and in demonstrating how various algorithms work. This chapter gives you a tour of the basic concepts and lingo, in case you haven’t dealt with graphs before.
Appendix D: Hints for Exercises. Just what the title says
