xiv Preface You should find it easy to organize your course around just the chapters you need.We have made chapters relatively self-contained,so that you need not worry about an unexpected and unnecessary dependence of one chapter on another.Each chapter presents the easier material first and the more difficult material later,with section boundaries marking natural stopping points.In an undergraduate course, you might use only the earlier sections from a chapter;in a graduate course,you might cover the entire chapter. We have included 957 exercises and 158 problems.Each section ends with exer- cises,and each chapter ends with problems.The exercises are generally short ques- tions that test basic mastery of the material.Some are simple self-check thought exercises,whereas others are more substantial and are suitable as assigned home- work.The problems are more elaborate case studies that often introduce new ma- terial;they often consist of several questions that lead the student through the steps required to arrive at a solution. Departing from our practice in previous editions of this book,we have made publicly available solutions to some,but by no means all,of the problems and ex- ercises.Our Web site,http://mitpress.mit.edu/algorithms/,links to these solutions. You will want to check this site to make sure that it does not contain the solution to an exercise or problem that you plan to assign.We expect the set of solutions that we post to grow slowly over time,so you will need to check it each time you teach the course. We have starred (*)the sections and exercises that are more suitable for graduate students than for undergraduates.A starred section is not necessarily more diffi- cult than an unstarred one,but it may require an understanding of more advanced mathematics.Likewise,starred exercises may require an advanced background or more than average creativity. To the student We hope that this textbook provides you with an enjoyable introduction to the field of algorithms.We have attempted to make every algorithm accessible and interesting.To help you when you encounter unfamiliar or difficult algorithms,we describe each one in a step-by-step manner.We also provide careful explanations of the mathematics needed to understand the analysis of the algorithms.If you already have some familiarity with a topic,you will find the chapters organized so that you can skim introductory sections and proceed quickly to the more advanced material. This is a large book,and your class will probably cover only a portion of its material.We have tried,however,to make this a book that will be useful to you now as a course textbook and also later in your career as a mathematical desk reference or an engineering handbook.xiv Preface You should find it easy to organize your course around just the chapters you need. We have made chapters relatively self-contained, so that you need not worry about an unexpected and unnecessary dependence of one chapter on another. Each chapter presents the easier material first and the more difficult material later, with section boundaries marking natural stopping points. In an undergraduate course, you might use only the earlier sections from a chapter; in a graduate course, you might cover the entire chapter. We have included 957 exercises and 158 problems. Each section ends with exer￾cises, and each chapter ends with problems. The exercises are generally short ques￾tions that test basic mastery of the material. Some are simple self-check thought exercises, whereas others are more substantial and are suitable as assigned home￾work. The problems are more elaborate case studies that often introduce new ma￾terial; they often consist of several questions that lead the student through the steps required to arrive at a solution. Departing from our practice in previous editions of this book, we have made publicly available solutions to some, but by no means all, of the problems and ex￾ercises. Our Web site, http://mitpress.mit.edu/algorithms/, links to these solutions. You will want to check this site to make sure that it does not contain the solution to an exercise or problem that you plan to assign. We expect the set of solutions that we post to grow slowly over time, so you will need to check it each time you teach the course. We have starred (?) the sections and exercises that are more suitable for graduate students than for undergraduates. A starred section is not necessarily more diffi- cult than an unstarred one, but it may require an understanding of more advanced mathematics. Likewise, starred exercises may require an advanced background or more than average creativity. To the student We hope that this textbook provides you with an enjoyable introduction to the field of algorithms. We have attempted to make every algorithm accessible and interesting. To help you when you encounter unfamiliar or difficult algorithms, we describe each one in a step-by-step manner. We also provide careful explanations of the mathematics needed to understand the analysis of the algorithms. If you already have some familiarity with a topic, you will find the chapters organized so that you can skim introductory sections and proceed quickly to the more advanced material. This is a large book, and your class will probably cover only a portion of its material. We have tried, however, to make this a book that will be useful to you now as a course textbook and also later in your career as a mathematical desk reference or an engineering handbook