Combinatorial Algorithm Toolbox : An Interactive Course on Discrete Mathematics

An Interactive Course on Discrete Mathematics

By: Winfried Hochstattler, Alexander Schliep

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Paperback | 1 February 2010

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202 Pages

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23.5 x 15.88 x 1.27

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Graph algorithms are easy to visualize and indeed there already exists a variety of packages to animate the dynamics when solving problems from graph theory. Still it can be difficult to understand the ideas behind the algorithm from the dynamic display alone.

CATBox consists of a software system for animating graph algorithms and a course book which we developed simultaneously. The software system presents both the algorithm and the graph and puts the user always in control of the actual code that is executed. In the course book, intended for readers at advanced undergraduate or graduate level, computer exercises and examples replace the usual static pictures of algorithm dynamics.

For this volume we have chosen solely algorithms for classical problems from combinatorial optimization, such as minimum spanning trees, shortest paths, maximum flows, minimum cost flows, weighted and unweighted matchings both for bipartite and non-bipartite graphs.

Find more information at http://schliep.org/CATBox/.

Industry Reviews

From the reviews: "Intend to provide both mathematical rigor as well as an interactive tool to assist the learning process. ... The text is proposed for use at the graduate or advanced undergraduate levels. ... theory and practice come together admirably. ... CATBox is an excellent tool for learning the presented algorithms and concepts. ... Students ... in both computer science and the methods of mathematical proof would be well-served by this text." (Keith Sinkhorn, The Mathematical Association of America, October, 2010) "This book is devoted to algorithms which solve some classical problems from discrete optimization. ... The authors ... attempt to make accessible a number of topics that are not often found in textbooks. ... The organization and presentation of the materials of the book makes it possible to use it both for readers at advanced undergraduate or graduate level and as a reference book. It is a very valuable book for successful application of real life problems from discrete optimization. I certainly recommend buying it." (Samir Kumar Neogy, Zentralblatt MATH, Vol. 1190, 2010)

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Discrete Problems from Applications	p. 1
Basics, Notation and Data Structures	p. 7
Definition of a Graph	p. 7
How to Represent a Graph on the Computer?	p. 9
Basic Terminology from Graph Theory	p. 10
Algorithms and Complexity	p. 11
Correctness	p. 13
Running Time	p. 14
Two Graph Traversals	p. 14
Minimum Spanning Trees	p. 19
Minimum Connected Subgraphs	p. 19
Trees	p. 20
Minimum Spanning Trees	p. 22
Kruskal's Algorithm	p. 25
Prim's Algorithm	p. 27
Remarks	p. 29
Some Additional Notation	p. 31
Linear Programming Duality	p. 33
The MST-Polytope	p. 33
Farkas' Lemma	p. 41
Duality Theorem of Linear Programming	p. 44
The Greedy as Primal-Dual Algorithm	p. 48
The Base Polytope	p. 50
Shortest Paths	p. 53
Introduction	p. 53
Dijkstra's Algorithm	p. 54
Some Considerations about Implementations	p. 56
Special Label Setting Methods for the Computation of Shortest Paths Between Two Vertices	p. 56
Label Setting Versus Label Correcting	p. 58
Detecting Negative Circuits	p. 61
Application	p. 63
An Analog Computer and LP Duality	p. 64
Maximal Flows	p. 69
Introduction	p. 69
The Algorithm of Ford and Fulkerson	p. 70
Max-Flow-Min-Cut	p. 72
Pathologies	p. 74
Edmonds-Karp Implementation	p. 76
Max-flow-Min-cut as Linear Programming Duality	p. 77
Preflow Push	p. 79
Preflow Push Considered as a Dual Algorithm	p. 83
FIFO-Implementation	p. 85
Minimum-Cost Flows	p. 89
Introduction	p. 89
Optimality Criteria	p. 90
First Algorithms	p. 95
Canceling Negative Circuits, the Primal Method	p. 95
Augmenting Shortest Paths, the Dual Method	p. 96
The Primal Dual Method	p. 98
Polynomial Time Algorithms	p. 99
Min Mean Circuit Canceling	p. 99
Capacity Rounding	p. 103
Cost Scaling	p. 104
Matching	p. 111
Bipartite Matching	p. 111
Augmenting Paths	p. 117
Non-Bipartite Graphs	p. 118
The Tutte-Berge Formula	p. 125
Weighted Matching	p. 129
Bipartite Weighted Matching	p. 130
Inverse Minimum Spanning Tree Problem	p. 137
Non-Bipartite Matching	p. 137
The Matching Polytope	p. 138
The Weighted Matching Algorithm	p. 144
Karyotyping and the Chinese Postman Problem	p. 150
Using Gato and Gred	p. 155
Easy Installation	p. 155
Installing from Source	p. 156
The Gato Application	p. 157
Running Algorithms in Gato	p. 161
Gred	p. 163
Editing Graphs with Gred	p. 166
A Brief Introduction to Reading Python	p. 169
Where to Obtain Python	p. 171
Visualizing Graph Algorithms with Gato	p. 173
Interesting Events During Runtime	p. 173
Rule-Based Visualization	p. 175
Animated Data Structures (ADS)	p. 176
Case Study: Ford-Fulkerson Algorithm	p. 178
Implementation of Gato	p. 180
ADS Defined in Gato	p. 181
References	p. 185
Index	p. 187
Table of Contents provided by Ingram. All Rights Reserved.

Combinatorial Algorithm Toolbox : An Interactive Course on Discrete Mathematics

An Interactive Course on Discrete Mathematics

At a Glance

Paperback

Industry Reviews

Shipping

How to return your order

Defective items

You Can Find This Book In

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