PythonAlgorithms,2ndEdition.pdf

Python

Algorithm

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Hetland

Shelve in

Programming Languages/General

User level:

Beginning–Intermediate

www.apress.com

SOURCE CODE ONLINE

BOOKS FOR PROFESSIONALS BY PROFESSIONALS

Python Algorithms

Python Algorithms, Second Edition, explains the Python approach to algorithm

analysis and design. Written by Magnus Lie Hetland, author of Beginning Python,

this book is sharply focused on classical algorithms, but it also gives a solid

understanding of fundamental algorithmic problem-solving techniques.

The book deals with some of the most important and challenging areas of

programming and computer science in a highly readable manner. It covers both

algorithmic theory and programming practice, demonstrating how theory is reflected

in real Python programs. Well-known algorithms and data structures that are built

into the Python language are explained, and the user is shown how to implement

and evaluate others that aren’t built into Python.

What You’ll Learn:

• Transform new problems to well-known algorithmic problems with efficient

solutions, or show that the problems belong to classes of problems

thought not to be efficiently solvable

• Analyze algorithms and Python programs using both mathematical tools

and basic experiments and benchmarks

• Understand several classical algorithms and data structures in depth,

and be able to implement these efficiently in Python

• Design and implement new algorithms for new problems, using time-tested

design principles and techniques

• Speed up implementations, using a plethora of tools for high-performance

computing in Python

SECOND

EDITION

9781484 200568

55499

ISBN 978-1-4842-0056-8

www.allitebooks.com

Contents at a Glance

About the Author �� xv

About the Technical Reviewer �� xvii

Acknowledgments �� xix

Preface �� xxi

Chapter 1 ■

: Introduction ��1

Chapter 2 ■ : The Basics ��9

Chapter 3 ■ : Counting 101 ��43

Chapter 4 ■ : Induction and Recursion �� and Reduction ��67

Chapter 5 ■ : Traversal: The Skeleton Key of Algorithmics ��93

Chapter 6 ■ : Divide, Combine, and Conquer ��115

Chapter 7 ■ : Greed Is Good? Prove It! ��139

Chapter 8 ■ : Tangled Dependencies and Memoization��163

Chapter 9 ■ : From A to B with Edsger and Friends ��187

Chapter 10 ■ : Matchings, Cuts, and Flows ��209

Chapter 11 ■ : Hard Problems and (Limited) Sloppiness ��227

www.allitebooks.com

Chapter 1

Introduction

1. Write down the problem.

2. Think real hard.

3. Write down the solution.

— “The Feynman Algorithm”

as described by Murray Gell-Mann

Consider the following problem: You are to visit all the cities, towns, and villages of, say, Sweden and then return

to your starting point. This might take a while (there are 24,978 locations to visit, after all), so you want to minimize

your route. You plan on visiting each location exactly once, following the shortest route possible. As a programmer,

you certainly don’t want to plot the route by hand. Rather, you try to write some code that will plan your trip for you.

For some reason, however, you can’t seem to get it right. A straightforward program works well for a smaller number

of towns and cities but seems to run forever on the actual problem, and improving the program turns out to be

surprisingly hard. How come?

Actually, in 2004, a team of five researchers

found such a tour of Sweden, after a number of other research teams

had tried and failed. The five-man team used cutting-edge software with lots of clever optimizations and tricks of

the trade, running on a cluster of 96 Xeon 2.6GHz workstations. Their software ran from March 2003 until May 2004,

before it finally printed out the optimal solution. Taking various interruptions into account, the team estimated that

the total CPU time spent was about 85 years!

Consider a similar problem: You want to get from Kashgar, in the westernmost region of China, to Ningbo, on the

east coast, following the shortest route possible.

Now, China has 3,583,715 km of roadways and 77,834 km of railways,

with millions of intersections to consider and a virtually unfathomable number of possible routes to follow. It might

seem that this problem is related to the previous one, yet this shortest path problem is one solved routinely, with no

appreciable delay, by GPS software and online map services. If you give those two cities to your favorite map service,

you should get the shortest route in mere moments. What’s going on here?

You will learn more about both of these problems later in the book; the first one is called the traveling salesman

(or salesrep) problem and is covered in Chapter 11, while so-called shortest path problems are primarily dealt with

in Chapter 9. I also hope you will gain a rather deep insight into why one problem seems like such a hard nut to

crack while the other admits several well-known, efficient solutions. More importantly, you will learn something

about how to deal with algorithmic and computational problems in general, either solving them efficiently, using

one of the several techniques and algorithms you encounter in this book, or showing that they are too hard and that

approximate solutions may be all you can hope for. This chapter briefly describes what the book is about—what you

can expect and what is expected of you. It also outlines the specific contents of the various chapters to come in case

you want to skip around.

David Applegate, Robert Bixby, Vašek Chvátal, William Cook, and Keld Helsgaun

Let’s assume that flying isn’t an option.

www.allitebooks.com

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