The study of random graphs was begun in the 1960s and now has a comprehensive literature. This excellent book by one of the top researchers in the field now joins the study of random graphs (and other random discrete objects) with mathematical logic. The methodologies involve probability, discrete structures and logic, with an emphasis on discrete structures.
The text covers random graphs from the basic to the advanced, including numerous exercises and recommendations for further reading.
In the ten years since the publication of the best-selling first edition, more than 1,000 graph theory papers have been published each year. Reflecting these advances, Handbook of Graph Theory, Second Edition provides comprehensive coverage of the main topics in pure and applied graph theory. This second edition—over 400 pages longer than its predecessor—incorporates 14 new sections. Each chapter includes lists of essential definitions and facts, accompanied by examples, tables, remarks, and, in some cases, conjectures and open problems. A bibliography at the end of each chapter provides an extensive guide to the research literature and pointers to monographs. In addition, a glossary is included in each chapter as well as at the end of each section. This edition also contains notes regarding terminology and notation. With 34 new contributors, this handbook is the most comprehensive single-source guide to graph theory. It emphasizes quick accessibility to topics for non-experts and enables easy cross-referencing among chapters.
This books gives an introduction to discrete mathematics for beginning undergraduates. One of original features of this book is that it begins with a presentation of the rules of logic as used in mathematics. Many examples of formal and informal proofs are given. With this logical framework firmly in place, the book describes the major axioms of set theory and introduces the natural numbers. The rest of the book is more standard. It deals with functions and relations, directed and undirected graphs, and an introduction to combinatorics. There is a section on public key cryptography and RSA, with complete proofs of Fermat's little theorem and the correctness of the RSA scheme, as well as explicit algorithms to perform modular arithmetic. The last chapter provides more graph theory. Eulerian and Hamiltonian cycles are discussed. Then, we study flows and tensions and state and prove the max flow min-cut theorem. We also discuss matchings, covering, bipartite graphs.
This book comprises a collection of high quality papers in selected topics of Discrete Mathematics, to celebrate the 60th birthday of Professor Jarik Nešetril. Leading experts have contributed survey and research papers in the areas of Algebraic Combinatorics, Combinatorial Number Theory, Game theory, Ramsey Theory, Graphs and Hypergraphs, Homomorphisms, Graph Colorings and Graph Embeddings.
These are the Proceedings of the International Colloquium of Mathematics and Computer Science held at the Vienna University of Technology, September 13-17, 2004. This colloquium is the third one in a now regularly established series following the first two venues in September 2000 and September 2002 in Ver sailles. The present issue is centered around Combinatorics and Random Struc tures, Graph Theory, Analysis of Algorithms, Trees, Probability, Combinatorial Stochastic Processes, and Applications. It contains invited papers, contributed papers (lectures) and short communications (posters). The contributions have been carefully reviewed for their scientific quality and originality by the Scientific Committee chaired by Michael Drmota (Vienna Uni versity of Technology, Austria) and composed of Brigitte Chauvin (Universite de Versailles, France), Luc Devroye (McGill University, Canada), Daniele Gardy (Uni versite de Versailles, France), Philippe Flajolet (INRIA Rocquencourt, France), Michal Karonski (Adam Mickiewicz University, Poland), Abdelkader Mokkadem (Universite de Versailles, France), Helmut Prodinger (University of Witwatersrand, South Africa), J. Michael Steele (University of Pennsylvania, Philadelphia, USA), Brigitte Vallee (Universite de Caen, France). We thank them and all anonymous referees for their impressive work."
What is the "most uniform" way of distributing n points in the unit square? How big is the "irregularity" necessarily present in any such distribution? This book is an accessible and lively introduction to the area of geometric discrepancy theory, with numerous exercises and illustrations. In separate, more specialized parts, it also provides a comprehensive guide to recent research.
For a long time computer scientists have distinguished between fast and slow algo rithms. Fast (or good) algorithms are the algorithms that run in polynomial time, which means that the number of steps required for the algorithm to solve a problem is bounded by some polynomial in the length of the input. All other algorithms are slow (or bad). The running time of slow algorithms is usually exponential. This book is about bad algorithms. There are several reasons why we are interested in exponential time algorithms. Most of us believe that there are many natural problems which cannot be solved by polynomial time algorithms. The most famous and oldest family of hard problems is the family of NP complete problems. Most likely there are no polynomial time al gorithms solving these hard problems and in the worst case scenario the exponential running time is unavoidable. Every combinatorial problem is solvable in ?nite time by enumerating all possi ble solutions, i. e. by brute force search. But is brute force search always unavoid able? De?nitely not. Already in the nineteen sixties and seventies it was known that some NP complete problems can be solved signi?cantly faster than by brute force search. Three classic examples are the following algorithms for the TRAVELLING SALESMAN problem, MAXIMUM INDEPENDENT SET, and COLORING.
For more than 35 years now, George B. Dantzig's Simplex-Method has been the most efficient mathematical tool for solving linear programming problems. It is proba bly that mathematical algorithm for which the most computation time on computers is spent. This fact explains the great interest of experts and of the public to understand the method and its efficiency. But there are linear programming problems which will not be solved by a given variant of the Simplex-Method in an acceptable time. The discrepancy between this (negative) theoretical result and the good practical behaviour of the method has caused a great fascination for many years. While the "worst-case analysis" of some variants of the method shows that this is not a "good" algorithm in the usual sense of complexity theory, it seems to be useful to apply other criteria for a judgement concerning the quality of the algorithm. One of these criteria is the average computation time, which amounts to an anal ysis of the average number of elementary arithmetic computations and of the number of pivot steps. A rigid analysis of the average behaviour may be very helpful for the decision which algorithm and which variant shall be used in practical applications. The subject and purpose of this book is to explain the great efficiency in prac tice by assuming certain distributions on the "real-world" -problems. Other stochastic models are realistic as well and so this analysis should be considered as one of many possibilities.
This newly expanded and updated second edition of the best-selling classic continues to take the "mystery" out of designing algorithms, and analyzing their efficacy and efficiency. Expanding on the first edition, the book now serves as the primary textbook of choice for algorithm design courses while maintaining its status as the premier practical reference guide to algorithms for programmers, researchers, and students. The reader-friendly Algorithm Design Manual provides straightforward access to combinatorial algorithms technology, stressing design over analysis. The first part, Techniques, provides accessible instruction on methods for designing and analyzing computer algorithms. The second part, Resources, is intended for browsing and reference, and comprises the catalog of algorithmic resources, implementations and an extensive bibliography. NEW to the second edition: • Doubles the tutorial material and exercises over the first edition • Provides full online support for lecturers, and a completely updated and improved website component with lecture slides, audio and video • Contains a unique catalog identifying the 75 algorithmic problems that arise most often in practice, leading the reader down the right path to solve them • Includes several NEW "war stories" relating experiences from real-world applications • Provides up-to-date links leading to the very best algorithm implementations available in C, C++, and Java
The new 6th edition of Applied Combinatorics builds on the previous editions with more in depth analysis of computer systems in order to help develop proficiency in basic discrete math problem solving. As one of the most widely used books in combinatorial problems, this edition explains how to reason and model combinatorically while stressing the systematic analysis of different possibilities, exploration of the logical structure of a problem, and ingenuity. Although important uses of combinatorics in computer science, operations research, and finite probability are mentioned, these applications are often used solely for motivation. Numerical examples involving the same concepts use more interesting settings such as poker probabilities or logical games.
Fun and exciting textbook on the mathematics underpinning the most dynamic areas of modern science and engineering.
Elements of Artificial Neural Networks provides a clearly organized general introduction, focusing on a broad range of algorithms, for students and others who want to use neural networks rather than simply study them. The authors, who have been developing and team teaching the material in a one-semester course over the past six years, describe most of the basic neural network models (with several detailed solved examples) and discuss the rationale and advantages of the models, as well as their limitations. The approach is practical and open-minded and requires very little mathematical or technical background. Written from a computer science and statistics point of view, the text stresses links to contiguous fields and can easily serve as a first course for students in economics and management. The opening chapter sets the stage, presenting the basic concepts in a clear and objective way and tackling important -- yet rarely addressed -- questions related to the use of neural networks in practical situations. Subsequent chapters on supervised learning (single layer and multilayer networks), unsupervised learning, and associative models are structured around classes of problems to which networks can be applied. Applications are discussed along with the algorithms. A separate chapter takes up optimization methods. The most frequently used algorithms, such as backpropagation, are introduced early on, right after perceptrons, so that these can form the basis for initiating course projects. Algorithms published as late as 1995 are also included. All of the algorithms are presented using block-structured pseudo-code, and exercises are provided throughout. Software implementing many commonly used neural network algorithms is available at the book's website. Transparency masters, including abbreviated text and figures for the entire book, are available for instructors using the text.
In the summer of 2006 two books attacking string theory, a prominent theory in physics, appeared: Peter Woit's "Not Even Wrong" and Lee Smolin's "The Trouble with Physics." A fierce public debate, much of it on weblogs, ensued. Gina is very curious about science blogs. Can they be useful for learning about or discussing science? What happens in these blogs and who participates in them? Gina is eager to learn the issues and to form her own opinion about the string theory controversy. She is equipped with some academic background, including in mathematics, and has some familiarity with academic life. Her knowledge of physics is derived mainly from popular accounts. Gina likes to debate and to argue. She is fascinated by questions about rationality and philosophy, and was exposed to various other scientific controversies in the past. This book uses the blog debate on string theory to discuss blogs, science, and mathematics. Meandering over various topics from children's dyscalculia to Chomskian linguistics, the reader may get some sense of the chaotic and often confusing scientific experience. The book tries to show the immense difficulty involved in getting the factual matters right, and interpreting fragmented and partial information. Contents: Not Even Wrong: The Blog of Peter Woitn-Category CaféAsymptotia Readership: The general public interested in science, especially those who read scientific blogs. Keywords: Blogosphere;Science Blogs;String TheoryReview: Key Features: It is an unusual combination of popular science, the story of a major scientific debate, the story of scientific blogs, and the story of the hero "Gina" who tries to explore and participate in these blogs
This book comprises the refereed proceedings of the International Conference, AIM/CCPE 2012, held in Bangalore, India, in April 2012. The papers presented were carefully reviewed and selected from numerous submissions and focus on the various aspects of research and development activities in computer science, information technology, computational engineering, mobile communication, control and instrumentation, communication system, power electronics and power engineering.