Combinatorial Geometry And Its Algorithmic Applications

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Combinatorial Geometry and Its Algorithmic Applications

Author : János Pach
Publisher : American Mathematical Soc.
ISBN 13 : 0821846914
Total Pages : 251 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Combinatorial Geometry and Its Algorithmic Applications by : János Pach

Download or read book Combinatorial Geometry and Its Algorithmic Applications written by János Pach and published by American Mathematical Soc.. This book was released on 2009 with total page 251 pages. Available in PDF, EPUB and Kindle. Book excerpt: "Based on a lecture series given by the authors at a satellite meeting of the 2006 International Congress of Mathematicians and on many articles written by them and their collaborators, this volume provides a comprehensive up-to-date survey of several core areas of combinatorial geometry. It describes the beginnings of the subject, going back to the nineteenth century (if not to Euclid), and explains why counting incidences and estimating the combinatorial complexity of various arrangements of geometric objects became the theoretical backbone of computational geometry in the 1980s and 1990s. The combinatorial techniques outlined in this book have found applications in many areas of computer science from graph drawing through hidden surface removal and motion planning to frequency allocation in cellular networks. "Combinatorial Geometry and Its Algorithmic Applications" is intended as a source book for professional mathematicians and computer scientists as well as for graduate students interested in combinatorics and geometry. Most chapters start with an attractive, simply formulated, but often difficult and only partially answered mathematical question, and describes the most efficient techniques developed for its solution. The text includes many challenging open problems, figures, and an extensive bibliography."--BOOK JACKET.

Algorithms in Combinatorial Geometry

Author : Herbert Edelsbrunner
Publisher : Springer Science & Business Media
ISBN 13 : 3642615686
Total Pages : 423 pages
Book Rating : 4.6/5 (426 download)

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Book Synopsis Algorithms in Combinatorial Geometry by : Herbert Edelsbrunner

Download or read book Algorithms in Combinatorial Geometry written by Herbert Edelsbrunner and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 423 pages. Available in PDF, EPUB and Kindle. Book excerpt: Computational geometry as an area of research in its own right emerged in the early seventies of this century. Right from the beginning, it was obvious that strong connections of various kinds exist to questions studied in the considerably older field of combinatorial geometry. For example, the combinatorial structure of a geometric problem usually decides which algorithmic method solves the problem most efficiently. Furthermore, the analysis of an algorithm often requires a great deal of combinatorial knowledge. As it turns out, however, the connection between the two research areas commonly referred to as computa tional geometry and combinatorial geometry is not as lop-sided as it appears. Indeed, the interest in computational issues in geometry gives a new and con structive direction to the combinatorial study of geometry. It is the intention of this book to demonstrate that computational and com binatorial investigations in geometry are doomed to profit from each other. To reach this goal, I designed this book to consist of three parts, acorn binatorial part, a computational part, and one that presents applications of the results of the first two parts. The choice of the topics covered in this book was guided by my attempt to describe the most fundamental algorithms in computational geometry that have an interesting combinatorial structure. In this early stage geometric transforms played an important role as they reveal connections between seemingly unrelated problems and thus help to structure the field.

Geometric Algorithms and Combinatorial Optimization

Author : Martin Grötschel
Publisher : Springer Science & Business Media
ISBN 13 : 3642978819
Total Pages : 374 pages
Book Rating : 4.6/5 (429 download)

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Book Synopsis Geometric Algorithms and Combinatorial Optimization by : Martin Grötschel

Download or read book Geometric Algorithms and Combinatorial Optimization written by Martin Grötschel and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 374 pages. Available in PDF, EPUB and Kindle. Book excerpt: Historically, there is a close connection between geometry and optImization. This is illustrated by methods like the gradient method and the simplex method, which are associated with clear geometric pictures. In combinatorial optimization, however, many of the strongest and most frequently used algorithms are based on the discrete structure of the problems: the greedy algorithm, shortest path and alternating path methods, branch-and-bound, etc. In the last several years geometric methods, in particular polyhedral combinatorics, have played a more and more profound role in combinatorial optimization as well. Our book discusses two recent geometric algorithms that have turned out to have particularly interesting consequences in combinatorial optimization, at least from a theoretical point of view. These algorithms are able to utilize the rich body of results in polyhedral combinatorics. The first of these algorithms is the ellipsoid method, developed for nonlinear programming by N. Z. Shor, D. B. Yudin, and A. S. NemirovskiI. It was a great surprise when L. G. Khachiyan showed that this method can be adapted to solve linear programs in polynomial time, thus solving an important open theoretical problem. While the ellipsoid method has not proved to be competitive with the simplex method in practice, it does have some features which make it particularly suited for the purposes of combinatorial optimization. The second algorithm we discuss finds its roots in the classical "geometry of numbers", developed by Minkowski. This method has had traditionally deep applications in number theory, in particular in diophantine approximation.

New Trends in Discrete and Computational Geometry

Author : Janos Pach
Publisher : Springer Science & Business Media
ISBN 13 : 3642580432
Total Pages : 342 pages
Book Rating : 4.6/5 (425 download)

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Book Synopsis New Trends in Discrete and Computational Geometry by : Janos Pach

Download or read book New Trends in Discrete and Computational Geometry written by Janos Pach and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 342 pages. Available in PDF, EPUB and Kindle. Book excerpt: Discrete and computational geometry are two fields which in recent years have benefitted from the interaction between mathematics and computer science. The results are applicable in areas such as motion planning, robotics, scene analysis, and computer aided design. The book consists of twelve chapters summarizing the most recent results and methods in discrete and computational geometry. All authors are well-known experts in these fields. They give concise and self-contained surveys of the most efficient combinatorical, probabilistic and topological methods that can be used to design effective geometric algorithms for the applications mentioned above. Most of the methods and results discussed in the book have not appeared in any previously published monograph. In particular, this book contains the first systematic treatment of epsilon-nets, geometric tranversal theory, partitions of Euclidean spaces and a general method for the analysis of randomized geometric algorithms. Apart from mathematicians working in discrete and computational geometry this book will also be of great use to computer scientists and engineers, who would like to learn about the most recent results.

Handbook of Discrete and Computational Geometry

Author : Csaba D. Toth
Publisher : CRC Press
ISBN 13 : 1498711421
Total Pages : 1928 pages
Book Rating : 4.4/5 (987 download)

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Book Synopsis Handbook of Discrete and Computational Geometry by : Csaba D. Toth

Download or read book Handbook of Discrete and Computational Geometry written by Csaba D. Toth and published by CRC Press. This book was released on 2017-11-22 with total page 1928 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Handbook of Discrete and Computational Geometry is intended as a reference book fully accessible to nonspecialists as well as specialists, covering all major aspects of both fields. The book offers the most important results and methods in discrete and computational geometry to those who use them in their work, both in the academic world—as researchers in mathematics and computer science—and in the professional world—as practitioners in fields as diverse as operations research, molecular biology, and robotics. Discrete geometry has contributed significantly to the growth of discrete mathematics in recent years. This has been fueled partly by the advent of powerful computers and by the recent explosion of activity in the relatively young field of computational geometry. This synthesis between discrete and computational geometry lies at the heart of this Handbook. A growing list of application fields includes combinatorial optimization, computer-aided design, computer graphics, crystallography, data analysis, error-correcting codes, geographic information systems, motion planning, operations research, pattern recognition, robotics, solid modeling, and tomography.

Sampling in Combinatorial and Geometric Set Systems

Author : Nabil H. Mustafa
Publisher : American Mathematical Society
ISBN 13 : 1470461560
Total Pages : 251 pages
Book Rating : 4.4/5 (74 download)

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Book Synopsis Sampling in Combinatorial and Geometric Set Systems by : Nabil H. Mustafa

Download or read book Sampling in Combinatorial and Geometric Set Systems written by Nabil H. Mustafa and published by American Mathematical Society. This book was released on 2022-01-14 with total page 251 pages. Available in PDF, EPUB and Kindle. Book excerpt: Understanding the behavior of basic sampling techniques and intrinsic geometric attributes of data is an invaluable skill that is in high demand for both graduate students and researchers in mathematics, machine learning, and theoretical computer science. The last ten years have seen significant progress in this area, with many open problems having been resolved during this time. These include optimal lower bounds for epsilon-nets for many geometric set systems, the use of shallow-cell complexity to unify proofs, simpler and more efficient algorithms, and the use of epsilon-approximations for construction of coresets, to name a few. This book presents a thorough treatment of these probabilistic, combinatorial, and geometric methods, as well as their combinatorial and algorithmic applications. It also revisits classical results, but with new and more elegant proofs. While mathematical maturity will certainly help in appreciating the ideas presented here, only a basic familiarity with discrete mathematics, probability, and combinatorics is required to understand the material.

Graph Theory, Combinatorics and Algorithms

Author : Martin Charles Golumbic
Publisher : Springer Science & Business Media
ISBN 13 : 0387250360
Total Pages : 296 pages
Book Rating : 4.3/5 (872 download)

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Book Synopsis Graph Theory, Combinatorics and Algorithms by : Martin Charles Golumbic

Download or read book Graph Theory, Combinatorics and Algorithms written by Martin Charles Golumbic and published by Springer Science & Business Media. This book was released on 2006-03-30 with total page 296 pages. Available in PDF, EPUB and Kindle. Book excerpt: Graph Theory, Combinatorics and Algorithms: Interdisciplinary Applications focuses on discrete mathematics and combinatorial algorithms interacting with real world problems in computer science, operations research, applied mathematics and engineering. The book contains eleven chapters written by experts in their respective fields, and covers a wide spectrum of high-interest problems across these discipline domains. Among the contributing authors are Richard Karp of UC Berkeley and Robert Tarjan of Princeton; both are at the pinnacle of research scholarship in Graph Theory and Combinatorics. The chapters from the contributing authors focus on "real world" applications, all of which will be of considerable interest across the areas of Operations Research, Computer Science, Applied Mathematics, and Engineering. These problems include Internet congestion control, high-speed communication networks, multi-object auctions, resource allocation, software testing, data structures, etc. In sum, this is a book focused on major, contemporary problems, written by the top research scholars in the field, using cutting-edge mathematical and computational techniques.

Handbook of Combinatorics Volume 1

Author : Ronald L. Graham
Publisher : Elsevier
ISBN 13 : 9780444823465
Total Pages : 1124 pages
Book Rating : 4.8/5 (234 download)

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Book Synopsis Handbook of Combinatorics Volume 1 by : Ronald L. Graham

Download or read book Handbook of Combinatorics Volume 1 written by Ronald L. Graham and published by Elsevier. This book was released on 1995-12-11 with total page 1124 pages. Available in PDF, EPUB and Kindle. Book excerpt: Handbook of Combinatorics, Volume 1 focuses on basic methods, paradigms, results, issues, and trends across the broad spectrum of combinatorics. The selection first elaborates on the basic graph theory, connectivity and network flows, and matchings and extensions. Discussions focus on stable sets and claw free graphs, nonbipartite matching, multicommodity flows and disjoint paths, minimum cost circulations and flows, special proof techniques for paths and circuits, and Hamilton paths and circuits in digraphs. The manuscript then examines coloring, stable sets, and perfect graphs and embeddings and minors. The book takes a look at random graphs, hypergraphs, partially ordered sets, and matroids. Topics include geometric lattices, structural properties, linear extensions and correlation, dimension and posets of bounded degree, hypergraphs and set systems, stability, transversals, and matchings, and phase transition. The manuscript also reviews the combinatorial number theory, point lattices, convex polytopes and related complexes, and extremal problems in combinatorial geometry. The selection is a valuable reference for researchers interested in combinatorics.

Geometric Approximation Algorithms

Author : Sariel Har-Peled
Publisher : American Mathematical Soc.
ISBN 13 : 0821849115
Total Pages : 378 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Geometric Approximation Algorithms by : Sariel Har-Peled

Download or read book Geometric Approximation Algorithms written by Sariel Har-Peled and published by American Mathematical Soc.. This book was released on 2011 with total page 378 pages. Available in PDF, EPUB and Kindle. Book excerpt: Exact algorithms for dealing with geometric objects are complicated, hard to implement in practice, and slow. Over the last 20 years a theory of geometric approximation algorithms has emerged. These algorithms tend to be simple, fast, and more robust than their exact counterparts. This book is the first to cover geometric approximation algorithms in detail. In addition, more traditional computational geometry techniques that are widely used in developing such algorithms, like sampling, linear programming, etc., are also surveyed. Other topics covered include approximate nearest-neighbor search, shape approximation, coresets, dimension reduction, and embeddings. The topics covered are relatively independent and are supplemented by exercises. Close to 200 color figures are included in the text to illustrate proofs and ideas.

LEDA

Author : Kurt Mehlhorn
Publisher : Cambridge University Press
ISBN 13 : 9780521563291
Total Pages : 1050 pages
Book Rating : 4.5/5 (632 download)

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Book Synopsis LEDA by : Kurt Mehlhorn

Download or read book LEDA written by Kurt Mehlhorn and published by Cambridge University Press. This book was released on 1999-11-11 with total page 1050 pages. Available in PDF, EPUB and Kindle. Book excerpt: LEDA is a library of efficient data types and algorithms and a platform for combinatorial and geometric computing on which application programs can be built. In each of the core computer science areas of data structures, graph and network algorithms, and computational geometry, LEDA covers all (and more) that is found in the standard textbooks. LEDA is the first such library; it is written in C++ and is available on many types of machine. Whilst the software is freely available worldwide and is installed at hundreds of sites, this is the first book devoted to the library. Written by the main authors of LEDA, it is the definitive account, describing how the system is constructed and operates and how it can be used. The authors supply ample examples from a range of areas to show how the library can be used in practice, making the book essential for all workers in algorithms, data structures and computational geometry.

Combinatorial and Computational Geometry

Author : Jacob E. Goodman
Publisher : Cambridge University Press
ISBN 13 : 9780521848626
Total Pages : 640 pages
Book Rating : 4.8/5 (486 download)

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Book Synopsis Combinatorial and Computational Geometry by : Jacob E. Goodman

Download or read book Combinatorial and Computational Geometry written by Jacob E. Goodman and published by Cambridge University Press. This book was released on 2005-08-08 with total page 640 pages. Available in PDF, EPUB and Kindle. Book excerpt: This 2005 book deals with interest topics in Discrete and Algorithmic aspects of Geometry.

Handbook of Discrete and Computational Geometry, Second Edition

Author : Csaba D. Toth
Publisher : CRC Press
ISBN 13 : 1420035312
Total Pages : 1557 pages
Book Rating : 4.4/5 (2 download)

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Book Synopsis Handbook of Discrete and Computational Geometry, Second Edition by : Csaba D. Toth

Download or read book Handbook of Discrete and Computational Geometry, Second Edition written by Csaba D. Toth and published by CRC Press. This book was released on 2004-04-13 with total page 1557 pages. Available in PDF, EPUB and Kindle. Book excerpt: While high-quality books and journals in this field continue to proliferate, none has yet come close to matching the Handbook of Discrete and Computational Geometry, which in its first edition, quickly became the definitive reference work in its field. But with the rapid growth of the discipline and the many advances made over the past seven years, it's time to bring this standard-setting reference up to date. Editors Jacob E. Goodman and Joseph O'Rourke reassembled their stellar panel of contributors, added manymore, and together thoroughly revised their work to make the most important results and methods, both classic and cutting-edge, accessible in one convenient volume. Now over more then 1500 pages, the Handbook of Discrete and Computational Geometry, Second Edition once again provides unparalleled, authoritative coverage of theory, methods, and applications. Highlights of the Second Edition: Thirteen new chapters: Five on applications and others on collision detection, nearest neighbors in high-dimensional spaces, curve and surface reconstruction, embeddings of finite metric spaces, polygonal linkages, the discrepancy method, and geometric graph theory Thorough revisions of all remaining chapters Extended coverage of computational geometry software, now comprising two chapters: one on the LEDA and CGAL libraries, the other on additional software Two indices: An Index of Defined Terms and an Index of Cited Authors Greatly expanded bibliographies

Lectures on Discrete Geometry

Author : Jiri Matousek
Publisher : Springer Science & Business Media
ISBN 13 : 1461300398
Total Pages : 491 pages
Book Rating : 4.4/5 (613 download)

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Book Synopsis Lectures on Discrete Geometry by : Jiri Matousek

Download or read book Lectures on Discrete Geometry written by Jiri Matousek and published by Springer Science & Business Media. This book was released on 2013-12-01 with total page 491 pages. Available in PDF, EPUB and Kindle. Book excerpt: The main topics in this introductory text to discrete geometry include basics on convex sets, convex polytopes and hyperplane arrangements, combinatorial complexity of geometric configurations, intersection patterns and transversals of convex sets, geometric Ramsey-type results, and embeddings of finite metric spaces into normed spaces. In each area, the text explains several key results and methods.

Thirty-three Miniatures

Author : Jiří Matoušek
Publisher : American Mathematical Soc.
ISBN 13 : 0821849778
Total Pages : 196 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Thirty-three Miniatures by : Jiří Matoušek

Download or read book Thirty-three Miniatures written by Jiří Matoušek and published by American Mathematical Soc.. This book was released on 2010 with total page 196 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains a collection of clever mathematical applications of linear algebra, mainly in combinatorics, geometry, and algorithms. Each chapter covers a single main result with motivation and full proof in at most ten pages and can be read independently of all other chapters (with minor exceptions), assuming only a modest background in linear algebra. The topics include a number of well-known mathematical gems, such as Hamming codes, the matrix-tree theorem, the Lovasz bound on the Shannon capacity, and a counterexample to Borsuk's conjecture, as well as other, perhaps less popular but similarly beautiful results, e.g., fast associativity testing, a lemma of Steinitz on ordering vectors, a monotonicity result for integer partitions, or a bound for set pairs via exterior products. The simpler results in the first part of the book provide ample material to liven up an undergraduate course of linear algebra. The more advanced parts can be used for a graduate course of linear-algebraic methods or for seminar presentations. Table of Contents: Fibonacci numbers, quickly; Fibonacci numbers, the formula; The clubs of Oddtown; Same-size intersections; Error-correcting codes; Odd distances; Are these distances Euclidean?; Packing complete bipartite graphs; Equiangular lines; Where is the triangle?; Checking matrix multiplication; Tiling a rectangle by squares; Three Petersens are not enough; Petersen, Hoffman-Singleton, and maybe 57; Only two distances; Covering a cube minus one vertex; Medium-size intersection is hard to avoid; On the difficulty of reducing the diameter; The end of the small coins; Walking in the yard; Counting spanning trees; In how many ways can a man tile a board?; More bricks--more walls?; Perfect matchings and determinants; Turning a ladder over a finite field; Counting compositions; Is it associative?; The secret agent and umbrella; Shannon capacity of the union: a tale of two fields; Equilateral sets; Cutting cheaply using eigenvectors; Rotating the cube; Set pairs and exterior products; Index. (STML/53)

Computational Geometry

Author : Franco P. Preparata
Publisher : Springer Science & Business Media
ISBN 13 : 1461210984
Total Pages : 413 pages
Book Rating : 4.4/5 (612 download)

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Book Synopsis Computational Geometry by : Franco P. Preparata

Download or read book Computational Geometry written by Franco P. Preparata and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 413 pages. Available in PDF, EPUB and Kindle. Book excerpt: From the reviews: "This book offers a coherent treatment, at the graduate textbook level, of the field that has come to be known in the last decade or so as computational geometry. ... ... The book is well organized and lucidly written; a timely contribution by two founders of the field. It clearly demonstrates that computational geometry in the plane is now a fairly well-understood branch of computer science and mathematics. It also points the way to the solution of the more challenging problems in dimensions higher than two." #Mathematical Reviews#1 "... This remarkable book is a comprehensive and systematic study on research results obtained especially in the last ten years. The very clear presentation concentrates on basic ideas, fundamental combinatorial structures, and crucial algorithmic techniques. The plenty of results is clever organized following these guidelines and within the framework of some detailed case studies. A large number of figures and examples also aid the understanding of the material. Therefore, it can be highly recommended as an early graduate text but it should prove also to be essential to researchers and professionals in applied fields of computer-aided design, computer graphics, and robotics." #Biometrical Journal#2

Geometric Methods and Optimization Problems

Author : Vladimir Boltyanski
Publisher : Springer Science & Business Media
ISBN 13 : 9780792354543
Total Pages : 448 pages
Book Rating : 4.3/5 (545 download)

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Book Synopsis Geometric Methods and Optimization Problems by : Vladimir Boltyanski

Download or read book Geometric Methods and Optimization Problems written by Vladimir Boltyanski and published by Springer Science & Business Media. This book was released on 1998-12-31 with total page 448 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book focuses on three disciplines of applied mathematics: control theory, location science and computational geometry. The authors show how methods and tools from convex geometry in a wider sense can help solve various problems from these disciplines. More precisely they consider mainly the tent method (as an application of a generalized separation theory of convex cones) in nonclassical variational calculus, various median problems in Euclidean and other Minkowski spaces (including a detailed discussion of the Fermat-Torricelli problem) and different types of partitionings of topologically complicated polygonal domains into a minimum number of convex pieces. Figures are used extensively throughout the book and there is also a large collection of exercises. Audience: Graduate students, teachers and researchers.

Combinatorial Methods with Computer Applications

Author : Jonathan L. Gross
Publisher : CRC Press
ISBN 13 : 1584887435
Total Pages : 664 pages
Book Rating : 4.5/5 (848 download)

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Book Synopsis Combinatorial Methods with Computer Applications by : Jonathan L. Gross

Download or read book Combinatorial Methods with Computer Applications written by Jonathan L. Gross and published by CRC Press. This book was released on 2007-11-16 with total page 664 pages. Available in PDF, EPUB and Kindle. Book excerpt: Combinatorial Methods with Computer Applications provides in-depth coverage of recurrences, generating functions, partitions, and permutations, along with some of the most interesting graph and network topics, design constructions, and finite geometries. Requiring only a foundation in discrete mathematics, it can serve as the textbook in a combinatorial methods course or in a combined graph theory and combinatorics course. After an introduction to combinatorics, the book explores six systematic approaches within a comprehensive framework: sequences, solving recurrences, evaluating summation expressions, binomial coefficients, partitions and permutations, and integer methods. The author then focuses on graph theory, covering topics such as trees, isomorphism, automorphism, planarity, coloring, and network flows. The final chapters discuss automorphism groups in algebraic counting methods and describe combinatorial designs, including Latin squares, block designs, projective planes, and affine planes. In addition, the appendix supplies background material on relations, functions, algebraic systems, finite fields, and vector spaces. Paving the way for students to understand and perform combinatorial calculations, this accessible text presents the discrete methods necessary for applications to algorithmic analysis, performance evaluation, and statistics as well as for the solution of combinatorial problems in engineering and the social sciences.