1st Edition

# A Guide to Algorithm Design Paradigms, Methods, and Complexity Analysis

380 Pages 71 B/W Illustrations
by CRC Press

380 Pages 71 B/W Illustrations
by CRC Press

Also available as eBook on:

Presenting a complementary perspective to standard books on algorithms, A Guide to Algorithm Design: Paradigms, Methods, and Complexity Analysis provides a roadmap for readers to determine the difficulty of an algorithmic problem by finding an optimal solution or proving complexity results. It gives a practical treatment of algorithmic complexity and guides readers in solving algorithmic problems.

Divided into three parts, the book offers a comprehensive set of problems with solutions as well as in-depth case studies that demonstrate how to assess the complexity of a new problem.

• Part I helps readers understand the main design principles and design efficient algorithms.
• Part II covers polynomial reductions from NP-complete problems and approaches that go beyond NP-completeness.
• Part III supplies readers with tools and techniques to evaluate problem complexity, including how to determine which instances are polynomial and which are NP-hard.

Drawing on the authors’ classroom-tested material, this text takes readers step by step through the concepts and methods for analyzing algorithmic complexity. Through many problems and detailed examples, readers can investigate polynomial-time algorithms and NP-completeness and beyond.

Polynomial-Time Algorithms: Exercises
Introduction to Complexity
On the complexity to compute xn
Asymptotic notations: O, o, Θ, and Ω

Divide-and-Conquer
Strassen’s algorithm
Master theorem
Solving recurrences

Greedy Algorithms
Motivating example: the sports hall
Designing greedy algorithms
Graph coloring
Theory of matroids

Dynamic Programming
The coin changing problem
The knapsack problem
Designing dynamic-programming algorithms

Amortized Analysis
Methods for amortized analysis

Exercises, Solutions, and Bibliographic Notes appear at the end of each chapter in this section.

NP-Completeness and Beyond
NP-Completeness

A practical approach to complexity theory
Problem classes
NP-complete problems and reduction theory
Examples of NP-complete problems and reductions
Importance of problem definition
Strong NP-completeness
Why does it matter?

Exercises on NP-Completeness
Easy reductions
Scheduling problems
More involved reductions
2-PARTITION is NP-complete

Beyond NP-Completeness
Approximation results
Polynomial problem instances
Linear programming
Randomized algorithms
Branch-and-bound and backtracking

Exercises Going beyond NP-Completeness
Approximation results
Dealing with NP-complete problems

Reasoning on Problem Complexity
Reasoning to Assess a Problem Complexity

Basic reasoning
Set of problems with polynomial-time algorithms
Set of NP-complete problems

Chains-on-Chains Partitioning
Optimal algorithms for homogeneous resources
Variants of the problem
Extension to a clique of heterogeneous resources
Conclusion

Replica Placement in Tree Networks
Access policies
Complexity results
Variants of the replica placement problem
Conclusion

Packet Routing
MEDP: Maximum edge-disjoint paths
PRVP: Packet routing with variable-paths
Conclusion

Matrix Product, or Tiling the Unit Square
Problem motivation
NP-completeness
A guaranteed heuristic
Related problems

Online Scheduling
Flow time optimization
Competitive analysis
Makespan optimization
Conclusion

Bibliography

Index

### Biography

Yves Robert, École Normale Supérieure de Lyon, Institut Universitaire de France, and Université de Lyon, France

Anne Benoit and Frederic Vivien, École Normale Supérieure de Lyon, France

"This well-written book takes a fresh look at a classical subject, with features that will please both instructors and students. The book provides an insightful introduction to algorithm design with detailed coverage of a broad range of topics. The nice selection of exercises and the well-chosen case studies will be very useful for teachers of undergraduate courses. The sections devoted to the methodology of algorithm design should help students acquire a deeper understanding of advanced notions in complexity theory and approximation. A really nice job in both scope and style!"
—Denis Trystram, Distinguished Professor, Grenoble Institute of Technology

"Graduate students and instructors alike will find this book an invaluable resource. The subject matter is presented following a logical progression that makes students understand not only the principles behind algorithm design and analysis, but also the raison d’etre and the practical relevance of these principles. And yet, the book can also be used as a compendium of knowledge that provides the practitioner a quiver of algorithmic arrows. Finally, the extensive sets of motivating examples and exercises will prove instrumental for any instructor willing to develop an engaging course on advanced algorithms."
—Henri Casanova, Associate Professor, Department of Information and Computer Sciences, University of Hawai'i at Manoa

"This book is a great technical arsenal for every graduate student and post-graduate researcher. By providing a treasure trove of concrete algorithmic examples, the book trains the reader to recognize clues that indicate the complexity of a broad range of algorithmic problems, while supplying a battery of techniques for solving a particular problem in hand. The book is also a true source of inspiration for instructors looking for a material to teach advanced algorithms courses."
—Umit Catalyurek, Professor, Ohio State University

"This book is unique among texts on algorithmics in its emphasis on how to ‘think algorithmically’ rather than just how to solve specific (classes of) algorithmic problems. The authors skillfully engage the reader in a journey of algorithmic self-discovery as they cover a broad spectrum of issues, from the very basic (computing powers, coin changing) through the quite advanced (NP-completeness, polynomial-time approximation schemes). The authors emphasize algorithmic topics that have proven useful in ‘applied’ situations … . I shall be very happy to have this text on my bookshelf as a reference on methods as well as results."
—Arnold L. Rosenberg, Research Professor, Northeastern University, and Distinguished University Professor Emeritus, University of Massachusetts Amherst

"This book presents a well-balanced approach to theory and algorithms and introduces difficult concepts using rich motivating examples. It demonstrates the applicability of fundamental principles and analysis techniques to practical problems facing computer scientists and engineers. You do not have to be a theoretician to enjoy and learn from this book."
—Rami Melhem, Professor of Computer Science, University of Pittsburgh