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Ryuhei Uehara - First Course in Algorithms Through Puzzles

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Ryuhei Uehara First Course in Algorithms Through Puzzles
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This textbook introduces basic algorithms and explains their analytical methods. All algorithms and methods introduced in this book are well known and frequently used in real programs. Intended to be self-contained, the contents start with the basic models, and no prerequisite knowledge is required. This book is appropriate for undergraduate students in computer science, mathematics, and engineering as a textbook, and is also appropriate for self-study by beginners who are interested in the fascinating field of algorithms.

More than 40 exercises are distributed throughout the text, and their difficulty levels are indicated. Solutions and comments for all the exercises are provided in the last chapter. These detailed solutions will enable readers to follow the authors steps to solve problems and to gain a better understanding of the contents.

Although details of the proofs and the analyses of algorithms are also provided, the mathematical descriptions in this book are not beyond the range of high school mathematics. Some famous real puzzles are also used to describe the algorithms. These puzzles are quite suitable for explaining the basic techniques of algorithms, which show how to solve these puzzles.

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Ryuhei Uehara First Course in Algorithms Through Puzzles Ryuhei Uehara - photo 1
Ryuhei Uehara
First Course in Algorithms Through Puzzles
Ryuhei Uehara Japan Advanced Institute of Science and Technology Nomi - photo 2
Ryuhei Uehara
Japan Advanced Institute of Science and Technology, Nomi, Ishikawa, Japan
ISBN 978-981-13-3187-9 e-ISBN 978-981-13-3188-6
https://doi.org/10.1007/978-981-13-3188-6
Library of Congress Control Number: 2018960741
Springer Nature Singapore Pte Ltd. 2019
This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.
The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd.

The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

This book is an introduction to algorithms . What is an algorithm ? In short, algorithm means a way of solving a problem. When people encounter a problem, the approach to solving it depends on who is solving it, and then the efficiency varies accordingly. Lately, mobile devices have become quite smarter, planning your route when you provide your destination, and running an application when you talk to it. Inside this tiny device, there exists a computer that is smarter than the old large-scale computers, and it runs some neat algorithms to solve your problems. Smart algorithms use clever tricks to reduce computational time and the amount of memory needed.

One may think that from the viewpoint of the progress of hardware devices, is an improvement such as this insignificant? However, you would be wrong. For example, take integer programming. Without going into details, it is a general framework to solve mathematical optimization programs, and many realistic problems can be solved using this model. The running time of programs for solving integer programming has been improved 10,000,000,000 times over the last two decades. Surprisingly, the contribution of hardware is 2000 times, and the contribution of software, that is, the algorithm, is 4,75,000 times. It is not easy to see, but the improvement of the way of solving has been much more effective than the development of hardware.

In this book, I introduce and explain the basic algorithms and their analytical methods for undergraduate students in the Faculty of Information Science. This book starts with the basic models, and no prerequisite knowledge is required. All algorithms and methods in this book are well known and frequently used in real computing. This book aims to be self-contained; thus, it is not only a textbook, but also allows you to learn by yourself, or use as a reference book for beginners. On the other hand, I provide some smart tips for non-beginner readers.

Exercise : Exercises appear in the text and are not collected at the end of sections. If you find an exercise when you read this book, I would like to ask you to take a break, tackle the exercise, and check the answer and comments. For every exercise, an answer and comments are given in this book.

Each exercise is marked with
Picture 3
marks, and those with many cups are more difficult than those with fewer ones. They say that a mathematician is a machine for turning coffee into theorems (this quotation is often attributed to Paul Erds; however, he ascribed it to Alfrd Rnyi). In this regard, I hope you will enjoy them with cups of coffee.

Exercises are provided in various ways in your textbooks and reference books. In this book, all solutions for exercises are given. This is rather rare. Actually, this is not that easy from the authors viewpoints. Some textbooks seem to be missing the answers because of the authors laziness. Even if the author considers a problem trivial, it is not necessarily the case for readers, and beginners tend to fail to follow in such small steps. Moreover, although it seems trivial at a glance, difficulties may start to appear when we try to solve some problems. For both readers and authors, it is tragic that beginners fail to learn because of authors prejudgments.

Even if it seems easy to solve a problem, solving it by yourself enables you to learn many more things than you expect. I am hoping you will try to tackle the exercises by yourselves. If you have no time to solve them, I strongly recommend to check their solutions carefully before proceeding to the next topic.

Analyses and Proofs : In this book, I provide details of the proofs and the analyses of algorithms. Some unfamiliar readers could find some of the mathematical descriptions difficult. However, they are not beyond the range of high school mathematics although some readers may hate mathematical induction, exponential functions, or logarithmic functions. Although I believe that ordinary high school students should be able to follow the main ideas of algorithms and the pleasure they bring, you may not lie down on your couch to follow the analyses and proofs. However, I want to emphasize that there exists beauty you cannot taste without such understanding. The reason why I do not omit these tough parts is that I would like to invite you to appreciate this deep beauty beyond the obstacles you will encounter. Each equation is described in detail, and it is not as hard once you indeed try to follow.

Algorithms are fun. Useful algorithms provide a pleasant feeling much the same as well-designed puzzles. In this book, I have included some famous real puzzles to describe the algorithms. They are quite suitable for explaining the basic techniques of algorithms, which also show us how to solve these puzzles. Through learning algorithms, I hope you will enjoy acquiring knowledge in such a pleasant way.

Ryuhei Uehara
Nomi, Japan
October 2018
Contents
Afterword
Index
Footnotes

R. Bixby, Z. Gu, and Ed. Rothberg, Presolve for Linear and Mixed-Integer Programming, RAMP Symposium, 2012, Japan.

Paul Erds(19131996) :

He was one of the most productive mathematicians of the twentieth century, and wrote more than 1400 journal papers with more than 500 collaborators. Erd numbers are a part of the folklore of mathematicians that indicate the distance between themselves and Erds. Erds himself has Erds number 0, and his coauthors have Erds number 1. For example, I have Erds number 2 because I have collaborated with a mathematician who has Erds number 1.

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