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| big o discrete math: Foundations of Discrete Mathematics with Algorithms and Programming R. Balakrishnan, Sriraman Sridharan, 2018-10-26 Discrete Mathematics has permeated the whole of mathematics so much so it has now come to be taught even at the high school level. This book presents the basics of Discrete Mathematics and its applications to day-to-day problems in several areas. This book is intended for undergraduate students of Computer Science, Mathematics and Engineering. A number of examples have been given to enhance the understanding of concepts. The programming languages used are Pascal and C. |
| big o discrete math: Discrete Mathematics with Proof Eric Gossett, 2009-06-22 A Trusted Guide to Discrete Mathematics with Proof?Now in a Newly Revised Edition Discrete mathematics has become increasingly popular in recent years due to its growing applications in the field of computer science. Discrete Mathematics with Proof, Second Edition continues to facilitate an up-to-date understanding of this important topic, exposing readers to a wide range of modern and technological applications. The book begins with an introductory chapter that provides an accessible explanation of discrete mathematics. Subsequent chapters explore additional related topics including counting, finite probability theory, recursion, formal models in computer science, graph theory, trees, the concepts of functions, and relations. Additional features of the Second Edition include: An intense focus on the formal settings of proofs and their techniques, such as constructive proofs, proof by contradiction, and combinatorial proofs New sections on applications of elementary number theory, multidimensional induction, counting tulips, and the binomial distribution Important examples from the field of computer science presented as applications including the Halting problem, Shannon's mathematical model of information, regular expressions, XML, and Normal Forms in relational databases Numerous examples that are not often found in books on discrete mathematics including the deferred acceptance algorithm, the Boyer-Moore algorithm for pattern matching, Sierpinski curves, adaptive quadrature, the Josephus problem, and the five-color theorem Extensive appendices that outline supplemental material on analyzing claims and writing mathematics, along with solutions to selected chapter exercises Combinatorics receives a full chapter treatment that extends beyond the combinations and permutations material by delving into non-standard topics such as Latin squares, finite projective planes, balanced incomplete block designs, coding theory, partitions, occupancy problems, Stirling numbers, Ramsey numbers, and systems of distinct representatives. A related Web site features animations and visualizations of combinatorial proofs that assist readers with comprehension. In addition, approximately 500 examples and over 2,800 exercises are presented throughout the book to motivate ideas and illustrate the proofs and conclusions of theorems. Assuming only a basic background in calculus, Discrete Mathematics with Proof, Second Edition is an excellent book for mathematics and computer science courses at the undergraduate level. It is also a valuable resource for professionals in various technical fields who would like an introduction to discrete mathematics. |
| big o discrete math: Discrete Mathematics and Its Applications Kenneth H. Rosen, 2018-05 A precise, relevant, comprehensive approach to mathematical concepts... |
| big o discrete math: Fundamentals of Discrete Math for Computer Science Tom Jenkyns, Ben Stephenson, 2012-10-16 This textbook provides an engaging and motivational introduction to traditional topics in discrete mathematics, in a manner specifically designed to appeal to computer science students. The text empowers students to think critically, to be effective problem solvers, to integrate theory and practice, and to recognize the importance of abstraction. Clearly structured and interactive in nature, the book presents detailed walkthroughs of several algorithms, stimulating a conversation with the reader through informal commentary and provocative questions. Features: no university-level background in mathematics required; ideally structured for classroom-use and self-study, with modular chapters following ACM curriculum recommendations; describes mathematical processes in an algorithmic manner; contains examples and exercises throughout the text, and highlights the most important concepts in each section; selects examples that demonstrate a practical use for the concept in question. |
| big o discrete math: Discrete Mathematics for Computer Science Gary Haggard, John Schlipf, Sue Whitesides, 2006 Master the fundamentals of discrete mathematics with DISCRETE MATHEMATICS FOR COMPUTER SCIENCE with Student Solutions Manual CD-ROM! An increasing number of computer scientists from diverse areas are using discrete mathematical structures to explain concepts and problems and this mathematics text shows you how to express precise ideas in clear mathematical language. Through a wealth of exercises and examples, you will learn how mastering discrete mathematics will help you develop important reasoning skills that will continue to be useful throughout your career. |
| big o discrete math: Discrete Mathematics Oscar Levin, 2016-08-16 This gentle introduction to discrete mathematics is written for first and second year math majors, especially those who intend to teach. The text began as a set of lecture notes for the discrete mathematics course at the University of Northern Colorado. This course serves both as an introduction to topics in discrete math and as the introduction to proof course for math majors. The course is usually taught with a large amount of student inquiry, and this text is written to help facilitate this. Four main topics are covered: counting, sequences, logic, and graph theory. Along the way proofs are introduced, including proofs by contradiction, proofs by induction, and combinatorial proofs. The book contains over 360 exercises, including 230 with solutions and 130 more involved problems suitable for homework. There are also Investigate! activities throughout the text to support active, inquiry based learning. While there are many fine discrete math textbooks available, this text has the following advantages: It is written to be used in an inquiry rich course. It is written to be used in a course for future math teachers. It is open source, with low cost print editions and free electronic editions. |
| big o discrete math: Algorithms Unlocked Thomas H. Cormen, 2013-03-01 For anyone who has ever wondered how computers solve problems, an engagingly written guide for nonexperts to the basics of computer algorithms. Have you ever wondered how your GPS can find the fastest way to your destination, selecting one route from seemingly countless possibilities in mere seconds? How your credit card account number is protected when you make a purchase over the Internet? The answer is algorithms. And how do these mathematical formulations translate themselves into your GPS, your laptop, or your smart phone? This book offers an engagingly written guide to the basics of computer algorithms. In Algorithms Unlocked, Thomas Cormen—coauthor of the leading college textbook on the subject—provides a general explanation, with limited mathematics, of how algorithms enable computers to solve problems. Readers will learn what computer algorithms are, how to describe them, and how to evaluate them. They will discover simple ways to search for information in a computer; methods for rearranging information in a computer into a prescribed order (“sorting”); how to solve basic problems that can be modeled in a computer with a mathematical structure called a “graph” (useful for modeling road networks, dependencies among tasks, and financial relationships); how to solve problems that ask questions about strings of characters such as DNA structures; the basic principles behind cryptography; fundamentals of data compression; and even that there are some problems that no one has figured out how to solve on a computer in a reasonable amount of time. |
| big o discrete math: Computational Complexity Sanjeev Arora, Boaz Barak, 2009-04-20 New and classical results in computational complexity, including interactive proofs, PCP, derandomization, and quantum computation. Ideal for graduate students. |
| big o discrete math: Practical Discrete Mathematics Ryan T. White, Archana Tikayat Ray, 2021-02-22 A practical guide simplifying discrete math for curious minds and demonstrating its application in solving problems related to software development, computer algorithms, and data science Key FeaturesApply the math of countable objects to practical problems in computer scienceExplore modern Python libraries such as scikit-learn, NumPy, and SciPy for performing mathematicsLearn complex statistical and mathematical concepts with the help of hands-on examples and expert guidanceBook Description Discrete mathematics deals with studying countable, distinct elements, and its principles are widely used in building algorithms for computer science and data science. The knowledge of discrete math concepts will help you understand the algorithms, binary, and general mathematics that sit at the core of data-driven tasks. Practical Discrete Mathematics is a comprehensive introduction for those who are new to the mathematics of countable objects. This book will help you get up to speed with using discrete math principles to take your computer science skills to a more advanced level. As you learn the language of discrete mathematics, you'll also cover methods crucial to studying and describing computer science and machine learning objects and algorithms. The chapters that follow will guide you through how memory and CPUs work. In addition to this, you'll understand how to analyze data for useful patterns, before finally exploring how to apply math concepts in network routing, web searching, and data science. By the end of this book, you'll have a deeper understanding of discrete math and its applications in computer science, and be ready to work on real-world algorithm development and machine learning. What you will learnUnderstand the terminology and methods in discrete math and their usage in algorithms and data problemsUse Boolean algebra in formal logic and elementary control structuresImplement combinatorics to measure computational complexity and manage memory allocationUse random variables, calculate descriptive statistics, and find average-case computational complexitySolve graph problems involved in routing, pathfinding, and graph searches, such as depth-first searchPerform ML tasks such as data visualization, regression, and dimensionality reductionWho this book is for This book is for computer scientists looking to expand their knowledge of discrete math, the core topic of their field. University students looking to get hands-on with computer science, mathematics, statistics, engineering, or related disciplines will also find this book useful. Basic Python programming skills and knowledge of elementary real-number algebra are required to get started with this book. |
| big o discrete math: Algorithm and Data Structures M.M Raghuwanshi, 2016-01-05 ALGORITHMS AND DATA STRUCTURES is primarily designed for use in a first undergraduate course on algorithms, but it can also be used as the basis for an introductory graduate course, for researchers, or computer professionals who want to get and sense for how they might be able to use particular data structure and algorithm design techniques in the context of their own work.The goal of this book is to convey this approach to algorithms, as a design process that begins with problems arising across the full range of computing applications, builds on an understanding of algorithm design techniques, and results in the development of efficient solutions to these problems. It seek to explore the role of algorithmic ideas in computer science generally, and relate these ideas to the range of precisely formulated problems for which we can design and analyze algorithm. |
| big o discrete math: A History of Algorithms Jean-Luc Chabert, 2012-12-06 The development of computing has reawakened interest in algorithms. Often neglected by historians and modern scientists, algorithmic procedures have been instrumental in the development of fundamental ideas: practice led to theory just as much as the other way round. The purpose of this book is to offer a historical background to contemporary algorithmic practice. |
| big o discrete math: Introductory Discrete Mathematics V. K . Balakrishnan, 2012-04-30 This concise, undergraduate-level text focuses on combinatorics, graph theory with applications to some standard network optimization problems, and algorithms. More than 200 exercises, many with complete solutions. 1991 edition. |
| big o discrete math: Applied Asymptotic Analysis Peter David Miller, 2006 This book is a survey of asymptotic methods set in the current applied research context of wave propagation. It stresses rigorous analysis in addition to formal manipulations. Asymptotic expansions developed in the text are justified rigorously, and students are shown how to obtain solid error estimates for asymptotic formulae. The book relates examples and exercises to subjects of current research interest, such as the problem of locating the zeros of Taylor polynomials of entirenonvanishing functions and the problem of counting integer lattice points in subsets of the plane with various geometrical properties of the boundary. The book is intended for a beginning graduate course on asymptotic analysis in applied mathematics and is aimed at students of pure and appliedmathematics as well as science and engineering. The basic prerequisite is a background in differential equations, linear algebra, advanced calculus, and complex variables at the level of introductory undergraduate courses on these subjects. The book is ideally suited to the needs of a graduate student who, on the one hand, wants to learn basic applied mathematics, and on the other, wants to understand what is needed to make the various arguments rigorous. Down here in the Village, this is knownas the Courant point of view!! --Percy Deift, Courant Institute, New York Peter D. Miller is an associate professor of mathematics at the University of Michigan at Ann Arbor. He earned a Ph.D. in Applied Mathematics from the University of Arizona and has held positions at the Australian NationalUniversity (Canberra) and Monash University (Melbourne). His current research interests lie in singular limits for integrable systems. |
| big o discrete math: Street-Fighting Mathematics Sanjoy Mahajan, 2010-03-05 An antidote to mathematical rigor mortis, teaching how to guess answers without needing a proof or an exact calculation. In problem solving, as in street fighting, rules are for fools: do whatever works—don't just stand there! Yet we often fear an unjustified leap even though it may land us on a correct result. Traditional mathematics teaching is largely about solving exactly stated problems exactly, yet life often hands us partly defined problems needing only moderately accurate solutions. This engaging book is an antidote to the rigor mortis brought on by too much mathematical rigor, teaching us how to guess answers without needing a proof or an exact calculation. In Street-Fighting Mathematics, Sanjoy Mahajan builds, sharpens, and demonstrates tools for educated guessing and down-and-dirty, opportunistic problem solving across diverse fields of knowledge—from mathematics to management. Mahajan describes six tools: dimensional analysis, easy cases, lumping, picture proofs, successive approximation, and reasoning by analogy. Illustrating each tool with numerous examples, he carefully separates the tool—the general principle—from the particular application so that the reader can most easily grasp the tool itself to use on problems of particular interest. Street-Fighting Mathematics grew out of a short course taught by the author at MIT for students ranging from first-year undergraduates to graduate students ready for careers in physics, mathematics, management, electrical engineering, computer science, and biology. They benefited from an approach that avoided rigor and taught them how to use mathematics to solve real problems. Street-Fighting Mathematics will appear in print and online under a Creative Commons Noncommercial Share Alike license. |
| big o discrete math: Algorithms from THE BOOK Kenneth Lange, 2020-05-04 Algorithms are a dominant force in modern culture, and every indication is that they will become more pervasive, not less. The best algorithms are undergirded by beautiful mathematics. This text cuts across discipline boundaries to highlight some of the most famous and successful algorithms. Readers are exposed to the principles behind these examples and guided in assembling complex algorithms from simpler building blocks. Written in clear, instructive language within the constraints of mathematical rigor, Algorithms from THE BOOK includes a large number of classroom-tested exercises at the end of each chapter. The appendices cover background material often omitted from undergraduate courses. Most of the algorithm descriptions are accompanied by Julia code, an ideal language for scientific computing. This code is immediately available for experimentation. Algorithms from THE BOOK is aimed at first-year graduate and advanced undergraduate students. It will also serve as a convenient reference for professionals throughout the mathematical sciences, physical sciences, engineering, and the quantitative sectors of the biological and social sciences. |
| big o discrete math: Concrete Mathematics Ronald L. Graham, Donald E. Knuth, Oren Patashnik, 1994-02-28 This book introduces the mathematics that supports advanced computer programming and the analysis of algorithms. The primary aim of its well-known authors is to provide a solid and relevant base of mathematical skills - the skills needed to solve complex problems, to evaluate horrendous sums, and to discover subtle patterns in data. It is an indispensable text and reference not only for computer scientists - the authors themselves rely heavily on it! - but for serious users of mathematics in virtually every discipline. Concrete Mathematics is a blending of CONtinuous and disCRETE mathematics. More concretely, the authors explain, it is the controlled manipulation of mathematical formulas, using a collection of techniques for solving problems. The subject matter is primarily an expansion of the Mathematical Preliminaries section in Knuth's classic Art of Computer Programming, but the style of presentation is more leisurely, and individual topics are covered more deeply. Several new topics have been added, and the most significant ideas have been traced to their historical roots. The book includes more than 500 exercises, divided into six categories. Complete answers are provided for all exercises, except research problems, making the book particularly valuable for self-study. Major topics include: Sums Recurrences Integer functions Elementary number theory Binomial coefficients Generating functions Discrete probability Asymptotic methods This second edition includes important new material about mechanical summation. In response to the widespread use of the first edition as a reference book, the bibliography and index have also been expanded, and additional nontrivial improvements can be found on almost every page. Readers will appreciate the informal style of Concrete Mathematics. Particularly enjoyable are the marginal graffiti contributed by students who have taken courses based on this material. The authors want to convey not only the importance of the techniques presented, but some of the fun in learning and using them. |
| big o discrete math: The Algorithm Design Manual Steven S Skiena, 2009-04-05 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 |
| big o discrete math: Discrete Mathematics with Applications Thomas Koshy, 2004-01-19 This approachable text studies discrete objects and the relationsips that bind them. It helps students understand and apply the power of discrete math to digital computer systems and other modern applications. It provides excellent preparation for courses in linear algebra, number theory, and modern/abstract algebra and for computer science courses in data structures, algorithms, programming languages, compilers, databases, and computation.* Covers all recommended topics in a self-contained, comprehensive, and understandable format for students and new professionals * Emphasizes problem-solving techniques, pattern recognition, conjecturing, induction, applications of varying nature, proof techniques, algorithm development and correctness, and numeric computations* Weaves numerous applications into the text* Helps students learn by doing with a wealth of examples and exercises: - 560 examples worked out in detail - More than 3,700 exercises - More than 150 computer assignments - More than 600 writing projects* Includes chapter summaries of important vocabulary, formulas, and properties, plus the chapter review exercises* Features interesting anecdotes and biographies of 60 mathematicians and computer scientists* Instructor's Manual available for adopters* Student Solutions Manual available separately for purchase (ISBN: 0124211828) |
| big o discrete math: The Essence of Discrete Mathematics Neville Dean, 1997 Presenting a gentle introduction to all the basics of discrete mathematics, this book introduces sets, propositional logic, predicate logic, and mathematical models. It discusses relations, including homogeneous relations. |
| big o discrete math: Applied Discrete Structures Ken Levasseur, Al Doerr, 2012-02-25 ''In writing this book, care was taken to use language and examples that gradually wean students from a simpleminded mechanical approach and move them toward mathematical maturity. We also recognize that many students who hesitate to ask for help from an instructor need a readable text, and we have tried to anticipate the questions that go unasked. The wide range of examples in the text are meant to augment the favorite examples that most instructors have for teaching the topcs in discrete mathematics. To provide diagnostic help and encouragement, we have included solutions and/or hints to the odd-numbered exercises. These solutions include detailed answers whenever warranted and complete proofs, not just terse outlines of proofs. Our use of standard terminology and notation makes Applied Discrete Structures a valuable reference book for future courses. Although many advanced books have a short review of elementary topics, they cannot be complete. The text is divided into lecture-length sections, facilitating the organization of an instructor's presentation.Topics are presented in such a way that students' understanding can be monitored through thought-provoking exercises. The exercises require an understanding of the topics and how they are interrelated, not just a familiarity with the key words. An Instructor's Guide is available to any instructor who uses the text. It includes: Chapter-by-chapter comments on subtopics that emphasize the pitfalls to avoid; Suggested coverage times; Detailed solutions to most even-numbered exercises; Sample quizzes, exams, and final exams. This textbook has been used in classes at Casper College (WY), Grinnell College (IA), Luzurne Community College (PA), University of the Puget Sound (WA).''-- |
| big o discrete math: The Probabilistic Method Noga Alon, Joel H. Spencer, 2015-11-02 Praise for the Third Edition “Researchers of any kind of extremal combinatorics or theoretical computer science will welcome the new edition of this book.” - MAA Reviews Maintaining a standard of excellence that establishes The Probabilistic Method as the leading reference on probabilistic methods in combinatorics, the Fourth Edition continues to feature a clear writing style, illustrative examples, and illuminating exercises. The new edition includes numerous updates to reflect the most recent developments and advances in discrete mathematics and the connections to other areas in mathematics, theoretical computer science, and statistical physics. Emphasizing the methodology and techniques that enable problem-solving, The Probabilistic Method, Fourth Edition begins with a description of tools applied to probabilistic arguments, including basic techniques that use expectation and variance as well as the more advanced applications of martingales and correlation inequalities. The authors explore where probabilistic techniques have been applied successfully and also examine topical coverage such as discrepancy and random graphs, circuit complexity, computational geometry, and derandomization of randomized algorithms. Written by two well-known authorities in the field, the Fourth Edition features: Additional exercises throughout with hints and solutions to select problems in an appendix to help readers obtain a deeper understanding of the best methods and techniques New coverage on topics such as the Local Lemma, Six Standard Deviations result in Discrepancy Theory, Property B, and graph limits Updated sections to reflect major developments on the newest topics, discussions of the hypergraph container method, and many new references and improved results The Probabilistic Method, Fourth Edition is an ideal textbook for upper-undergraduate and graduate-level students majoring in mathematics, computer science, operations research, and statistics. The Fourth Edition is also an excellent reference for researchers and combinatorists who use probabilistic methods, discrete mathematics, and number theory. Noga Alon, PhD, is Baumritter Professor of Mathematics and Computer Science at Tel Aviv University. He is a member of the Israel National Academy of Sciences and Academia Europaea. A coeditor of the journal Random Structures and Algorithms, Dr. Alon is the recipient of the Polya Prize, The Gödel Prize, The Israel Prize, and the EMET Prize. Joel H. Spencer, PhD, is Professor of Mathematics and Computer Science at the Courant Institute of New York University. He is the cofounder and coeditor of the journal Random Structures and Algorithms and is a Sloane Foundation Fellow. Dr. Spencer has written more than 200 published articles and is the coauthor of Ramsey Theory, Second Edition, also published by Wiley. |
| big o discrete math: Discrete and Computational Geometry Satyan L. Devadoss, Joseph O'Rourke, 2011-04-11 An essential introduction to discrete and computational geometry Discrete geometry is a relatively new development in pure mathematics, while computational geometry is an emerging area in applications-driven computer science. Their intermingling has yielded exciting advances in recent years, yet what has been lacking until now is an undergraduate textbook that bridges the gap between the two. Discrete and Computational Geometry offers a comprehensive yet accessible introduction to this cutting-edge frontier of mathematics and computer science. This book covers traditional topics such as convex hulls, triangulations, and Voronoi diagrams, as well as more recent subjects like pseudotriangulations, curve reconstruction, and locked chains. It also touches on more advanced material, including Dehn invariants, associahedra, quasigeodesics, Morse theory, and the recent resolution of the Poincaré conjecture. Connections to real-world applications are made throughout, and algorithms are presented independently of any programming language. This richly illustrated textbook also features numerous exercises and unsolved problems. The essential introduction to discrete and computational geometry Covers traditional topics as well as new and advanced material Features numerous full-color illustrations, exercises, and unsolved problems Suitable for sophomores in mathematics, computer science, engineering, or physics Rigorous but accessible An online solutions manual is available (for teachers only) |
| big o discrete math: Discrete Mathematics Babu Ram, 2012 Discrete Mathematics will be of use to any undergraduate as well as post graduate courses in Computer Science and Mathematics. The syllabi of all these courses have been studied in depth and utmost care has been taken to ensure that all the essential topics in discrete structures are adequately emphasized. The book will enable the students to develop the requisite computational skills needed in software engineering. |
| big o discrete math: Fundamentals of Discrete Math for Computer Science Tom Jenkyns, Ben Stephenson, 2012-08-28 This textbook provides an engaging and motivational introduction to traditional topics in discrete mathematics, in a manner specifically designed to appeal to computer science students. The text empowers students to think critically, to be effective problem solvers, to integrate theory and practice, and to recognize the importance of abstraction. Clearly structured and interactive in nature, the book presents detailed walkthroughs of several algorithms, stimulating a conversation with the reader through informal commentary and provocative questions. Features: no university-level background in mathematics required; ideally structured for classroom-use and self-study, with modular chapters following ACM curriculum recommendations; describes mathematical processes in an algorithmic manner; contains examples and exercises throughout the text, and highlights the most important concepts in each section; selects examples that demonstrate a practical use for the concept in question. |
| big o discrete math: Algorithms and Complexity Herbert S. Wilf, 2020-09-30 This book is an introductory textbook on the design and analysis of algorithms. The author uses a careful selection of a few topics to illustrate the tools for algorithm analysis. Recursive algorithms are illustrated by Quicksort, FFT, fast matrix multiplications, and others. Algorithms associated with the network flow problem are fundamental in many areas of graph connectivity, matching theory, etc. Algorithms in number theory are discussed with some applications to public key encryption. This second edition will differ from the present edition mainly in that solutions to most of the exercises will be included. |
| big o discrete math: An Introduction to Discrete Mathematics Steven Roman, 1989 Intended for a one-term course in discrete mathematics, to prepare freshmen and sophomores for further work in computer science as well as mathematics. Sets, proof techniques, logic, combinatorics, and graph theory are covered in concise form. All topics are motivated by concrete examples, often emphasizing the interplay between computer science and mathematics. Examples also illustrate all definitions. Applications and references cover a wide variety of realistic situations. Coverage of mathematical induction includes the stroung form of induction, and new sections have been added on nonhomogeneous recurrence relations and the essentials of probability. |
| big o discrete math: Introduction To Algorithms Thomas H Cormen, Charles E Leiserson, Ronald L Rivest, Clifford Stein, 2001 An extensively revised edition of a mathematically rigorous yet accessible introduction to algorithms. |
| big o discrete math: Mathematics Edward R. Scheinerman, 2006 Master the fundamentals of discrete mathematics and proof-writing with MATHEMATICS: A DISCRETE INTRODUCTION! With a wealth of learning aids and a clear presentation, the mathematics text teaches you not only how to write proofs, but how to think clearly and present cases logically beyond this course. Though it is presented from a mathematician's perspective, you will learn the importance of discrete mathematics in the fields of computer science, engineering, probability, statistics, operations research, and other areas of applied mathematics. Tools such as Mathspeak, hints, and proof templates prepare you to succeed in this course. |
| big o discrete math: Mathematics under the Microscope Alexandre Borovik, 2010 Discusses, from a working mathematician's point of view, the mystery of mathematical intuition: Why are certain mathematical concepts more intuitive than others? And to what extent does the 'small scale' structure of mathematical concepts and algorithms reflect the workings of the human brain? |
| big o discrete math: Problems on Algorithms Ian Parberry, 1995 With approximately 600 problems and 35 worked examples, this supplement provides a collection of practical problems on the design, analysis and verification of algorithms. The book focuses on the important areas of algorithm design and analysis: background material; algorithm design techniques; advanced data structures and NP-completeness; and miscellaneous problems. Algorithms are expressed in Pascal-like pseudocode supported by figures, diagrams, hints, solutions, and comments. |
| big o discrete math: Our Mathematical Universe Max Tegmark, 2015-02-03 Max Tegmark leads us on an astonishing journey through past, present and future, and through the physics, astronomy and mathematics that are the foundation of his work, most particularly his hypothesis that our physical reality is a mathematical structure and his theory of the ultimate multiverse. In a dazzling combination of both popular and groundbreaking science, he not only helps us grasp his often mind-boggling theories, but he also shares with us some of the often surprising triumphs and disappointments that have shaped his life as a scientist. Fascinating from first to last—this is a book that has already prompted the attention and admiration of some of the most prominent scientists and mathematicians. |
| big o discrete math: Foundations of Data Science Avrim Blum, John Hopcroft, Ravindran Kannan, 2020-01-23 This book provides an introduction to the mathematical and algorithmic foundations of data science, including machine learning, high-dimensional geometry, and analysis of large networks. Topics include the counterintuitive nature of data in high dimensions, important linear algebraic techniques such as singular value decomposition, the theory of random walks and Markov chains, the fundamentals of and important algorithms for machine learning, algorithms and analysis for clustering, probabilistic models for large networks, representation learning including topic modelling and non-negative matrix factorization, wavelets and compressed sensing. Important probabilistic techniques are developed including the law of large numbers, tail inequalities, analysis of random projections, generalization guarantees in machine learning, and moment methods for analysis of phase transitions in large random graphs. Additionally, important structural and complexity measures are discussed such as matrix norms and VC-dimension. This book is suitable for both undergraduate and graduate courses in the design and analysis of algorithms for data. |
| big o discrete math: How to Prove It Daniel J. Velleman, 2006-01-16 Many students have trouble the first time they take a mathematics course in which proofs play a significant role. This new edition of Velleman's successful text will prepare students to make the transition from solving problems to proving theorems by teaching them the techniques needed to read and write proofs. The book begins with the basic concepts of logic and set theory, to familiarize students with the language of mathematics and how it is interpreted. These concepts are used as the basis for a step-by-step breakdown of the most important techniques used in constructing proofs. The author shows how complex proofs are built up from these smaller steps, using detailed 'scratch work' sections to expose the machinery of proofs about the natural numbers, relations, functions, and infinite sets. To give students the opportunity to construct their own proofs, this new edition contains over 200 new exercises, selected solutions, and an introduction to Proof Designer software. No background beyond standard high school mathematics is assumed. This book will be useful to anyone interested in logic and proofs: computer scientists, philosophers, linguists, and of course mathematicians. |
| big o discrete math: An Introduction to Measure Theory Terence Tao, 2021-09-03 This is a graduate text introducing the fundamentals of measure theory and integration theory, which is the foundation of modern real analysis. The text focuses first on the concrete setting of Lebesgue measure and the Lebesgue integral (which in turn is motivated by the more classical concepts of Jordan measure and the Riemann integral), before moving on to abstract measure and integration theory, including the standard convergence theorems, Fubini's theorem, and the Carathéodory extension theorem. Classical differentiation theorems, such as the Lebesgue and Rademacher differentiation theorems, are also covered, as are connections with probability theory. The material is intended to cover a quarter or semester's worth of material for a first graduate course in real analysis. There is an emphasis in the text on tying together the abstract and the concrete sides of the subject, using the latter to illustrate and motivate the former. The central role of key principles (such as Littlewood's three principles) as providing guiding intuition to the subject is also emphasized. There are a large number of exercises throughout that develop key aspects of the theory, and are thus an integral component of the text. As a supplementary section, a discussion of general problem-solving strategies in analysis is also given. The last three sections discuss optional topics related to the main matter of the book. |
| big o discrete math: Mathematics and Computation Avi Wigderson, 2019-10-29 From the winner of the Turing Award and the Abel Prize, an introduction to computational complexity theory, its connections and interactions with mathematics, and its central role in the natural and social sciences, technology, and philosophy Mathematics and Computation provides a broad, conceptual overview of computational complexity theory—the mathematical study of efficient computation. With important practical applications to computer science and industry, computational complexity theory has evolved into a highly interdisciplinary field, with strong links to most mathematical areas and to a growing number of scientific endeavors. Avi Wigderson takes a sweeping survey of complexity theory, emphasizing the field’s insights and challenges. He explains the ideas and motivations leading to key models, notions, and results. In particular, he looks at algorithms and complexity, computations and proofs, randomness and interaction, quantum and arithmetic computation, and cryptography and learning, all as parts of a cohesive whole with numerous cross-influences. Wigderson illustrates the immense breadth of the field, its beauty and richness, and its diverse and growing interactions with other areas of mathematics. He ends with a comprehensive look at the theory of computation, its methodology and aspirations, and the unique and fundamental ways in which it has shaped and will further shape science, technology, and society. For further reading, an extensive bibliography is provided for all topics covered. Mathematics and Computation is useful for undergraduate and graduate students in mathematics, computer science, and related fields, as well as researchers and teachers in these fields. Many parts require little background, and serve as an invitation to newcomers seeking an introduction to the theory of computation. Comprehensive coverage of computational complexity theory, and beyond High-level, intuitive exposition, which brings conceptual clarity to this central and dynamic scientific discipline Historical accounts of the evolution and motivations of central concepts and models A broad view of the theory of computation's influence on science, technology, and society Extensive bibliography |
| big o discrete math: Mathematics of Public Key Cryptography Steven D. Galbraith, 2012-03-15 This advanced graduate textbook gives an authoritative and insightful description of the major ideas and techniques of public key cryptography. |
| big o discrete math: Ask a Manager Alison Green, 2018-05-01 From the creator of the popular website Ask a Manager and New York’s work-advice columnist comes a witty, practical guide to 200 difficult professional conversations—featuring all-new advice! There’s a reason Alison Green has been called “the Dear Abby of the work world.” Ten years as a workplace-advice columnist have taught her that people avoid awkward conversations in the office because they simply don’t know what to say. Thankfully, Green does—and in this incredibly helpful book, she tackles the tough discussions you may need to have during your career. You’ll learn what to say when • coworkers push their work on you—then take credit for it • you accidentally trash-talk someone in an email then hit “reply all” • you’re being micromanaged—or not being managed at all • you catch a colleague in a lie • your boss seems unhappy with your work • your cubemate’s loud speakerphone is making you homicidal • you got drunk at the holiday party Praise for Ask a Manager “A must-read for anyone who works . . . [Alison Green’s] advice boils down to the idea that you should be professional (even when others are not) and that communicating in a straightforward manner with candor and kindness will get you far, no matter where you work.”—Booklist (starred review) “The author’s friendly, warm, no-nonsense writing is a pleasure to read, and her advice can be widely applied to relationships in all areas of readers’ lives. Ideal for anyone new to the job market or new to management, or anyone hoping to improve their work experience.”—Library Journal (starred review) “I am a huge fan of Alison Green’s Ask a Manager column. This book is even better. It teaches us how to deal with many of the most vexing big and little problems in our workplaces—and to do so with grace, confidence, and a sense of humor.”—Robert Sutton, Stanford professor and author of The No Asshole Rule and The Asshole Survival Guide “Ask a Manager is the ultimate playbook for navigating the traditional workforce in a diplomatic but firm way.”—Erin Lowry, author of Broke Millennial: Stop Scraping By and Get Your Financial Life Together |
| big o discrete math: What Can Be Computed? John MacCormick, 2018-05-01 An accessible and rigorous textbook for introducing undergraduates to computer science theory What Can Be Computed? is a uniquely accessible yet rigorous introduction to the most profound ideas at the heart of computer science. Crafted specifically for undergraduates who are studying the subject for the first time, and requiring minimal prerequisites, the book focuses on the essential fundamentals of computer science theory and features a practical approach that uses real computer programs (Python and Java) and encourages active experimentation. It is also ideal for self-study and reference. The book covers the standard topics in the theory of computation, including Turing machines and finite automata, universal computation, nondeterminism, Turing and Karp reductions, undecidability, time-complexity classes such as P and NP, and NP-completeness, including the Cook-Levin Theorem. But the book also provides a broader view of computer science and its historical development, with discussions of Turing's original 1936 computing machines, the connections between undecidability and Gödel's incompleteness theorem, and Karp's famous set of twenty-one NP-complete problems. Throughout, the book recasts traditional computer science concepts by considering how computer programs are used to solve real problems. Standard theorems are stated and proven with full mathematical rigor, but motivation and understanding are enhanced by considering concrete implementations. The book's examples and other content allow readers to view demonstrations of—and to experiment with—a wide selection of the topics it covers. The result is an ideal text for an introduction to the theory of computation. An accessible and rigorous introduction to the essential fundamentals of computer science theory, written specifically for undergraduates taking introduction to the theory of computation Features a practical, interactive approach using real computer programs (Python in the text, with forthcoming Java alternatives online) to enhance motivation and understanding Gives equal emphasis to computability and complexity Includes special topics that demonstrate the profound nature of key ideas in the theory of computation Lecture slides and Python programs are available at whatcanbecomputed.com |
| big o discrete math: Numerical Algorithms Justin Solomon, 2015-06-24 Numerical Algorithms: Methods for Computer Vision, Machine Learning, and Graphics presents a new approach to numerical analysis for modern computer scientists. Using examples from a broad base of computational tasks, including data processing, computational photography, and animation, the textbook introduces numerical modeling and algorithmic desig |
| big o discrete math: The Minimum You Need to Know about Logic to Work in IT Roland Hughes, 2007 This book is part of aaThe Minimum You Need to Knowaa family of books by Logikal Solutions. As the family expands they will cover an increasing variety of topics. This book is designed to be used as a text book for classes in logic from high school to college level. It should be one of the first courses you have on IT and this should be one of the first books you read when starting in IT. Not only does this book cover flow charting and pseudocode, it teaches the reader to think before they start mapping out the logic to solve a problem. The author of this book is an industry veteran with nearly 20 years in the field. It has been his experience that recent graduates, from any country, are nearly useless at problem solving. If they cannot point, click, and drag, they cannot solve the problem. This book is an attempt to teach them how to solve the problem. An instructoraas guide is available for schools looking to make this book the basis of coursework. |
Big O notation - MIT
f(n) = O(g(n)) see above f(n) = o(g(n)) see above < f(n) = (g(n)) g(n)=O(f(n)) f(n) = (g(n)) g(n)=o(f(n)) > f(n) = (g(n)) f(n)=O(g(n)) and g(n)=O(f(n)) = The notations and are … See more
Complexity Theory Intro CS311H: Discrete Mathematics …
Complexity Theory Intro CS311H: Discrete Mathematics Asymptotic Analysis Big-O Notation. CS311H: Discrete Mathematics Asymptotic Analysis. Instructor: Is l Dillig, CS311H: Discrete …
Big O and Little o Notation - Carnegie Mellon University
g(n)=o(f (n)) as “g(n) is ultimately negligible compared to f (n)”. There are some rules for arithmetic with big-O symbols: • If g(n)=O(f (n)), then cg(n)=O(f (n)) for any constant c.
Discrete MathematicsDiscrete Mathematics CS 2610
Big-O Notation Also, the function f(x) = 3x2 + 2x + 3 is O(x3) What about O(x4) ? In fact, the function Cg(x) is an upper bound for f(f( ), y gx), but not necessarily the tightest bound. When …
Big-O Notation - Stanford University
Big-O Notation Big-O notation. is a way of quantifying the rate at which some quantity grows. For example: A square of side length . r. has area O(r. 2). A circle of radius . r. has area O(r. 2). …
Last Revised: February 2009 - University of Arizona
Big-O (O()) is one of five standard asymptotic notations. In practice, Big-O is used as a tight upper-bound on the growth of an algorithm’s effort (this effort is described by the function f(n)), …
Chapter 14 Big-O - University of Illinois Urbana-Champaign
As the input x gets large, the behavior of the function is dominated by the term with the fastest growth (the first term in this case). Let’s look at some basic functions and try to put them into …
Lecture 17: Discrete Mathematics - MathCity.org
Difference between big-O and little-o: The difference between the earlier definition for the big-O notation and the present definition of little-o, is that while the former has to be true for at least …
Lecture Notes on “Big-O” Notation - Jamie Morgenstern
In this lecture, we will make formal these notions, and try to give some examples and moti-vation for a framework for analysis of algorithms and programs known as. asymptotic complexity.
Discrete Math Big O - courses.grainger.illinois.edu
Introduce the concept of asymptotic relationships and Big O Practice proofs on efficiency bounding Discuss possible benchmarks on algorithm performance
The Growth of Functions - Virginia Tech
The Big-O Notation Definition: Let f and g be functions from N to R.
CS106B Handout Big O Complexity - Stanford University
When using big-O notation, the goal is to provide a qualitative insight as to how changes in N affect how many units of computation are performed for large amounts of data.
Seven Rules for Big-O and - WPI
Here are seven rules that you can use to solve problems involving big-O and They will solve the big majority of the big- O and comparisons you’ll need in this course (and for a long way beyond).
“Big Oh” notation in terms of limits - Massachusetts Institute …
Little “o” Used to indicate that < : lim 𝑛→∞ ( ) ( ) =0⇒ ( )= ( ( )) Note: ) ( = ( ( ))⇒ ( )=𝑂( ( )), because Big O is used to indicate that ≤ Little omega: 𝜔 Used to indicate that > : lim 𝑛→∞ ( ) ( ) =∞⇒ ( )=𝜔( ( ))
CompSci 230 Announcements Discrete Math for Computer …
Big-O Notation Definition: Let f and g be functions from the set of integers or the set of real numbers to the set of real numbers. We say that f(x) is O(g(x)) if there are constants C and k …
Lecture 3: Analysing Complexity of Algorithms - Big Oh, Big
To prove that some g(n) is O(f(n)) means to show that for g and f such constants c and n0 exist. The constants c and n0 are interdependent. g(n) is O(f(n)) i the graph of g(n) is always below …
Big O Notation - math.torontomu.ca
Big O notation is especially useful when analyzing the efficiency of algorithms. In this case n is the size of the input and f(n) is the running time of the algorithm relative to input size.
Big-O Notation - Stanford University
Nuances of Big-O Notation Big-O notation is designed to capture the rate at which a quantity grows. It does not capture information about leading coefficients: the area of a square of side …
Discrete Structures for Computer Science - people.cs.pitt.edu
We take Theorem 3 in Appendix 2 to see that logb. As mentioned before, big-O notation is used to estimate the solve a problem using a specified procedure or algorithm. The often include the …
Big-O De nition - Computer Science and Engineering
IBig-O is actually Omicron, but it su ces to write \O" IIntuition: f is ( asymptotically ) less than or equal to g. IBig-O gives an asymptotic upper bound. IntroductionI. In practice, speci c …
Big O notation - MIT
Big O notation (with a capital letter O, not a zero), also called Landau's symbol, is a symbolism used in complexity theory, computer science, and mathematics to describe the asymptotic behavior of …
Complexity Theory Intro CS311H: Discrete Mathematics …
Complexity Theory Intro CS311H: Discrete Mathematics Asymptotic Analysis Big-O Notation. CS311H: Discrete Mathematics Asymptotic Analysis. Instructor: Is l Dillig, CS311H: Discrete …
Big O and Little o Notation - Carnegie Mellon University
g(n)=o(f (n)) as “g(n) is ultimately negligible compared to f (n)”. There are some rules for arithmetic with big-O symbols: • If g(n)=O(f (n)), then cg(n)=O(f (n)) for any constant c.
Discrete MathematicsDiscrete Mathematics CS 2610
Big-O Notation Also, the function f(x) = 3x2 + 2x + 3 is O(x3) What about O(x4) ? In fact, the function Cg(x) is an upper bound for f(f( ), y gx), but not necessarily the tightest bound. When …
Big-O Notation - Stanford University
Big-O Notation Big-O notation. is a way of quantifying the rate at which some quantity grows. For example: A square of side length . r. has area O(r. 2). A circle of radius . r. has area O(r. 2). …
Last Revised: February 2009 - University of Arizona
Big-O (O()) is one of five standard asymptotic notations. In practice, Big-O is used as a tight upper-bound on the growth of an algorithm’s effort (this effort is described by the function f(n)), even …
Chapter 14 Big-O - University of Illinois Urbana-Champaign
As the input x gets large, the behavior of the function is dominated by the term with the fastest growth (the first term in this case). Let’s look at some basic functions and try to put them into …
Lecture 17: Discrete Mathematics - MathCity.org
Difference between big-O and little-o: The difference between the earlier definition for the big-O notation and the present definition of little-o, is that while the former has to be true for at least …
Lecture Notes on “Big-O” Notation - Jamie Morgenstern
In this lecture, we will make formal these notions, and try to give some examples and moti-vation for a framework for analysis of algorithms and programs known as. asymptotic complexity.
Discrete Math Big O - courses.grainger.illinois.edu
Introduce the concept of asymptotic relationships and Big O Practice proofs on efficiency bounding Discuss possible benchmarks on algorithm performance
The Growth of Functions - Virginia Tech
The Big-O Notation Definition: Let f and g be functions from N to R.
CS106B Handout Big O Complexity - Stanford University
When using big-O notation, the goal is to provide a qualitative insight as to how changes in N affect how many units of computation are performed for large amounts of data.
Seven Rules for Big-O and - WPI
Here are seven rules that you can use to solve problems involving big-O and They will solve the big majority of the big- O and comparisons you’ll need in this course (and for a long way beyond).
“Big Oh” notation in terms of limits - Massachusetts Institute …
Little “o” Used to indicate that < : lim 𝑛→∞ ( ) ( ) =0⇒ ( )= ( ( )) Note: ) ( = ( ( ))⇒ ( )=𝑂( ( )), because Big O is used to indicate that ≤ Little omega: 𝜔 Used to indicate that > : lim 𝑛→∞ ( ) ( ) =∞⇒ ( )=𝜔( ( ))
CompSci 230 Announcements Discrete Math for Computer …
Big-O Notation Definition: Let f and g be functions from the set of integers or the set of real numbers to the set of real numbers. We say that f(x) is O(g(x)) if there are constants C and k …
Lecture 3: Analysing Complexity of Algorithms - Big Oh, …
To prove that some g(n) is O(f(n)) means to show that for g and f such constants c and n0 exist. The constants c and n0 are interdependent. g(n) is O(f(n)) i the graph of g(n) is always below or …
Big O Notation - math.torontomu.ca
Big O notation is especially useful when analyzing the efficiency of algorithms. In this case n is the size of the input and f(n) is the running time of the algorithm relative to input size.
Big-O Notation - Stanford University
Nuances of Big-O Notation Big-O notation is designed to capture the rate at which a quantity grows. It does not capture information about leading coefficients: the area of a square of side length r …
Discrete Structures for Computer Science - people.cs.pitt.edu
We take Theorem 3 in Appendix 2 to see that logb. As mentioned before, big-O notation is used to estimate the solve a problem using a specified procedure or algorithm. The often include the …
Big-O De nition - Computer Science and Engineering
IBig-O is actually Omicron, but it su ces to write \O" IIntuition: f is ( asymptotically ) less than or equal to g. IBig-O gives an asymptotic upper bound. IntroductionI. In practice, speci c hardware, …
