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\(\def \AE {\unicode {x00C6}}\)
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\(\def \AA {\unicode {x00C5}}\)
\(\def \o {\unicode {x00F8}}\)
\(\def \O {\unicode {x00D8}}\)
\(\def \l {\unicode {x0142}}\)
\(\def \L {\unicode {x0141}}\)
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\(\def \SS {\unicode {x1E9E}}\)
\(\def \dag {\unicode {x2020}}\)
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\(\def \P {\unicode {x00B6}}\)
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\(\def \pounds {\unicode {x00A3}}\)
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\( \newcommand {\multicolumn }[3]{#3}\)
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Preface
Quite exciting, this computer magic!
— Viv Savage, This is Spinal Tap
One man’s “magic” is another man’s engineering.
— Robert A. Heinlein
This is a book about turning your ideas into working programs. It explains how to model problems from a wide range of applications, how to design appropriate data structures, how to implement the required algorithms, and
how to combine everything into efficient computer programs. We start with a few simple problems related to the visualization of fractals, continue with programs that are able to solve classical puzzles such as Sudoku, peg solitaire, or Sokoban, and finally discuss how programs like ZIP are able to compress
files to a fraction of their original size and how computer algebra systems handle arbitrarily large integers. For each of these problems we will develop complete, working programs and discuss their implementations.
My main goal was to write the kind of book I would have loved to read when I was learning programming back in high school, a book that goes beyond the toy problems discussed in ordinary programming books but doesn’t require the mathematical sophistication (or access to teaching assistants) that is
assumed by many university textbooks. Here are a few potential target audiences and how they might benefit from reading this book:
-
• If you are a high-school student or a self-taught programmer, this book will teach you a wide range of programming techniques and important concepts from computer science. You may find the first few chapters relatively easy, but some of the
topics discussed in later chapters will put your programming skills to the test: it’s intentionally a book that grows with you as you become a better programmer.
-
• If you are an undergraduate student, this books can serve as a complement to more rigorous courses on programming and algorithms and will teach you how to apply the concepts taught in class. If your courses make you think that programming is dry
and abstract, read on for practical applications that are both fun and instructive.
-
• And finally, as an expert programmer or professional computer scientist, you will probably appreciate the many programming problems and computational puzzles collected in this book. You will also find that reimplementing the example
programs is a great way to learn and evaluate new programming languages.
Ideally, before reading this book, you should already know how to program and how to compile and run your code. In addition, we will assume that you are familiar with fundamental concepts such as recursion, arrays, linked lists, and hash tables. A high school level of mathematics is sufficient to
understand most of the material, but we will make occasional use of more advanced topics such as linear algebra, induction, big-Oh notation, or computational complexity; See Appendix A for an overview of some of the notations
used in this book.
The programming language we will use to write all example programs is Java, mainly because it is widely used and taught and strikes a good balance between high-level conveniences and low-level features. If you are coming from another language, you will find a summary of Java’s main features
in Appendix B.
The book’s official home page can be found at
In addition, the book’s GitHub page at
hosts the source code of all example programs and serves as the official bug tracker. Reports of any errors in the text or the examples are greatly appreciated!
All code examples are licensed under the terms of the MIT license (https://opensource.org/license/mit). You are free (encouraged, even!) to use this code in your own programs, extend it to your liking, translate it into other
programming languages, and publish the resulting programs.
Organization
Thematically, the book consists of the following five parts.
Iteration
In the first two chapters, we explore one of the simplest yet most powerful ideas in computer science: iteration, that is, performing a certain sequence of operations repeatedly. To illustrate different aspects of iteration, we will mainly study fractals — mathematical objects that have detail at every level of magnification and that are almost impossible to visualize without the use of computers. In Chapter 1 we will discuss iterated function systems, which are procedures for generating geometric shapes that consist of countless smaller copies of themselves. To visualize these shapes, we employ the chaos game, a simple algorithm that demonstrates how even simple loops pack a surprising amount of computational punch.
In Chapter 2 we turn to another famous fractal called the Mandelbrot set. To visualize the Mandelbrot set, we will need to learn about complex numbers and how to use iteration to evaluate infinite sequences and compute the pixels of raster images. We also discuss a variation of the Mandelbrot set called the Buddhabrot.
Recursion
In the next two chapters we turn to recursion, an alternative way of expressing repetition in computer programs. Unlike iteration, which regards computations as long, linear sequences of operations, recursion decomposes them into a tree-shaped hierarchy of smaller and smaller computations. In
Chapter 3 we explain how to use recursion to construct a fractal known as the Pythagoras tree.
Chapter 4 discusses two other geometric objects that can be defined recursively: Hilbert curves and Bézier curves. The former are an example of a so-called space-filling curve, a curve that is coiled up so tightly that it ultimately converges toward a solid object in the plane. The Bézier curves covered in the second half of the chapter are widely used in computer
graphics; among other things, we they are used to describe the smooth curves in typefaces and vector graphics. Bézier curves have an unexpected recursive structure, which is the basis of several elegant algorithms for working with them.
Bits
We then turn our attention to bits, the fundamental unit of storage (and computation) inside a computer. We start with an overview of programming techniques for working with bits in Chapter 5. We will discuss how to read and modify individual
bits and groups of bits, learn about two’s complement representation for storing signed integers, and develop a wide range of useful techniques for performing computations that involve powers of 2 and binary logarithms.
In Chapter 6 we then use some of these bit-processing techniques to study a classic puzzle called peg solitaire. We will see how to use bits to represent the possible states of this puzzle and how to use bit operations to generate all possible moves and solutions. We will also study backtracking, a simple, recursive method for systematically generating chains of moves (or arbitrary other objects), as well as techniques for speeding it up.
Graph traversal
We continue with games and puzzles in the next two chapters, but this time the focus is on problems that can be modeled and solved using graphs and graph algorithms. As we shall discuss, graphs are a general model for describing and analyzing the relationships between various kinds of objects — including the relationships between the possible states of many games and puzzles. In Chapter 7, we discuss the basics of graph theory, how to model simple puzzles in this framework, and how to solve them algorithmically using elementary techniques such as breadth-first search. In Chapter 8, we tackle a significantly more challenging game and study how to find optimal solutions for the classic computer game Sokoban. Our investigation of Sokoban will naturally lead us to the notion of a weighted graph and Dijkstra’s algorithm, an important generalization of breadth-first search.
Data compression
The next two chapters focus on data compression, the art and science of reducing the number of bits needed to store or transmit information. We start in Chapter 9 with an overview of binary codes. A binary code can be thought of as a procedure for translating a sequence of objects (for example, the bytes in a file) to a string of bits. The main question is: How can we design binary codes that use as few bits as possible but still allow us to recover the original
sequence of objects? We will discuss an important class of codes known as prefix codes and a fundamental technique for constructing efficient prefix codes known as Huffman coding.
In Chapter 10 we discuss the implementation of LZ77, a general-purpose compression algorithm that is the basis of popular file formats such as zip and png. From an algorithmic perspective, LZ77 is fairly simple, but its implementation is far from trivial. We will
discuss how to make the algorithm run quickly, how to design an efficient bit encoding for the output, and how to combine LZ77 with Huffman coding to achieve even better compression rates.
Arithmetic
We conclude in Chapter 11 with an in-depth discussion of big integers, a data type that allows us to compute with arbitrarily large integers. Conceptually, big integers are closely related to the familiar decimal numbers, and the standard arithmetic procedures for adding, multiplying, and dividing decimal numbers have natural
generalizations to big integers. There are several crucial differences, however. On the one hand, we must contend with computer-specific challenges such as memory management and integer overflow. On the other hand, the use of computers allow us improve some of the algorithms in unique ways, for
example by exploiting certain arithmetic tricks or by using algorithmic techniques that are hard to carry out by hand.
Code Examples
Object-oriented programming languages like Java organize complex programs by dividing them into multiple classes with well-defined responsibilities. Each class consists of fields that hold some of the program’s state and methods that operate on this state and perform some of the program’s
computations. Classes can be nested inside other classes, and the top-level classes are organized into packages, which can be nested inside other packages.
This kind of hierarchical organization makes it possible to divide large programs into manageable, self-contained units, but it’s not a good match for presenting code in a book. We will therefore split complex classes and their nested parts into multiple smaller listings, which we then discuss separately.
Listing 0.1 illustrates how this would look for a class Hello with one field greeting and some additional code that isn’t shown. The label before the listing tells us that the code is part of a class
called Hello that is defined in a package called preface. In general, the source code for every chapter can be found in a separate package; for example, the classes in Chapter 1 are all defined in a package called ch1.
Listing 0.1preface / Hello
public class Hello {
private String greeting;
// ...
}
The comment ‘// ...’ on line 4 of Listing 0.1 indicates that some nested parts of the class have been omitted. In this case, we have moved the implementation of a method called printGreeting() to Listing 0.2. The label above the listing tells us that the code also belongs to the class preface/Hello.
Listing 0.2preface / Hello
public void printGreeting() {
System.out.println(greeting);
}
Exercises and Problems
More than 130 exercises are included in the book, most of which are easy and primarily designed to review or reinforce the material presented in each section. Exercises with a mathematical flavor are marked with a star (\(\star \)).
Exercise 0.1.How many Java programmers does it take to change a light bulb?
\(\star \) Exercise 0.2.Assume that one programmer can change a light bulb in one minute. How long does it take \(n\) programmers to change \(n\) light bulbs?
In addition, every chapter ends with a selection of problems, which are larger exercises that expand on the material covered in the chapter in nontrivial ways. Some of this book’s best material can be found in these problems, so be sure to give them a try! Complete answers to most exercises and
problems are provided in Appendix C.