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| algorithms in combinatorial geometry: Algorithms in Combinatorial Geometry Herbert Edelsbrunner, 1987-07-31 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. |
| algorithms in combinatorial geometry: Geometric Algorithms and Combinatorial Optimization Martin Grötschel, Laszlo Lovasz, Alexander Schrijver, 2012-12-06 Since the publication of the first edition of our book, geometric algorithms and combinatorial optimization have kept growing at the same fast pace as before. Nevertheless, we do not feel that the ongoing research has made this book outdated. Rather, it seems that many of the new results build on the models, algorithms, and theorems presented here. For instance, the celebrated Dyer-Frieze-Kannan algorithm for approximating the volume of a convex body is based on the oracle model of convex bodies and uses the ellipsoid method as a preprocessing technique. The polynomial time equivalence of optimization, separation, and membership has become a commonly employed tool in the study of the complexity of combinatorial optimization problems and in the newly developing field of computational convexity. Implementations of the basis reduction algorithm can be found in various computer algebra software systems. On the other hand, several of the open problems discussed in the first edition are still unsolved. For example, there are still no combinatorial polynomial time algorithms known for minimizing a submodular function or finding a maximum clique in a perfect graph. Moreover, despite the success of the interior point methods for the solution of explicitly given linear programs there is still no method known that solves implicitly given linear programs, such as those described in this book, and that is both practically and theoretically efficient. In particular, it is not known how to adapt interior point methods to such linear programs. |
| algorithms in combinatorial geometry: Combinatorial and Computational Geometry Jacob E. Goodman, Janos Pach, Emo Welzl, 2005-08-08 This 2005 book deals with interest topics in Discrete and Algorithmic aspects of Geometry. |
| algorithms in combinatorial geometry: Combinatorial Algorithms Donald L. Kreher, Douglas R. Stinson, 2020-09-24 This textbook thoroughly outlines combinatorial algorithms for generation, enumeration, and search. Topics include backtracking and heuristic search methods applied to various combinatorial structures, such as: Combinations Permutations Graphs Designs Many classical areas are covered as well as new research topics not included in most existing texts, such as: Group algorithms Graph isomorphism Hill-climbing Heuristic search algorithms This work serves as an exceptional textbook for a modern course in combinatorial algorithms, providing a unified and focused collection of recent topics of interest in the area. The authors, synthesizing material that can only be found scattered through many different sources, introduce the most important combinatorial algorithmic techniques - thus creating an accessible, comprehensive text that students of mathematics, electrical engineering, and computer science can understand without needing a prior course on combinatorics. |
| algorithms in combinatorial geometry: Computational Geometry Franco P. Preparata, Michael I. Shamos, 2012-12-06 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 |
| algorithms in combinatorial geometry: Combinatorial Geometry DIMACS (Group), J. Pach, P. K. Agarwal, 1991 |
| algorithms in combinatorial geometry: Geometry of Cuts and Metrics Michel Marie Deza, Monique Laurent, 1997-05-20 Cuts and metrics are well-known objects that arise - independently, but with many deep and fascinating connections - in diverse fields: in graph theory, combinatorial optimization, geometry of numbers, combinatorial matrix theory, statistical physics, VLSI design etc. This book presents a wealth of results, from different mathematical disciplines, in a unified comprehensive manner, and establishes new and old links, which cannot be found elsewhere. It provides a unique and invaluable source for researchers and graduate students. From the Reviews: This book is definitely a milestone in the literature of integer programming and combinatorial optimization. It draws from the Interdisciplinarity of these fields [...]. With knowledge about the relevant terms, one can enjoy special subsections without being entirely familiar with the rest of the chapter. This makes it not only an interesting research book but even a dictionary. [...] The longer one works with it, the more beautiful it becomes. Optima 56, 1997. |
| algorithms in combinatorial geometry: Algorithms in Structural Molecular Biology Bruce R. Donald, 2023-08-15 An overview of algorithms important to computational structural biology that addresses such topics as NMR and design and analysis of proteins.Using the tools of information technology to understand the molecular machinery of the cell offers both challenges and opportunities to computational scientists. Over the past decade, novel algorithms have been developed both for analyzing biological data and for synthetic biology problems such as protein engineering. This book explains the algorithmic foundations and computational approaches underlying areas of structural biology including NMR (nuclear magnetic resonance); X-ray crystallography; and the design and analysis of proteins, peptides, and small molecules. Each chapter offers a concise overview of important concepts, focusing on a key topic in the field. Four chapters offer a short course in algorithmic and computational issues related to NMR structural biology, giving the reader a useful toolkit with which to approach the fascinating yet thorny computational problems in this area. A recurrent theme is understanding the interplay between biophysical experiments and computational algorithms. The text emphasizes the mathematical foundations of structural biology while maintaining a balance between algorithms and a nuanced understanding of experimental data. Three emerging areas, particularly fertile ground for research students, are highlighted: NMR methodology, design of proteins and other molecules, and the modeling of protein flexibility. The next generation of computational structural biologists will need training in geometric algorithms, provably good approximation algorithms, scientific computation, and an array of techniques for handling noise and uncertainty in combinatorial geometry and computational biophysics. This book is an essential guide for young scientists on their way to research success in this exciting field. |
| algorithms in combinatorial geometry: Combinatorial Algorithms, Luděk Kučera, 1991 Combinatorial Algorithms is devoted to the solution of problems presented by the theory of graphs. This area of problems has been growing dramatically. Until now, the majority of results could only be found in specialized journals, technical reports and conference proceedings. Here for the first time, the subject is dealt with in a systematic manner in one book. Although directed primarily to students of computer science, it will also be useful to programmers and other workers in the area of computers. |
| algorithms in combinatorial geometry: Geometric Approximation Algorithms Sariel Har-Peled, 2011 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. |
| algorithms in combinatorial geometry: Handbook of Computational Geometry J.R. Sack, J. Urrutia, 1999-12-13 Computational Geometry is an area that provides solutions to geometric problems which arise in applications including Geographic Information Systems, Robotics and Computer Graphics. This Handbook provides an overview of key concepts and results in Computational Geometry. It may serve as a reference and study guide to the field. Not only the most advanced methods or solutions are described, but also many alternate ways of looking at problems and how to solve them. |
| algorithms in combinatorial geometry: Combinatorial Optimization and Graph Algorithms Takuro Fukunaga, Ken-ichi Kawarabayashi, 2017-10-02 Covering network designs, discrete convex analysis, facility location and clustering problems, matching games, and parameterized complexity, this book discusses theoretical aspects of combinatorial optimization and graph algorithms. Contributions are by renowned researchers who attended NII Shonan meetings on this essential topic. The collection contained here provides readers with the outcome of the authors’ research and productive meetings on this dynamic area, ranging from computer science and mathematics to operations research. Networks are ubiquitous in today's world: the Web, online social networks, and search-and-query click logs can lead to a graph that consists of vertices and edges. Such networks are growing so fast that it is essential to design algorithms to work for these large networks. Graph algorithms comprise an area in computer science that works to design efficient algorithms for networks. Here one can work on theoretical or practical problems where implementation of an algorithm for large networks is needed. In two of the chapters, recent results in graph matching games and fixed parameter tractability are surveyed. Combinatorial optimization is an intersection of operations research and mathematics, especially discrete mathematics, which deals with new questions and new problems, attempting to find an optimum object from a finite set of objects. Most problems in combinatorial optimization are not tractable (i.e., NP-hard). Therefore it is necessary to design an approximation algorithm for them. To tackle these problems requires the development and combination of ideas and techniques from diverse mathematical areas including complexity theory, algorithm theory, and matroids as well as graph theory, combinatorics, convex and nonlinear optimization, and discrete and convex geometry. Overall, the book presents recent progress in facility location, network design, and discrete convex analysis. |
| algorithms in combinatorial geometry: Forbidden Configurations in Discrete Geometry David Eppstein, 2018-05-17 Unifies discrete and computational geometry by using forbidden patterns of points to characterize many of its problems. |
| algorithms in combinatorial geometry: Digital and Discrete Geometry Li M. Chen, 2014-12-12 This book provides comprehensive coverage of the modern methods for geometric problems in the computing sciences. It also covers concurrent topics in data sciences including geometric processing, manifold learning, Google search, cloud data, and R-tree for wireless networks and BigData. The author investigates digital geometry and its related constructive methods in discrete geometry, offering detailed methods and algorithms. The book is divided into five sections: basic geometry; digital curves, surfaces and manifolds; discretely represented objects; geometric computation and processing; and advanced topics. Chapters especially focus on the applications of these methods to other types of geometry, algebraic topology, image processing, computer vision and computer graphics. Digital and Discrete Geometry: Theory and Algorithms targets researchers and professionals working in digital image processing analysis, medical imaging (such as CT and MRI) and informatics, computer graphics, computer vision, biometrics, and information theory. Advanced-level students in electrical engineering, mathematics, and computer science will also find this book useful as a secondary text book or reference. Praise for this book: This book does present a large collection of important concepts, of mathematical, geometrical, or algorithmical nature, that are frequently used in computer graphics and image processing. These concepts range from graphs through manifolds to homology. Of particular value are the sections dealing with discrete versions of classic continuous notions. The reader finds compact definitions and concise explanations that often appeal to intuition, avoiding finer, but then necessarily more complicated, arguments... As a first introduction, or as a reference for professionals working in computer graphics or image processing, this book should be of considerable value. - Prof. Dr. Rolf Klein, University of Bonn. |
| algorithms in combinatorial geometry: Combinatorial Algebraic Topology Dimitry Kozlov, 2007-12-29 This volume is the first comprehensive treatment of combinatorial algebraic topology in book form. The first part of the book constitutes a swift walk through the main tools of algebraic topology. Readers - graduate students and working mathematicians alike - will probably find particularly useful the second part, which contains an in-depth discussion of the major research techniques of combinatorial algebraic topology. Although applications are sprinkled throughout the second part, they are principal focus of the third part, which is entirely devoted to developing the topological structure theory for graph homomorphisms. |
| algorithms in combinatorial geometry: Combinatorial Geometry János Pach, Pankaj K. Agarwal, 2011-10-18 A complete, self-contained introduction to a powerful and resurgingmathematical discipline . Combinatorial Geometry presents andexplains with complete proofs some of the most important resultsand methods of this relatively young mathematical discipline,started by Minkowski, Fejes Toth, Rogers, and Erd???s. Nearly halfthe results presented in this book were discovered over the pasttwenty years, and most have never before appeared in any monograph.Combinatorial Geometry will be of particular interest tomathematicians, computer scientists, physicists, and materialsscientists interested in computational geometry, robotics, sceneanalysis, and computer-aided design. It is also a superb textbook,complete with end-of-chapter problems and hints to their solutionsthat help students clarify their understanding and test theirmastery of the material. Topics covered include: * Geometric number theory * Packing and covering with congruent convex disks * Extremal graph and hypergraph theory * Distribution of distances among finitely many points * Epsilon-nets and Vapnik--Chervonenkis dimension * Geometric graph theory * Geometric discrepancy theory * And much more |
| algorithms in combinatorial geometry: Computational Geometry Mark de Berg, Marc van Krefeld, Mark Overmars, Otfried Cheong, 2013-04-17 Computational geometry emerged from the field of algorithms design and anal ysis in the late 1970s. It has grown into a recognized discipline with its own journals, conferences, and a large community of active researchers. The suc cess of the field as a research discipline can on the one hand be explained from the beauty of the problems studied and the solutions obtained, and, on the other hand, by the many application domains-computer graphics, geographic in formation systems (GIS), robotics, and others-in which geometric algorithms playafundamental role. For many geometric problems the early algorithmic solutions were either slow or difficult to understand and implement. In recent years a number of new algorithmic techniques have been developed that improved and simplified many of the previous approaches. In this textbook we have tried to make these modem algorithmic solutions accessible to a large audience. The book has been written as a textbook for a course in computational geometry, but it can also be used for self-study. |
| algorithms in combinatorial geometry: Probabilistic Methods for Algorithmic Discrete Mathematics Michel Habib, 1998-08-19 The book gives an accessible account of modern pro- babilistic methods for analyzing combinatorial structures and algorithms. Each topic is approached in a didactic manner but the most recent developments are linked to the basic ma- terial. Extensive lists of references and a detailed index will make this a useful guide for graduate students and researchers. Special features included: - a simple treatment of Talagrand inequalities and their applications - an overview and many carefully worked out examples of the probabilistic analysis of combinatorial algorithms - a discussion of the exact simulation algorithm (in the context of Markov Chain Monte Carlo Methods) - a general method for finding asymptotically optimal or near optimal graph colouring, showing how the probabilistic method may be fine-tuned to explit the structure of the underlying graph - a succinct treatment of randomized algorithms and derandomization techniques |
| algorithms in combinatorial geometry: Leda Kurt Mehlhorn, 1999 |
| algorithms in combinatorial geometry: Combinatorial And Global Optimization Rainer E Burkard, Athanasios Migdalas, Panos M Pardalos, 2002-04-05 Combinatorial and global optimization problems appear in a wide range of applications in operations research, engineering, biological science, and computer science. In combinatorial optimization and graph theory, many approaches have been developed that link the discrete universe to the continuous universe through geometric, analytic, and algebraic techniques. Such techniques include global optimization formulations, semidefinite programming, and spectral theory. Recent major successes based on these approaches include interior point algorithms for linear and discrete problems, the celebrated Goemans-Williamson relaxation of the maximum cut problem, and the Du-Hwang solution of the Gilbert-Pollak conjecture. Since integer constraints are equivalent to nonconvex constraints, the fundamental difference between classes of optimization problems is not between discrete and continuous problems but between convex and nonconvex optimization problems. This volume is a selection of refereed papers based on talks presented at a conference on “Combinatorial and Global Optimization” held at Crete, Greece. |
| algorithms in combinatorial geometry: Handbook of Discrete and Computational Geometry Csaba D. Toth, Joseph O'Rourke, Jacob E. Goodman, 2017-11-22 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. |
| algorithms in combinatorial geometry: Combinatorial Geometry and Its Algorithmic Applications János Pach, Micha Sharir, 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. |
| algorithms in combinatorial geometry: Algorithms in Real Algebraic Geometry Saugata Basu, Richard Pollack, Marie-Françoise Coste-Roy, 2007-04-21 The algorithmic problems of real algebraic geometry such as real root counting, deciding the existence of solutions of systems of polynomial equations and inequalities, finding global maxima or deciding whether two points belong in the same connected component of a semi-algebraic set appear frequently in many areas of science and engineering. In this textbook the main ideas and techniques presented form a coherent and rich body of knowledge. Mathematicians will find relevant information about the algorithmic aspects. Researchers in computer science and engineering will find the required mathematical background. Being self-contained the book is accessible to graduate students and even, for invaluable parts of it, to undergraduate students. This second edition contains several recent results, on discriminants of symmetric matrices, real root isolation, global optimization, quantitative results on semi-algebraic sets and the first single exponential algorithm computing their first Betti number. |
| algorithms in combinatorial geometry: Computational Geometry in C Joseph O'Rourke, 1998-10-13 This is the revised and expanded 1998 edition of a popular introduction to the design and implementation of geometry algorithms arising in areas such as computer graphics, robotics, and engineering design. The basic techniques used in computational geometry are all covered: polygon triangulations, convex hulls, Voronoi diagrams, arrangements, geometric searching, and motion planning. The self-contained treatment presumes only an elementary knowledge of mathematics, but reaches topics on the frontier of current research, making it a useful reference for practitioners at all levels. The second edition contains material on several new topics, such as randomized algorithms for polygon triangulation, planar point location, 3D convex hull construction, intersection algorithms for ray-segment and ray-triangle, and point-in-polyhedron. The code in this edition is significantly improved from the first edition (more efficient and more robust), and four new routines are included. Java versions for this new edition are also available. All code is accessible from the book's Web site (http://cs.smith.edu/~orourke/) or by anonymous ftp. |
| algorithms in combinatorial geometry: Geometry and Topology for Mesh Generation Herbert Edelsbrunner, 2001-05-28 The book combines topics in mathematics (geometry and topology), computer science (algorithms), and engineering (mesh generation). The original motivation for these topics was the difficulty faced (both conceptually and in the technical execution) in any attempt to combine elements of combinatorial and of numerical algorithms. Mesh generation is a topic where a meaningful combination of these different approaches to problem solving is inevitable. The book develops methods from both areas that are amenable to combination, and explains recent breakthrough solutions to meshing that fit into this category.The book should be an ideal graduate text for courses on mesh generation. The specific material is selected giving preference to topics that are elementary, attractive, lend themselves to teaching, useful, and interesting. |
| algorithms in combinatorial geometry: Discrete and Computational Geometry Boris Aronov, Saugata Basu, Janos Pach, Micha Sharir, 2012-12-06 An impressive collection of original research papers in discrete and computational geometry, contributed by many leading researchers in these fields, as a tribute to Jacob E. Goodman and Richard Pollack, two of the ‘founding fathers’ of the area, on the occasion of their 2/3 x 100 birthdays. The topics covered by the 41 papers provide professionals and graduate students with a comprehensive presentation of the state of the art in most aspects of discrete and computational geometry, including geometric algorithms, study of arrangements, geometric graph theory, quantitative and algorithmic real algebraic geometry, with important connections to algebraic geometry, convexity, polyhedral combinatorics, the theory of packing, covering, and tiling. The book serves as an invaluable source of reference in this discipline. |
| algorithms in combinatorial geometry: Sparsity Jaroslav Nešetřil, Patrice Ossona de Mendez, 2012-04-24 This is the first book devoted to the systematic study of sparse graphs and sparse finite structures. Although the notion of sparsity appears in various contexts and is a typical example of a hard to define notion, the authors devised an unifying classification of general classes of structures. This approach is very robust and it has many remarkable properties. For example the classification is expressible in many different ways involving most extremal combinatorial invariants. This study of sparse structures found applications in such diverse areas as algorithmic graph theory, complexity of algorithms, property testing, descriptive complexity and mathematical logic (homomorphism preservation,fixed parameter tractability and constraint satisfaction problems). It should be stressed that despite of its generality this approach leads to linear (and nearly linear) algorithms. Jaroslav Nešetřil is a professor at Charles University, Prague; Patrice Ossona de Mendez is a CNRS researcher et EHESS, Paris. This book is related to the material presented by the first author at ICM 2010. |
| algorithms in combinatorial geometry: Discrete and Computational Geometry, 2nd Edition Satyan L. Devadoss, Joseph O'Rourke, 2025-07-08 The essential introduction to discrete and computational geometry—now fully updated and expanded Discrete and Computational Geometry bridges the theoretical world of discrete geometry with the applications-driven realm of computational geometry, offering a comprehensive yet accessible introduction to this cutting-edge frontier of mathematics and computer science. Beginning with polygons and ending with polyhedra, it explains how to capture the shape of data given by a set of points, from convex hulls and triangulations to Voronoi diagrams, geometric duality, chains, linkages, and alpha complexes. Connections to real-world applications are made throughout, and algorithms are presented independent of any programming language. Now fully updated and expanded, this richly illustrated textbook is an invaluable learning tool for students in mathematics, computer science, engineering, and physics. Now with new sections on duality and on computational topology Project suggestions at the end of every chapter Covers traditional topics as well as new and advanced material Features numerous full-color illustrations, exercises, and fully updated unsolved problems Uniquely designed for a one-semester class Accessible to college sophomores with minimal background Also suitable for more advanced students Online solutions manual (available to instructors) |
| algorithms in combinatorial geometry: Art Gallery Theorems and Algorithms Joseph O'Rourke, 1987 Art gallery theorems and algorithms are so called because they relate to problems involving the visibility of geometrical shapes and their internal surfaces. This book explores generalizations and specializations in these areas. Among the presentations are recently discovered theorems on orthogonal polygons, polygons with holes, exterior visibility, visibility graphs, and visibility in three dimensions. The author formulates many open problems and offers several conjectures, providing arguments which may be followed by anyone familiar with basic graph theory and algorithms. This work may be applied to robotics and artificial intelligence as well as other fields, and will be especially useful to computer scientists working with computational and combinatorial geometry. |
| algorithms in combinatorial geometry: Combinatorial Geometry and Its Algorithmic Applications Jâanos Pach, 2014-05-21 |
| algorithms in combinatorial geometry: Combinatorial Algorithms Herbert S. Wilf, 1989-01-01 Covers key recent advances in combinatorial algorithms. |
| algorithms in combinatorial geometry: New Trends in Discrete and Computational Geometry Janos Pach, 2012-12-06 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. |
| algorithms in combinatorial geometry: Combinatorial Optimization Bernhard Korte, Jens Vygen, 2009-09-02 This well-written textbook on combinatorial optimization puts special emphasis on theoretical results and algorithms with provably good performance, in contrast to heuristics. The book contains complete (but concise) proofs, as well as many deep results, some of which have not appeared in any previous books. |
| algorithms in combinatorial geometry: Combinatorial Geometry and Its Algorithmic Applications János Pach, Micha Sharir, 2009 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: Computing in Euclidean Geometry Ding-Zhu Du, Frank Hwang, 1995 This book is a collection of surveys and exploratory articles about recent developments in the field of computational Euclidean geometry. Topics covered include the history of Euclidean geometry, Voronoi diagrams, randomized geometric algorithms, computational algebra, triangulations, machine proofs, topological designs, finite-element mesh, computer-aided geometric designs and Steiner trees. This second edition contains three new surveys covering geometric constraint solving, computational geometry and the exact computation paradigm. |
| algorithms in combinatorial geometry: Combinatorial Optimization Alexander Schrijver, 2003-02-12 This book offers an in-depth overview of polyhedral methods and efficient algorithms in combinatorial optimization.These methods form a broad, coherent and powerful kernel in combinatorial optimization, with strong links to discrete mathematics, mathematical programming and computer science. In eight parts, various areas are treated, each starting with an elementary introduction to the area, with short, elegant proofs of the principal results, and each evolving to the more advanced methods and results, with full proofs of some of the deepest theorems in the area. Over 4000 references to further research are given, and historical surveys on the basic subjects are presented. |
| algorithms in combinatorial geometry: Combinatorial Optimization William Cook, László Lovász, Paul D. Seymour, 1995 This book grew out of the fourth Special Year at DIMACS, which was devoted to the subject of combinatorial optimization. During the special year, a number of workshops, small and large, dealt with various aspects of this theme. Organizers of the workshops and selected participants were asked to write surveys about the hottest results and ideas in their fields. Therefore, this book is not a set of conference proceedings but rather a carefully refereed collection of invited survey articles written by outstanding researchers. Aimed at researchers in discrete mathematics, operations research, and the theory of computing, this book offers an in-depth look at many topics not treated in textbooks. |
| algorithms in combinatorial geometry: Geometric Folding Algorithms Erik D. Demaine, Joseph O'Rourke, 2008-08-21 How can linkages, pieces of paper, and polyhedra be folded? The authors present hundreds of results and over 60 unsolved 'open problems' in this comprehensive look at the mathematics of folding, with an emphasis on algorithmic or computational aspects. Folding and unfolding problems have been implicit since Albrecht Dürer in the early 1500s, but have only recently been studied in the mathematical literature. Over the past decade, there has been a surge of interest in these problems, with applications ranging from robotics to protein folding. A proof shows that it is possible to design a series of jointed bars moving only in a flat plane that can sign a name or trace any other algebraic curve. One remarkable algorithm shows you can fold any straight-line drawing on paper so that the complete drawing can be cut out with one straight scissors cut. Aimed primarily at advanced undergraduate and graduate students in mathematics or computer science, this lavishly illustrated book will fascinate a broad audience, from high school students to researchers. |
| algorithms in combinatorial geometry: Beyond Planar Graphs Seok-Hee Hong, Takeshi Tokuyama, 2020-09-30 This book is the first general and extensive review on the algorithmics and mathematical results of beyond planar graphs. Most real-world data sets are relational and can be modelled as graphs consisting of vertices and edges. Planar graphs are fundamental for both graph theory and graph algorithms and are extensively studied. Structural properties and fundamental algorithms for planar graphs have been discovered. However, most real-world graphs, such as social networks and biological networks, are non-planar. To analyze and visualize such real-world networks, it is necessary to solve fundamental mathematical and algorithmic research questions on sparse non-planar graphs, called beyond planar graphs.This book is based on the National Institute of Informatics (NII) Shonan Meeting on algorithmics on beyond planar graphs held in Japan in November, 2016. The book consists of 13 chapters that represent recent advances in various areas of beyond planar graph research. The main aims and objectives of this book include 1) to timely provide a state-of-the-art survey and a bibliography on beyond planar graphs; 2) to set the research agenda on beyond planar graphs by identifying fundamental research questions and new research directions; and 3) to foster cross-disciplinary research collaboration between computer science (graph drawing and computational geometry) and mathematics (graph theory and combinatorics). New algorithms for beyond planar graphs will be in high demand by practitioners in various application domains to solve complex visualization problems. This book therefore will be a valuable resource for researchers in graph theory, algorithms, and theoretical computer science, and will stimulate further deep scientific investigations into many areas of beyond planar graphs. |
Algorithm - Wikipedia
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This course is an introduction to mathematical modeling of computational problems, as well as common algorithms, algorithmic paradigms, and data structures used to solve these problems. …
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Algorithm - Wikipedia
Algorithms are used as specifications for performing calculations and data processing. More advanced algorithms can use conditionals to divert the code execution through various routes …
What is an Algorithm | Introduction to Algorithms
Apr 29, 2025 · Algorithms are necessary for solving complex problems efficiently and effectively. They help to automate processes and make them more reliable, faster, and easier to perform. …
What Is an Algorithm? | Definition & Examples - Scribbr
Aug 9, 2023 · Algorithms are fundamental tools for problem-solving in both the digital world and many real-life scenarios. Each time we try to solve a problem by breaking it down into smaller, …
What is an algorithm? | TechTarget
Jul 29, 2024 · An algorithm is a procedure used for solving a problem or performing a computation. Algorithms act as an exact list of instructions that conduct specified actions step …
Algorithm | Definition, Types, & Facts | Britannica
Apr 22, 2025 · Algorithms exist for many such infinite classes of questions; Euclid’s Elements, published about 300 bce, contained one for finding the greatest common divisor of two natural …
Algorithms, 4th Edition by Robert Sedgewick and Kevin Wayne
Sep 26, 2024 · The textbook Algorithms, 4th Edition by Robert Sedgewick and Kevin Wayne [ Amazon · Pearson · InformIT] surveys the most important algorithms and data structures in …
What is an algorithm? Definition, structure and examples
Dec 11, 2024 · In machine learning, algorithms refer to the set of instructions to build a machine learning model. Models are at the core of machine learning and they can recognize patterns, …
Introduction to Algorithms | Electrical Engineering and Computer ...
This course is an introduction to mathematical modeling of computational problems, as well as common algorithms, algorithmic paradigms, and data structures used to solve these problems. …
What is an Algorithm? Definition, Types, Implementation
Sep 28, 2023 · Understanding the different types of algorithms can help in selecting the most appropriate one for solving a specific problem. Broadly, we can categorize algorithms based on …
What Is an Algorithm? | HowStuffWorks
Mar 5, 2024 · In today's digital age, algorithms are the invisible architects shaping the landscape of technology and information. These complex sets of rules and instructions underpin almost …
